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Vladimir A. Zorich
Mathematical Analysis II
�
Springer
Vladimir A. Zorich
Moscow State University Department of Mathematics (Mech-Math) Vorobievy Gory Moscow 119992 Russia Translator: Roger Cooke Burlington, Vermont USA e-mail:
[email protected] Title of Russian edition: Matematicheskij Analiz (Part II, 4th corrected edition, Moscow, 2002) MCCME (Moscow Center for Continuous Mathematical Education Publ.)
Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2ooo ): Primary ooAo5 Secondary:26-0I,40-01,42-01,54-01,58-oi
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Prefaces
Preface to the English Edition An entire generation of mathematicians has grown up during the time be tween the appearance of the first edition of this textbook and the publication of the fourth edition, a translation of which is before you. The book is famil iar to many people, who either attended the lectures on which it is based or studied out of it, and who now teach others in universities all over the world. I am glad that it has become accessible to English-speaking readers. This textbook consists of two parts. It is aimed primarily at university students and teachers specializing in mathematics and natural sciences, and at all those who wish to see both the rigorous mathematical theory and examples of its effective use in the solution of real problems of natural science. The textbook exposes classical analysis as it is today, as an integral part of Mathematics in its interrelations with other modern mathematical courses such as algebra, differential geometry, differential equations, complex and functional analysis. The two chapters with which this second book begins, summarize and explain in a general form essentially all most important results of the first volume concerning continuous and differentiable functions, as well as differ ential calculus. The presence of these two chapters makes the second book formally independent of the first one. This assumes, however, that the reader is sufficiently well prepared to get by without introductory considerations of the first part, which preceded the resulting formalism discussed here. This second book, containing both the differential calculus in its generalized form and integral calculus of functions of several variables, developed up to the general formula of Newton-Leibniz-Stokes, thus acquires a certain unity and becomes more self-contained. More complete information on the textbook and some recommendations for its use in teaching can be found in the translations of the prefaces to the first and second Russian editions. Moscow,
2003
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VI
Prefaces
Preface to the Fourth Russian Edition In the fourth edition all misprints that the author is aware of have been corrected. Moscow,
2002
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Zorich
Preface to the Third Russian Edition The third edition differs from the second only in local corrections ( although in one case it also involves the correction of a proof ) and in the addition of some problems that seem to me to be useful. Moscow,
2001
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Preface to the Second Russian Edition In addition to the correction of all the misprints in the first edition of which the author is aware, the differences between the second edition and the first edition of this book are mainly the following. Certain sections on individual topics - for example, Fourier series and the Fourier transform - have been recast ( for the better, I hope ) . We have included several new examples of applications and new substantive problems relating to various parts of the theory and sometimes significantly extending it . Test questions are given, as well as questions and problems from the midterm examinations. The list of further readings has been expanded. Further information on the material and some characteristics of this sec ond part of the course are given below in the preface to the first edition. Moscow,
1998
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Preface to the First Russian Edition The preface to the first part contained a rather detailed characterization of the course as a whole, and hence I confine myself here to some remarks on the content of the second part only. The basic material of the present volume consists on the one hand of multiple integrals and line and surface integrals, leading to the generalized Stokes ' formula and some examples of its application, and on the other hand the machinery of series and integrals depending on a parameter, including
Preface to the First Russian Edition
VII
Fourier series, the Fourier transform, and the presentation of asymptotic expansions. Thus, this Part 2 basically conforms to the curriculum of the second year of study in the mathematics departments of universities. So as not to impose rigid restrictions on the order of presentation of these two major topics during the two semesters, I have discussed them practically independently of each other. Chapters 9 and 10, with which this book begins, reproduce in compressed and generalized form, essentially all of the most important results that were obtained in the first part concerning continuous and differentiable functions. These chapters are starred and written as an appendix to Part 1. This ap pendix contains, however, many concepts that play a role in any exposition of analysis to mathematicians. The presence of these two chapters makes the second book formally independent of the first , provided the reader is suffi ciently well prepared to get by without the numerous examples and introduc tory considerations that , in the first part , preceded the formalism discussed here. The main new material in the book, which is devoted to the integral calculus of several variables, begins in Chapter 11. One who has completed the first part may begin the second part of the course at this point without any loss of continuity in the ideas. The language of differential forms is explained and used in the discussion of the theory of line and surface integrals. All the basic geometric concepts and analytic constructions that later form a scale of abstract definitions lead ing to the generalized Stokes ' formula are first introduced by using elementary material. Chapter 15 is devoted to a similar summary exposition of the integration of differential forms on manifolds. I regard this chapter as a very desirable and systematizing supplement to what was expounded and explained using specific objects in the mandatory Chapters 11 - 14. The section on series and integrals depending on a parameter gives, along with the traditional material, some elementary information on asymptotic series and asymptotics of integrals (Chap. 19), since, due to its effectiveness, the latter is an unquestionably useful piece of analytic machinery. For convenience in orientation, ancillary material or sections that may be omitted on a first reading, are starred. The numbering of the chapters and figures in this book continues the numbering of the first part . Biographical information is given here only for those scholars not men tioned in the first part. As before, for the convenience of the reader, and to shorten the text , the end of a proof is denoted by D. Where convenient , definitions are introduced by the special symbols := or =: (equality by definition) , in which the colon stands on the side of the object being defined.
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Continuing the tradition of Part 1, a great deal of attention has been paid to both the lucidity and logical clarity of the mathematical construc tions themselves and the demonstration of substantive applications in natural science for the theory developed. Moscow,
1982
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Table of Content s
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*Continuous Mappings ( General Theory )
................. 1 1 1 5 7 7 8 9.2 9 9 13 13 14 9.3 15 15 16 18 9.4 19 20 9.5 21 21 24 27 28 9.6 28 30 33 9. 7 34 40 *Differential Calculus from a General Viewpoint . . . . . . . . . . 41 10.1 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 10.1.1 Some Examples of Vector Spaces in Analysis . . . . . . . . . 41 10.1.2 Norms in Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 10.1.3 Inner Products in Vector Spaces . . . . . . . . . . . . . . . . . . . . 45 9.1
Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Open and Closed Subsets of a Metric Space . . . . . . . . . . 9.1.3 Subspaces of a Metric space . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 The Direct Product of Metric Spaces . . . . . . . . . . . . . . . 9.1.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Subspaces of a Topological Space . . . . . . . . . . . . . . . . . . . 9.2.3 The Direct Product of Topological Spaces . . . . . . . . . . . 9.2.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Definition and General Properties of Compact Sets . . . 9.3.2 Metric Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Connected Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Complete Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Basic Definitions and Examples . . . . . . . . . . . . . . . . . . . . 9.5.2 The Completion of a Metric Space . . . . . . . . . . . . . . . . . . 9.5.3 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Mappings of Topological Spaces . . . . . . . . . . . . . . . . 9.6.1 The Limit of a Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Continuous Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . The Contraction Mapping Principle . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
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10.1.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 10.2 Linear and Multilinear Transformations . . . . . . . . . . . . . . . . . . . 49 10.2.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 49 10.2.2 The Norm of a Transformation . . . . . . . . . . . . . . . . . . . . . 51 10.2.3 The Space o f Continuous Transformations . . . . . . . . . . . 56 10.2.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 10.3 The Differential of a Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.3.1 Mappings Differentiable at a Point . . . . . . . . . . . . . . . . . 61 10.3.2 The General Rules for Differentiation . . . . . . . . . . . . . . . 62 10.3.3 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 10.3.4 The Partial Derivatives of a Mapping . . . . . . . . . . . . . . . 70 10.3.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 10.4 The Finite-increment (Mean-value) Theorem . . . . . . . . . . . . . . . 73 10.4.1 The Finite-increment Theorem . . . . . . . . . . . . . . . . . . . . . 73 10.4.2 Some Applications of the Finite-increment Theorem . . 75 10.4.3 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 10.5 Higher-order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 10.5.1 Definition of the nth Differential . . . . . . . . . . . . . . . . . . . 80 10.5.2 Derivative with Respect to a Vector . . . . . . . . . . . . . . . . 81 10.5.3 Symmetry of the Higher-order Differentials . . . . . . . . . . 82 10.5.4 Some Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 10.5.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 10.6 Taylor ' s Formula and the Study of Extrema . . . . . . . . . . . . . . . . 86 10.6.1 Taylor ' s Formula for Mappings . . . . . . . . . . . . . . . . . . . . . 86 10.6.2 Methods of Studying Interior Extrema . . . . . . . . . . . . . . 87 10.6.3 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 10.6.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 10.7 The General Implicit Function Theorem . . . . . . . . . . . . . . . . . . . 96 10.7.1 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 11.1 The Riemann Integral over an n-Dimensional Interval . . . . . . . 107 11.1.1 Definition of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 107 11.1.2 The Lebesgue Criterion for Riemann Integrability . . . . 109 11.1.3 The Darboux Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 11.1.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 11.2 The Integral over a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 11.2.1 Admissible Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 11.2.2 The Integral over a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 11.2.3 The Measure (Volume) of an Admissible Set . . . . . . . . . 119 11.2.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 11.3 General Properties of the Integral . . . . . . . . . . . . . . . . . . . . . . . 122 11.3.1 The Integral as a Linear Functional . . . . . . . . . . . . . . . . . 122 11.3.2 Additivity of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 122 11.3.3 Estimates for the Integral . . . . . . . . . . . . . . . . . . . . . . . . . 123 .
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11.3.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 11.4 Reduction of a Multiple Integral to an Iterated Integral . . . . . 127 11.4.1 Fubini ' s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 11.4.2 Some Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 11.4.3 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 11.5 Change of Variable in a Multiple Integral . . . . . . . . . . . . . . . . . . 135 11.5.1 Statement of the Problem. Heuristic Considerations . . . 135 11.5.2 Measurable Sets and Smooth Mappings . . . . . . . . . . . . . 136 11.5.3 The One-dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . 138 11.5.4 The Case of an Elementary Diffeomorphism in !Rn . . . . 140 11.5.5 Composite Mappings and Change of Variable . . . . . . . . 142 11.5.6 Additivity of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 142 11.5. 7 Generalizations of the Change of Variable Formula . . . 143 11.5.8 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11.6 Improper Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 11.6.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 11.6.2 The Comparison Test . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 153 11.6.3 Change of Variable in an Improper Integral . . . . . . . . . . 156 11.6.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Surfaces and Differential Forms in !Rn . . . . . . . . . . . . . . . . . . . . . 161 12.1 Surfaces in !Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12.1.1 Problems and Exercises . . . . . . . . . . . . . . . . . . .. . . . . . . . 170 12.2 Orientation of a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 12.2.1 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 12.3 The Boundary of a Surface and its Orientation . . . . . . . . . . . . . 178 12.3.1 Surfaces with Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 12.3.2 The Induced Orientation on the Boundary . . . . . . . . . . . 181 12.3.3 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 12.4 The Area of a Surface in Euclidean Space . . . . . . . . . . . . . . . . . 185 12.4.1 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 12.5 Elementary Facts about Differential Forms . . . . . . .. . . . . . . . . . 195 12.5.1 Differential Forms: Definition and Examples . . . . . . . . . 195 12.5.2 Coordinate Expression of a Differential Form . . . . . . . . 199 12.5.3 The Exterior Differential of a Form . . . . . . . . . . . . . . . . . 201 12.5.4 Transformation under Mappings . . . . . . . . . . . . . . . . . . . . 204 12.5.5 Forms on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 12.5.6 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Line and Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 13.1 The Integral o f a Differential Form . . . . . . . . . . . . . . . . . . . . . . . 211 13.1.1 The Original Problems. Examples . . . . . . . . . . . . . . . . . . 211 13.1.2 Integral over an Oriented Surface . . . . . . . . . . . . . . . . . . . 218 13.1.3 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 13.2 The Volume Element. Integrals of First and Second Kind . . . . 226 .
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13.2.1 The Mass of a Lamina . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 13.2.2 The Area of a Surface as the Integral of a Form . . . . . . 227 13.2.3 The Volume Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 13.2.4 Cartesian Expression of the Volume Element . . . . . . . . . 230 13.2.5 Integrals of First and Second Kind . . . . . . . . . . . . . . . . . . 231 13.2.6 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 13.3 The Fundamental Integral Formulas of Analysis . . . . . . . . . . . . 237 13.3.1 Green ' s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 13.3.2 The Gauss-Ostrogradskii Formula . . . . . . . . . . . . . . . . . . 242 13.3.3 Stokes ' Formula in JR3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 13.3.4 The General Stokes Formula . . . . . . . . . . . . . . . . . . . . . . . 248 13.3.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Elements of Vector Analysis and Field Theory . . . . . . . . . . . . 257 14.1 The Differential Operations of Vector Analysis . . . . . . . . . . . . . 257 14.1.1 Scalar and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 257 14.1.2 Vector Fields and Forms in JR3 . . . . . . . . . . . . . . . . . . . . . 257 14.1.3 The Differential Operators grad, curl, div, and V' . . . . . 260 14.1.4 Some Differential Formulas of Vector Analysis . . . . . . . . 263 14.1.5 *Vector Operations in Curvilinear Coordinates . . . . . . . 265 14.1.6 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 14.2 The Integral Formulas of Field Theory . . . . . . . . . . . . . . . . . . . . 276 14.2.1 The Classical Integral Formulas in Vector Notation . . . 276 14.2.2 The Physical Interpretation of div, curl, and grad . . . . 278 14.2.3 Other Integral Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 14.2.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 14.3 Potential Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 14.3.1 The Potential of a Vector Field . . . . . . . . . . . . . . . . . . . . . 288 14.3.2 Necessary Condition for Existence of a Potential . . . . . 289 14.3.3 Criterion for a Field to be Potential . . . . . . . . . . . . . . . . 290 14.3.4 Topological Structure of a Domain and Potentials . . . . 293 14.3.5 Vector Potential. Exact and Closed Forms . . . . . . . . . . . 295 14.3.6 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 14.4 Examples of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 14.4.1 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 14.4.2 The Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . 304 14.4.3 Equations of Dynamics of Continuous Media . . . . . . . . 306 14.4.4 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 14.4.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 *Integration o f Differential Forms o n Manifolds . . . . . . . . . . . 313 15.1 A Brief Review of Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . 313 15.1.1 The Algebra of Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 15.1.2 The Algebra of Skew-symmetric Forms . . . . . . . . . . . 314 15.1.3 Linear Mappings and their Adjoints . . . . . . . . . . . . . . . 317 .
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15.1.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 15.2 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 15.2.1 Definition of a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 15.2.2 Smooth Manifolds and Smooth Mappings . . . . . . . . . . . 325 15.2.3 Orientation of a Manifold and its Boundary . . . . . . . . . . 328 15.2.4 Partitions of Unity. Manifolds as Surfaces . . . . . . . . . . . 331 15.2.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 15.3 Differential Forms and Integration on Manifolds . . . . . . . . . . . . 337 15.3.1 The Tangent Space to a Manifold at a Point . . . . . . . . . 337 15.3.2 Differential Forms on a Manifold . . . . . . . . . . . . . . . . . . . 340 15.3.3 The Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 15.3.4 The Integral of a Form over a Manifold . . . . . . . . . . . . . 343 15.3.5 Stokes ' Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 15.3.6 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 15.4 Closed and Exact Forms on Manifolds . . . . . . . . . . . . . . . . . . . . . 352 15.4.1 Poincare ' s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 15.4.2 Homology and Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 356 15.4.3 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 Uniform Convergence and Basic Operations of Analysis . . 363 16.1 Pointwise and Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . 363 16.1.1 Pointwise Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 16.1.2 Statement of the Fundamental Problems . . . . . . . . . . . . 364 16.1.3 Convergence of a Family Depending on a Parameter . . 366 16.1.4 The Cauchy Criterion for Uniform Convergence . . . . . . 369 16.1.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 16.2 Uniform Convergence of Series of Functions . . . . . . . . . . . . . . . . 372 16.2.1 Basic Definitions. Uniform Convergence of a Series . . . 372 16.2.2 Weierstrass ' M-test for Uniform Convergence . . . . . . . . 374 16.2.3 The Abel-Dirichlet Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 16.2.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 16.3 Functional Properties of a Limit Function . . . . . . . . . . . . . . . . . 381 16.3.1 Specifics of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 16.3.2 Conditions for Two Limiting Passages to Commute . . . 381 16.3.3 Continuity and Passage to the Limit . . . . . . . . . . . . . . . . 383 16.3.4 Integration and Passage to the Limit . . . . . . . . . . . . . . . . 386 16.3.5 Differentiation and Passage to the Limit . . . . . . . . . . . . 388 16.3.6 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 393 16.4 *Subsets of the Space of Continuous Functions . . . . . . . . . . . . 397 16.4.1 The Arzela-Ascoli Theorem . . . . . . . . . . . . . . . . . . . . . . . 397 16.4.2 The Metric Space C(K, Y) . . . . . . . . . . . . . . . . . . . . . . . . 399 16.4.3 Stone ' s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 16.4.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 .
.
.
.
XIV
Table of Contents
. . . . . . . . . . . . . . . . . . . . . . 407 407 407 408 409 413 413 17.2 415 415 423 426 429 434 17.3 437 437 439 442 443 445 17.4 449 449 451 454 460 471 17.5 476 476 477 478 483 493 Fourier Series and the Fourier Transform . . . . . . . . . . . . . . . . . 499 18.1 Basic General Concepts Connected with Fourier Series . . . . . . 499 18.1.1 Orthogonal Systems of Functions . . . . . . . . . . . . . . . . . . . 499 18.1.2 Fourier Coefficients and Fourier Series . . . . . . . . . . . . . . 506 18.1.3 *A Source of Orthogonal Systems . . . . . . . . . . . . . . . . . . 516 18.1.4 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 18.2 Trigonometric Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 18.2.1 Basic Types of Convergence of Fourier Series . . . . . . . . . 526 18.2.2 Pointwise Convergence of a Fourier Series . . . . . . . . . . . 531 18.2.3 Smoothness and Decrease of Fourier Coefficients . . . . . . 540 18.2.4 Completeness of the Trigonometric System . . . . . . . . . . 545 18.2.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 18.3 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
17 Integrals Depending on a Parameter
17.1
18
Proper Integrals Depending on a Parameter . . . . . . . . . . . . . . . . 17 .1.1 The Basic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2 Continuity of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . 17 .1.3 Differentiation of the Integral . . . . . . . . . . . . . . . . . . . . . . 17.1.4 Integration of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Improper Integrals Depending on a Parameter . . . . . . . . . . . . . 17.2.1 Uniform Convergence with Respect to a Parameter . . . 17.2.2 Continuity of an Integral Depending on a Parameter . . 17 .2.3 Differentiation with Respect to a Parameter . . . . . . . . . 17.2.4 Integration of an Improper Integral . . . . . . . . . . . . . . . . . 17.2.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . The Eulerian Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 The Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.3 Connection Between the Beta and Gamma Functions . 17.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Convolution and Generalized Functions . . . . . . . . . . . . . . . . . . . 17.4.1 Convolution in Physical Problems . . . . . . . . . . . . . . . . . . 17.4.2 General Properties of Convolution . . . . . . . . . . . . . . . . . . 17.4.3 Approximate Identities and Weierstrass ' Theorem . . . . 17.4.4 *Elementary Concepts Involving Distributions . . . . . . . 17.4.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Integrals Depending on a Parameter . . . . . . . . . . . . . . 17.5.1 Proper Multiple Integrals Depending on a Parameter . 17.5.2 Improper Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . 17.5.3 Improper Integrals with a Variable Singularity . . . . . . . 17.5.4 *Convolution in the Multidimensional Case . . . . . . . . . . 17.5.5 Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
Table of Contents
180301 180302 180303 1803.4 180305
Fourier Integral Representation 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Differential Properties and the Fourier Transform 0 0 0 0 0 Main Structural Properties 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Examples of Applications 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Problems and Exercises 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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560 573 576 581 587 19 Asymptotic Expansions 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 595 1901 Asymptotic Formulas and Asymptotic Series 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 597 1901.1 Basic Definitions 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 597 1901.2 General Facts about Asymptotic Series 0 0 0 0 0 0 0 0 0 0 0 0 0 0 602 1901.3 Asymptotic Power Series 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 606 1901.4 Problems and Exercises 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 609 1902 The Asymptotics of Integrals (Laplace ' s Method) 0 0 0 0 0 0 0 0 0 0 0 612 190201 The Idea of Laplace ' s Method 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 612 190202 The Localization Principle for a Laplace Integral 0 0 0 0 0 615 190203 Canonical Integrals and their Asymptotics 0 0 0 0 0 0 0 0 0 0 0 617 190204 Asymptotics of a Laplace Integral 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 621 190205 *Asymptotic Expansions of Laplace Integrals 0 0 0 0 0 0 0 0 624 190206 Problems and Exercises 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 635 Topics and Questions for Midterm Examinations 0 0 0 0 0 0 0 0 0 0 0 0 0 643 10 Series and Integrals Depending on a Parameter 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 643 20 Problems Recommended as Midterm Questions 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 644 30 Integral Calculus (Several Variables) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 646 40 Problems Recommended for Studying the Midterm Topics 0 0 0 0 0 64 7 Examination Topics 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 649 1. Series and Integrals Depending on a Parameter 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 649 20 Integral Calculus (Several Variables) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 651 References 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 653 1. Classic Works 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 653 1.1 Primary Sources 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 653 1.20 Major Comprehensive Expository Works 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 653 1030 Classical courses of analysis from the first half of the twentieth century 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 653 20 Textbooks 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 654 30 Classroom Materials 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 654 40 Further Reading 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 655 Index of Basic Notation 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 657 Subject Index 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 661 Name Index 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 679 0
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0
9 *Continuous Mappings
( General
Theory
)
In this chapter we shall generalize the properties of continuous mappings established earlier for numerical-valued functions and mappings of the type f : JRm --+ JRn and discuss them from a unified point of view. In the process we shall introduce a number of simple, yet important concepts that are used everywhere in mathematics.
9 . 1 Metric Spaces 9 . 1 . 1 Definition and Examples
X is said to be endowed with a metric or a metric space structure or to be a metric space if a function (9.1) d : X x X--tlR
Definition 1. A set
i s exhibited satisfying the following conditions: a) = Q {o} = = b) ( symmetry ) , + c) ( the triangle inequality ) , where are arbitrary elements of X.
d(x 1 ,X2 ) X1 X2 , d(x 1 ,x2 ) d(x2 ,x2 ) d(x 1 , x3 )::; d(x 1 , x2 ) d(x2 , x3 ) x 1 , X 2 , X3
I n that case, the function (9 . 1) i s called a metric or distance o n X. Thus a metric space is a pair (X, consisting of a set X and a metric defined on it. In accordance with geometric terminology the elements of X are called
d)
points.
x3 x 1
We remark that if we set in the triangle inequality and take = account of conditions a) and b ) in the definition of a metric, we find that
that is, a distance satisfying axioms a ) , b ) , and c ) is nonnegative.
2
9 *Continuous Mappings (General Theory)
Let us now consider some examples.
Example 1.
The set � of real numbers becomes a metric space if we set
d(x i , x 2 ) = l x 2 - x i I for any two numbers XI and x 2 , as we have always done.
Example 2. Other metrics can also be introduced on R A trivial metric, for example, is the discrete metric in which the distance between any two distinct points is 1. The following metric on � is much more substantive. Let x f---t f(x) be a nonnegative function defined for x 2: 0 and vanishing for x = 0. If this function is strictly convex upward, then, setting (9.2)
for points XI, x 2 E �' we obtain a metric on R Axioms a ) and b ) obviously hold here, and the triangle inequality follows from the easily verified fact that f is strictly monotonic and satisfies the following inequalities for 0 < a < b:
f(a + b) - f(b) < f(a) - f(O) = f(a) . In particular, one could set d(x i , x 2 ) = Jlx i - x 2 l or d(x i , x 2 )
=
1 In the latter case the distance between any two points of the line I�1X-x21 l-X 2 is less than 1. •
Example 3.
Besides the traditional distance
n
d(x i , x 2 ) = L l x i - x 21 2 between points XI = ( x�, . . . , x?) and introduce the distance
i=I
x2
=
(9.3)
( x� , . . . , x?}:) in �n , one can also
(9.4) where p 2: 1. The validity of the triangle inequality for the function follows from Minkowski ' s inequality ( see Subsect. 5.4.2).
(9.4)
Example 4. When we encounter a word with incorrect letters while reading a text, we can reconstruct the word without too much trouble by correcting the errors, provided the number of errors is not too large. However, correcting the error and obtaining the word is an operation that is sometimes ambiguous. For that reason, other conditions being equal, one must give preference to the interpretation of the incorrect text that requires the fewest corrections.
9.1 Metric Spaces
3
Accordingly, in coding theory the metric (9.4) with p = 1 is used on the set of all finite sequences of length n consisting of zeros and ones. Geometrically the set of such sequences can be interpreted as the set of vertices of the unit cube I = {x E JR. n i O :::; x i :::; 1, i = 1, ... ,n } in JR. n . The distance between two vertices is the number of interchanges of zeros and ones needed to obtain the coordinates of one vertex from the other. Each such interchange represents a passage along one edge of the cube. Thus this distance is the shortest path along the edges of the cube from one of the vertices under consideration to the other.
Example 5. In comparing the results of two series of n measurements of the same quantity the metric most commonly used is (9.4) with p = 2. The distance between points in this metric is usually called their mean-square deviation. Example 6.
As one can easily see, if we pass to the limit in we obtain the following metric in JR. n :
(9.4) as p ---+ + oo , (9.5)
Example 7. The set C[a, b] of functions that are continuous on a closed inter val becomes a metric space if we define the distance between two functions f and g to be (9.6) d(f, g) = max l f(x) - g(x) l . a�x�b
Axioms a) and b ) for a metric obviously hold, and the triangle inequality follows from the relations
lf(x) - h (x) :S lf(x) - g(x) l + l g(x) - h (x) l :::; d(f, g) + d(g, h ) , that is,
d(f, h ) = a�x�b max l f(x) - h (x) l :::; d(f, g) + d(g, h ) . The metric ( 9.6) - the so-called uniform or Chebyshev metric in C[a, b] is used when we wish to replace one function by another ( for example, a polynomial ) from which it is possible to compute the values of the first with a required degree of precision at any point x E [a, b] . The quantity d(f, g) is precisely a characterization of the precision of such an approximate computation. The metric ( 9.6) closely resembles the metric ( 9.5) in JR. n . Example 8. Like the metric (9.4), for p ;:: 1 we can introduce in C[a, b] the metric b 1/ dp (f, g) = j I f - gi P (x) dx P ( 9.7)
-
(
a
)
4
9 *Continuous Mappings (General Theory)
It follows from Minkowski ' s inequality for integrals, which can be obtained from Minkowski ' s inequality for the Riemann sums by passing to the limit, that this is indeed a metric for p 2': 1. The following special cases of the metric (9. 7) are especially important: p = 1, which is the integral metric; p = 2, the metric of mean-square devia tion; and p = + oo , the uniform metric. The space C[a, b] endowed with the metric (9.7) is often denoted Cp [a, b]. One can verify that Coo [a, b] is the space C[a, b] endowed with the metric
(9.6). Example 9.
The metric (9.7) could also have been used on the set R[a, b] of Riemann-integrable functions on the closed interval [a, b]. However, since the integral of the absolute value of the difference of two functions may vanish even when the two functions are not identically equal, axiom a ) will not hold in this case. Nevertheless, we know that the integral of a nonnegative function
Example 1 0. In the set C ( k ) [a, b] of functions defined on [a, b] and having continuous derivatives up to order k inclusive one can define the following metric: (9.8 ) d(f, g) = max { M0 , . . . , Mk } , where
Using the fact that ( 9.6 ) is a metric, one can easily verify that ( 9.8 ) is also a metric. Assume for example that f is the coordinate of a moving point consid ered as a function of time. If a restriction is placed on the allowable region where the point can be during the time interval [a, b] and the particle is not allowed to exceed a certain speed, and, in addition, we wish to have some assurance that the accelerations cannot exceed a certain level, it is natural to consider the set { a�x�b max l f(x) l , max l f'(x) l , max l f"(x) l } for a function a�x�b
a�x�b
f E C ( 2 ) [a, b] and using these characteristics, to regard two motions f and g as close together if the quantity ( 9.8 ) for them is small.
These examples show that a given set can be metrized in various ways. The choice of the metric to be introduced is usually controlled by the statement of the problem. At present we shall be interested in the most general properties of metric spaces, the properties that are inherent in all of them.
9.1 Metric Spaces 9 . 1 . 2 Open and Closed Subsets of a Metric Space
5
Let (X, d) be a metric space. In the general case, as was done for the case X = 1Rn in Sect. 7.1, one can also introduce the concept of a ball with center
at a given point, open set , closed set , neighborhood of a point , limit point of a set , and so forth. Let us now recall these concepts, which are basic for what is to follow. >
8 0 and a E X the set B(a, 8) = {x E X I d(a, x) 8} is called the ball with center a E X of radius 8 or the 8-neighborhood of the point a. Definition 2. For
<
This name is a convenient one in a general metric space, but it must not be identified with the traditional geometric image we are familiar with in IR 3 .
Example 1 1 . The unit ball in C[a, b] with center at the function that is iden tically 0 on [a, b] consists of the functions that are continuous on the closed interval [a, b] and whose absolute values are less than 1 on that interval. Example 1 2. Let X be the unit square in IR 2 for which the distance between two points is defined to be the distance between those same points in IR 2 . Then X is a metric space, while the square X considered as a metric space in its own right can be regarded as the ball of any radius � ../2/2 about its p
center.
It is clear that in this way one could construct balls of very peculiar shape. Hence the term ball should not be understood too literally. Definition 3. A set G c is G there exists a ball
X open in the metric space (X, d) if for each B(x, 8) such that B(x, 8) G. It obviously follows from this definition that X itself is an open set in (X,n d). The empty set 0 is also open. By the same reasoning as in the case of 1R one can prove that a ball B(a, r) and its exterior {x E X : d(a, x) � r} are open sets. ( See Examples 3 and 4 o f Sect. 7.1). Definition 4 . A set F X i s closed in (X, d) i f its complement X \ F is open in (X, d). In particular, we conclude from this definition that the closed ball B (a, r) := {x E X l d(a, x) � r} is a closed set in a metric space (X, d). The following proposition holds for open and closed sets in a metric space (X, d). point
xE
C
C
6
9 *Continuous Mappings (General Theory)
a) The union aEA U Ga of the sets in any system {Ga, E A} of sets Ga that are open in X is an open set in X. n b) The intersection in=l Gi of any finite number of sets that are open in X is an open set in X. a') The intersection aEA n Fa of the sets in any system {Fa, E A} of sets Fa that are closed in X is a closed set in X. n b') The union iU=l Fi of any finite number of sets that are closed in X is a closed set in X. The proof of Proposition 1 is a verbatim repetition of the proof of the corresponding proposition for open and closed sets in JR n , and we omit it . ( See Proposition 1 in Sect . 7.1.) Definition 5 . A n open set i n X containing the point x E X is called a neighborhood of the point x in X. Definition 6. Relative to a set E X, a point x E X is called an interior point of E if some neighborhood of it is contained in X, an exterior point of E if some neighborhood of it is contained in the complement of E in X, a boundary point of E i f i t i s neither interior nor exterior t o E (that is, a
Proposition 1 .
a
C
every neighborhood of the point contains both a point belonging to E and a point not belonging to E) . Example 1 3. All points of a ball B(a, r) are interior to it , and the set Cx B (a, r) = X \ B (a, r) consists of the points exterior to the ball B (a, r). In the case of JR n with the standard metric d the sphere S(a, r) := {x E Rn l d(a, x) = r > 0} is the set of boundary points of the ball B(a, r). 1 Definition 7. A point a E X is a limit point of the set E C X if the set En O ( a) is infinite for every neighborhood O ( a) of the point . Definition 8 . The union of the set E and the set of all its limit points is called the closure of the set E in X. As before, the closure of a set E C X will be denoted E. Proposition 2 . A
limit points. Thus
C
set F X is closed in X if and only if it contains all its
(F is closed in
X)
-¢==:>
(F = F in
X); .
We omit the proof, since it repeats the proof of the analogous proposition for the case X = ]R n discussed in Sect . 7.1. 1 I n connection with Example 13 see also Problem 2 at the end of this section.
9.1 Metric Spaces
7
9 . 1 .3 Subspaces of a Metric space
If (X, d) is a metric space and E is a subset of X, then, setting the distance between two points X I and x 2 of E equal to d(x i , x 2 ), that is, the distance between them in X, we obtain the metric space ( E, d), which is customarily called a subspace of the original space (X, d). Thus we adopt the following definition.
(XI , d i ) is a subspace of the metric space (X, d) if XI X and the equality d i (a, b) = d(a, b) holds for any pair of points a, b in XI . Since the ball B I (a, r) = {x E XI I d i (a, x) r} in a subspace (XI , d i ) of the metric space (X, d) is obviously the intersection B I (a, r) = XI nB(a, r) of the set XI X with the ball B(a, r) in X, it follows that every open set in XI has the form Definition 9. A metric space c
<
c
where G is an open set in
ai =xi n a,
X, and every clos ed set FI in XI has the form
where F is a closed set in X. It follows from what has just been said that the properties of a set in a metric space of being open or closed are relative properties and depend on the ambient space.
Example 14. The open interval l x l 1, = 0 of the x-axis in the plane IR2 with the standard metric in IR 2 is a metric space (XI , d i ), which, like any metric space, is closed as a subset of itself, since it contains all its limit points in X I · At the same time, it is obviously not closed in JR 2 = X. <
y
This same example shows that openness is also a relative concept .
Example 15. The set C[a, b] of continuous functions on the closed interval [a, b] with the metric (9. 7) is a subspace of the metric space Rp [a, b]. However, if we consider the metric (9.6) on C[a, b] rather than (9. 7), this is no longer
true.
9 . 1 .4 The Direct Product of Metric Spaces
If (XI , d i ) and (X2 , d2 ) are two metric spaces, one can introduce a metric d on the direct product X I X x2 . The commonest methods of introducing a metric in XI X X2 are the following. If (x i , X 2 ) E X I X X2 and (x � , x�) E X I X X2 , one may set
9 *Continuous Mappings (General Theory)
8
or or
d( (x 1 , x2 ) , (x�, x�)) = max { d1 (x 1 , x�) , d2 (x2 , x�) } . It is easy to see that we obtain a metric on X 1 x X2 in all of these cases. Definition 10. if (X 1 , d l ) and (X2 , d2 ) are two metric spaces, the space (X1 X2 , d), where d is a metric on X1 X2 introduced by any of the methods just indicated, will be called the direct product of the original metric spaces. Example The space IR2 can be regarded as the direct product of two copies of the metric space lR with its standard metric, and JR 3 is the direct product IR 2 x IR 1 of the spaces IR 2 and IR 1 = R X
X
16.
9 . 1 . 5 Problems and Exercises 1 . a) Extending Example 2, show that if f : JR + -+ JR + is a continuous function that is strictly convex upward and satisfies f(O) = 0, while (X, d) is a metric space, then one can introduce a new metric df on X by setting df ( x1, x2 ) = f ( d( x1, x2 )) . b) Show that on any metric space (X, d) one can introduce a metric d' (x1, x2 ) = !� 1 ;·1�!� ) in which the distance between the points will be less than 1.
2 . Let (X, d) be a metric space with the trivial (discrete) metric shown in Example 2, and let a E X. For this case, what are the sets B(a, 1/2) , B(a, 1) , B(a, 1) , B(a, 1) , B(a, 3/2) , and what are the sets {x E X l d(a, x) = 1/2}, {x E Xl d(a, x) = 1}, B (a, 1) \ B(a, 1) , B(a, 1) \ B(a, 1 )?
a) Is it true that the union of any family of closed sets is a closed set? b) Is every boundary point of a set a limit point of that set? c) Is it true that in any neighborhood of a boundary point of a set there are points in both the interior and exterior of that set? d) Show that the set of boundary points of any set is a closed set. 3.
4. a) Prove that if (Y, dy ) is a subspace of the metric space (X, dx ) , then for any open (resp. closed) set Gy (resp. Fy ) in Y there is an open (resp. closed) set Gx (resp. Fx ) in X such that Gy = Y n Gx , (resp. Fy = Y n Fx ). b) Verify that if the open sets c;, and G'{r in Y do not intersect, then the corresponding sets G'x. and G'{ in X can be chosen so that they also have no points in common.
9.2 Topological Spaces
9
5. Having a metric d on a set X, one may attempt to define the distance d (A, B) between sets A C X and B C X as follows: d (A, B) inf d(a, b) . =
aEA, bEB
a) Give an example of a metric space and two nonintersecting subsets of it A and B for which d (A, B) = 0. b) Show, following Hausdorff, that on the set of closed sets of a metric space (X, d) one can introduce the Hausdorff metric D by assuming that for A C X and Be X D(A,B) := max { supd (a,B) , supd (A, b) } . aEA
bEB
9 . 2 Topological Spaces For questions connected with the concept of the limit of a function or a mapping, what is essential in many cases is not the presence of any particular metric on the space, but rather the possibility of saying what a neighborhood of a point is. To convince oneself of that it suffices to recall that the very definition of a limit or the definition of continuity can be stated in terms of neighborhoods. Topological spaces are the mathematical objects on which the operation of passage to the limit and the concept of continuity can be studied in maximum generality. 9.2. 1 Basic Definitions
X is said to be endowed with the structure of a topo logical space or a topology or is said to be a topological space if a system T of subsets of X is exhibited (called open sets in X) possessing the following Definition 1. A set
properties:
a ) 0 E T; X E T. b) ('v'a E A ( Ta E T))
C
===}
) ( T; E T; i = 1, . . . , n )
U Ta E T.
aEA
===}
n
n T; E T.
i=l
Thus, a topological space is a pair (X, T) consisting of a set X and a system T of distinguished subsets of the set having the properties that T contains the empty set and the whole set X, the union of any number of sets of T is a set of T, and the intersection of any finite number of sets of T is a set of T. As one can see, in the axiom system a) , b) , c) for a topological space we have postulated precisely the properties of open sets that we already proved in the case of a metric space. Thus any metric space with the definition of open sets given above is a topological space.
9 *Continuous Mappings (General Theory)
10
T
Thus defining a topology on X means exhibiting a system of subsets of X satisfying the axioms a) , b ) , and c ) for a topological space. Defining a metric in X, as we have seen, automatically defines the topol ogy on X induced by that metric. It should be remarked, however, that different metrics on X may generate the same topology on that set. Example 1. Let X = �n (n 1). Consider the metric d 1 (x 1 , x 2 ) defined by relation (9.5) in Sect . 9.1, and the metric d2 (x 1 , x 2 ) defined by formula (9.3) in Sect . 9.1. >
The inequalities
obviously imply that every ball B(a, r) with center at an arbitrary point a E X, interpreted in the sense of one of these two metrics, contains a ball with the same center, interpreted in the sense of the other metric. Hence by definition of an open subset of a metric space, it follows that the two metrics induce the same topology on X. Nearly all the topological spaces that we shall make active use of in this course are metric spaces. One should not think, however, that every topolog ical space can be metrized, that is, endowed with a metric whose open sets will be the same as the open sets in the system T that defines the topology on X. The conditions under which this can be done form the content of the so-called metrization theorems.
Definition 2. If (X, T ) is a topological space, the sets of the system T are called the open sets, and their complements in X are called the closed sets of the topological space ( X, T ) .
A topology T on a set X is seldom defined by enumerating all the sets in the system T. More often the system T is defined by exhibiting only a certain set of subsets of X from which one can obtain any set in the system T through union and intersection. The following definition is therefore very important.
Definition 3 . A base of the topological space (X, ) ( an open base or base for the topology) is a family 'B of open subsets of X such that every open set
GE
T
T
is the union of some collection of elements of the family
'B.
Example 2. If (X, d) is a metric space and (x, T ) the topological space corre sponding to it , the set 'B = { B( a, r)} of all balls, where a E X and r > 0, is obviously a base of the topology T. Moreover, if we take the system 'B of all balls with positive rational radii r, this system is also a base for the topology. Thus a topology can be defined by describing only a base of that topology. As one can see from Example 2, a topological space may have many different bases for the topology.
9.2 Topological Spaces
11
Definition 4 . The minimal cardinality among all bases of a topological
space is called its
weight.
As a rule, we shall be dealing with topological spaces whose topologies admit a countable base (see, however, Problems 4 and 6).
Example 3 . If we take the system �of balls in IR k of all possible rational radii r = n 0 with centers at all possible rational points ( mnt, , . . . , mn kk ) E IR k , we obviously obtain a countable base for the standard topology of IR k . It is not difficult to verify that it is impossible to define the standard topology in IR k by exhibiting a finite system of open sets. Thus the standard topological space IR k has countable weight. Definition 5. A neighborhood of a point of a topological space (X, ) is an !.!.!:.
>
open set containing the point.
T
It is clear that if a topology T is defined on X, then for each point the system of its neighborhoods is defined. It is also clear that the system of all neighborhoods of all possible points of topological space can serve as a base for the topology of that space. Thus a topology can be introduced on X by describing the neighborhoods of the points of X. This is the way of defining the topology in X that was originally used in the definition of a topological space. 2 Notice, for example, that we have introduced the topology in a metric space itself essentially by saying what a 6-neighborhood of a point is. Let us give one more example.
Example 4. Consider the set C(IR, IR) of real-valued continuous functions de fined on the entire real line. Using this set as foundation, we shall construct a new set - the set of germs of continuous functions. We shall regard two functions f, g E C(IR, IR) as equivalent at the point a E IR if there is a neigh borhood U(a) of that point such that Vx E U(a) ( f(x) = g(x)). The relation just introduced really is an equivalence relation (it is reflexive, symmetric, and transitive) . An equivalence class of continuous functions at the point a E IR is called germ of continuous functions at that point . If f is one of the functions generating the germ at the point a, we shall denote the germ itself by the symbol fa · Now let us define a neighborhood of a germ. Let U(a) be a neighborhood of the point a and f a function defined on U(a) generating the germ fa at a. This same function generates its germ fx at any point x E U (a). The set { !x } of all germs corresponding to the points x E U (a) will be called a neighborhood of the germ fa · Taking the set of such neigh borhoods of all germs as the base of a topology, we turn the set of germs of continuous functions into a topological space. It is worthwhile to note that 2 The concepts of a metric space and a topological space were explicitly stated
early in the twentieth century. In 1906 the French mathematician M. Fnkhet (1878-1973) introduced the concept of a metric space, and in 1914 the German mathematician F. Hausdorff ( 1868-1942) defined a topological space.
9 *Continuous Mappings (General Theory)
12
a
X
Fig. 9 . 1 .
in the resulting topological space two different points (germs) not have disjoint neighborhoods (see Fig. 9.1)
fa and ga may
Hausdorff if the Hausdorff axiom holds any two distinct points of the space have nonintersecting neighborhoods. Example 5. Any metric space is obviously Hausdorff, since for any two points a, b E X such that d( a, b) 0 their spherical neighborhoods B (a, � d( a, b) ) , B ( b, � d( a, b) ) have no points in common. At the same time, as Example 4 shows, there do exist non-Hausdorff topological spaces. Perhaps the simplest example here is the topological space (X, ) with the trivial topology { 0 , X}. If X contains at least two distinct points, then (X, ) is obviously not Hausdorff. Moreover, the complement X \ x of a point in this space is not an open set.
Definition 6. A topological space is
in it:
>
T
T =
T
We shall be working exclusively with Hausdorff spaces. Definition 7. A set E C X is (everywhere) dense in a topological space (X, T ) if for any point E X and any neighborhood of it the intersection E n ( X) is nonempty. Example 6. If we consider the standard topology in IR, the set Q of rational numbers is everywhere dense in JR. Similarly the set Q n of rational points in IRn is dense in !Rn . One can show that in every topological space there is an everywhere dense set whose cardinality does not exceed the weight of the topological space. Definition 8. A metric space having a countable dense set is called a sepa rable space. Example 7. The metric space (!Rn , d) in any of the standard metrics is a separable space, since Qn is dense in it. Example 8. The metric space ( C ([O, 1], IR ) , d) with the metric defined by (9.6) is also separable. For, as follows from the uniform continuity of the functions f E C ( [0, 1], IR ) , the graph of any such function can be approxi mated as closely as desired by a broken line consisting of a finite number of segments whose nodes have rational coordinates. The set of such broken lines is countable. We shall be dealing mainly with separable spaces.
U
x
U(x)
9.2 Topological Spaces
13
We now remark that , since the definition of a neighborhood of a point in a topological space is verbally the same as the definition of a neighborhood of a point in a metric space, the concepts of interior point, exterior point, boundary point, and limit point of a set , and the concept of the closure of a set , all of which use only the concept of a neighborhood, can be carried over without any changes to the case of an arbitrary topological space. Moreover, as can be seen from the proof of Proposition 2 in Sect . 7.1, it is also true that a set in a Hausdorff space is closed if and only if it contains all its limit points. 9.2.2 Subspaces of a Topological Space
Let (X, Tx ) be a topological space and Y a subset of X. The topology TX makes it possible to define the following topology Ty in Y, called the induced or relative topology on Y C X. We define an open set in Y to be any set Gy of the form Gy = Y n Gx, where Gx is an open set in X. It is not difficult to verify that the system Ty of subsets of Y that arises in this way satisfies the axioms for open sets in a topological space. As one can see, the definition of open sets Gy in Y agrees with the one we obtained in Subsect. 9.1.3 for the case when Y is a subspace of a metric space X. Definition 9. A subset Y C X of a topological space (X, T ) with the topol ogy Ty induced on Y is called a subspace of the topological space X. It is clear that a set that is open in (Y, Ty ) is not necessarily open in (X, Tx ).
9.2.3 The Direct Product of Topological Spaces
If (X1, T1 ) and (X2, T2 ) are two topological spaces with systems of open sets T! = {GI } and T2 = {G2 }, we can introduce a topology on xl X x2 by taking as the base the sets of the form G1 x G2 . Definition 10. The topological space (Xl X x2, Tl X T2 ) whose topology has the base consisting of sets of the form G1 x G2, where Gi is an open set in the topological space (Xi, Ti ), i = 1, 2, is called the direct product of the topological spaces (X1, TI ) and (X2, T2 ) · 2 Example 9. If IR = JR 1 and JR are considered with their standard topologies, 2 then, as one can see, JR is the direct product JR 1 x JR 1 . For every open set in 2 JR can be obtained, for exmaple, as the union of "square" neighborhoods of all its points. And squares ( with sides parallel to the axes ) are the products of open intervals, which are open sets in R It should be noted that the sets G1 x G2, where G1 E T1 and G2 E T2, constitute only a base for the topology, not all the open sets in the direct product of topological spaces.
14
9 *Continuous Mappings (General Theory)
9 . 2 . 4 Problems and Exercises
Verify that if (X, d) is a metric space, then ( X, 1 � d ) is also a metric space, and the metrics d and 1 � d induce the same topology on X . (See also Problem 1 of the preceding section.) 1.
a) In the set N of natural numbers we define a neighborhood of the number N to be an arithmetic progression with difference d relatively prime to n. Is the resulting topological space Hausdorff? b) What is the topology of N, regarded as a subset of the set lR of real numbers with the standard topology? c) Describe all open subsets of R 2.
n E
3. If two topologies 71 and 72 are defined on the same set, we say that 72 is stronger than 71 if 71 C 72 , that is 72 contains all the sets in 71 and some additional open sets not in 71 . a) Are the two topologies on N considered in the preceding problem comparable? b) If we introduce a metric on the set C[O, 1 ] of continuous real-valued functions defined on the closed interval [0, 1] first by relation (9.6) of Sect. 9. 1 , and then by relation (9. 7) of the same section, two topologies generally arise on C[a, b] . Are they comparable?
4. a) Prove in detail that the space of germs of continuous functions defined in Example 4 is not Hausdorff. b) Explain why this topological space is not metrizable. c) What is the weight of this space?
a) State the axioms for a topological space in the language of closed sets. b) Verify that the closure of the closure of a set equals the closure of the set. c) Verify that the boundary of any set is a closed set. d) Show that if F is closed and G is open in (X, 7) , then the set G \ F is open in (X, 7). e) If (Y, Ty ) is a subspace of the topological space (X, 7) , and the set E is such that E C Y C X and E E 7X , then E E Ty . 5.
A topological space (X, 7 ) in which every point is a closed set is called a topo logical space in the strong sense or a 71 -space. Verify the following statements. a) Every Hausdorff space is a 71 -space (partly for this reason, Hausdorff spaces are sometimes called 72 -spaces) . b) Not every 71 -space is a 72 -space. (See Example 4) . c) The two-point space X = { a , b} with the open sets {0, X } is not a 71 -space. d) In a 71 -space a set F is closed if and only if it contains all its limit points.
6.
15
9.3 Compact Sets
7. a ) Prove that in any topological space there is an everywhere dense set whose cardinality does not exceed the weight of the space. b ) Verify that the following metric spaces are separable: C[a, b] , c
9.3 Compact Sets 9 . 3 . 1 Definition and General Properties of Compact Sets
Definition 1 . A set K in a topological space (X, T) is compact ( or bicom pact3 ) if from every covering of K by sets that are open in X one can select
a finite number of sets that cover K.
Example 1. An interval [a, b] of the set lR of real numbers in the standard topology is a compact set , as follows immediately from the lemma of Subsect . 2.1 . 3 asserting that one can select a finite covering from any covering of a closed interval by open intervals. In general an m-dimensional closed interval I m = {x E lR m l a i :::; x i :::; b i , i = 1, . . . , m } in JRm is a compact set , as was established in Subsect . 7.1.3. It was also proved in Subsect . 7.1.3 that a subset of JR m is compact if and only if it is closed and bounded. In contrast to the relative properties of being open and closed, the prop erty of compactness is absolute, in the sense that it is independent of the ambient space. More precisely, the following proposition holds.
subset K of a topological space (X, T) is a compact subset of X if and only if K is compact as a subset of itself with the topology induced from (X, T). Proof. This proposition follows from the definition of compactness and the fact that every set G that is open in K can be obtained as the intersection of K with some set G that is open in X. D Thus, if (X, Tx ) and ( Y, Ty) are two topological spaces that induce the same topology on K X n Y, then K is simultaneously compact or not compact in both X and Y. Example 2 . Let d be the standard metric on lR and I = { x E JR I 0 x 1} the unit interval in R The metric space (I, d) is closed ( in itself ) and bounded, but is not a compact set , since for example, it is not a compact subset of JR. Proposition 1 . A
K
x
C
<
<
3 The concept of compactness introduced by Definition 1 is sometimes called bi compactness in topology.
16
9 *Continuous Mappings (General Theory)
We now establish the most important properties of compact sets.
If K is a compact set in a Hausdorff space (X, ) then K is a closed subset of X . Proof. By the criterion for a set to be closed, it suffices to verify that every limit point of K , xo E X, belongs to K. Suppose x0 ¢. K. For each point x E K we construct an open neigh borhood G(x) such that x0 has a neighborhood disjoint from G(x). The set G(x), x E K, of all such neighborhoods forms an open covering of K, from which one can select a finite covering G(x i ), . . . , G(x n ) · Now if Oi (x0) is a n neighborhood of Xo such that G(xi) n Oi(xo) 0, the set O(x) n Oi(xo) i= l is also a neighborhood of x0 , and G(xi) n O(x0) 0 for all i = 1, . .. , n. But this means that K n O(x0) 0 , and then x 0 cannot be a limit point for K. Lemma 1. (Compact sets are closed) . r ,
=
=
D
:J
=
=
:J
:J
:J
If K1 K2 · · · Kn · · · is a nested sequence of nonempty compact sets, then the intersection in= Ki is nonempty. l Proof. By Lemma 1 the sets Gi K1 \ Ki , i 1, ... , n, . . . are open in K1 . If the intersection n Ki is empty, then the sequence G 1 G 2 · · · Gn · · · i= l Lemma 2 . (Nested compact sets. ) =
00
=
c
00
c
c
c
forms a covering of K1 . Extracting a finite covering from it, we find that some element G m of the sequence forms a covering of K 1 . But by hypothesis Km = K1 \ G m =/= 0. This contradiction completes the proof of Lemma 2.
D
Lemma 3 . (Closed subsets of compact sets. ) A
closed subset F of a compact set K is itself compact. Proof. Let {G0 , E A} be an open covering of F. Adjoining to this collection the open set G K \ F, we obtain an open covering of the entire compact set K. From this covering we can extract a finite covering of K. Since GnF = 0, it follows that the set {Ga , E A} contains a finite covering of F. D a
=
a
9 . 3 . 2 Metric Compact Sets
We shall establish below some properties of metric compact sets, that is, metric spaces that are compact sets with respect to the topology induced by the metric. C
E X is called an e-grid in the metric space (X, d) x E X there is a point e E E such that d(e, x) < e. Lemma 4. (Finite e-grids. ) If a metric space (K, d) is compact, then for every e 0 there exists a finite €-grid in X .
Definition 2 . The set
if for every point >
9.3 Compact Sets
17
Proof. For each point x E K we choose an open ball B ( x, e). From the open covering of K by these balls we select a finite covering B(xb e), . . . , B(x n , e). The points x 1 , . . . , X n obviously form the required e-grid. 0 In analysis, besides arguments that involve the extraction of a finite cov ering, one often encounters arguments in which a convergent subsequence is extracted from an arbitrary sequence. As it happens, the following proposi tion holds.
( ) A metric space ( K, d) is compact if and only if from each sequence of its points one can extract a subsequence that converges to a point of K . The convergence o f the sequence {x n } t o some point a E K, as before, means that for every neighborhood U(a) of the point a E K there exists an index N E N such that n E U(a) for n N. We shall discuss the concept of limit in more detail below in Sect. 9.6. Proposition 2 . Criterion for compactness in a metric space.
X
>
We preface the proof of Proposition 2 with two lemmas.
Lemma 5. If a metric space ( K, d) is such that from each sequence of its points one can select a subsequence that converges in K, then for every e 0 there exists a finite e-grid. Proof. If there were no finite e0 -grid for some e o 0, one could construct a sequence {x n } of points in K such that d(x n , ) e o for all n E N and all i E { 1 , . . . ,n 1 } . Obviously it is impossible to extract a convergent >
Xi
-
subsequence of this sequence. 0
>
>
Lemma 6. If the metric space ( K, d) is such that from each sequence of its points one can select a subsequence that converges in K, then every nested sequence of nonempty closed subsets of the space has a nonempty intersection. Proof. If F1 · · · Fn · · · is the sequence of closed sets, then choosing one point of each, we obtain a sequence x 1 , . . . , n , , from which we ex tract a convergent subsequence {x n J · The limit a E K of this sequence, by construction, necessarily belongs to each of the closed sets F , i E N. 0 :J
:J
:J
X
We can now prove Proposition 2.
•
.
.
i
Proof. We first verify that if (K, d) is compact and { Xn } a sequence of points in it , one can extract a subsequence that converges to some point of K. If the sequence {x n } has only a finite number of different values, the assertion is obvious. Therefore we may assume that the sequence {x n } has infinitely many different values. For = 1 / 1 , we construct a finite 1-grid and take a closed ball 1 ) that contains an infinite number of terms of the se quence. By Lemma 3 the ball 1 ) is itself a compact set, in which there exists a finite e 2 = 1 / 2-grid and a ball 1 / 2 ) containing infinitaly many
B(a 1 ,
e1
B(al >
B(a2 ,
18
9 *Continuous Mappings (General Theory)
elements of the sequence. In this way a nested sequence of compact sets B (at , 1 ) :J B (a2 , 1 / 2 ) :J · · · :J B (an , 1 / n ) :J · · · arises, and by Lemma 2 has a common point a E K. Choosing a point X n1 of the sequence {x n } in the ball B (a 1 , 1 ) , then a point X n 2 in B (a2 , 1 / 2 ) with n 2 > n 1 , and so on, we obtain a subsequence { X n ; } that converges to a by construction. We now prove the converse, that is, we verify that if from every sequence {xn } of points of the metric space ( K, d) one can select a subsequence that converges in K, then ( K, d) is compact. In fact , if there is some open covering { G a , a E A} of the space ( K, d) from which one cannot select a finite covering, then using Lemma 5 to con struct a finite 1-grid in K, we find a closed ball B(a 1 , 1 ) , that also cannot be covered by a finite collection of sets of the system {Ba , a E A}. The ball B (a 1 , 1 ) can now be regarded as the initial set , and, constructing a finite 1 / 2-grid in it , we find in it a ball B (a 2 , 1 / 2 ) that does not admit covering by a finite number of sets in the system {Go, a E A}. The resulting nested sequence of closed sets B (a 1 , 1 ) :J B(a 2 , 1 / 2 ) :J · · · :J B (an , 1 / n ) :J · · · has a common point a E K by Lemma 6, and the construction shows that there is only one such point . This point is covered by some set G0 0 of the system; and since G0 0 is open, all the sets B (an , 1 / n ) must be contained in G0 0 for sufficiently large values of n. This contradiction completes the proof of the proposition. 0 9.3.3 Problems and Exercises 1. A subset of a metric space is totally bounded if for every c > 0 it has a finite c-grid. a) Verify that total boundedness of a set is unaffected, whether one forms the grid from points of the set itself or from points of the ambient space. b) Show that a subset of a metric space is compact if and only if it is totally bounded and closed. c) Show by example that a closed bounded subset of a metric space is not always totally bounded, and hence not always compact. 2 . A subset of a topological space is relatively (or conditionally) compact if its closure is compact. Give examples of relatively compact subsets of IR.n . 3. A topological space is locally compact if each point of the space has a relatively compact neighborhood. Give examples of locally compact topological spaces that are not compact.
4. Show that for every locally compact, but not compact topological space (X, Tx ) there is a compact topological space (Y, Ty ) such that X C Y, Y \ X consists of a single point , and the space (X, rx ) is a subspace of the space (Y, Ty ) .
9.4 Connected Topological Spaces
19
9.4 Connected Topological Spaces Definition 1. A topological space (X, 7 ) is connected if it contains no open closed sets 4 except X itself and the empty set.
This definition will become more transparent to intuition if we recast it in the following form. A topological space is connected if and only if it cannot be represented as the union of two disjoint nonempty closed sets (or two disjoint nonempty open sets) . Definition 2. A set E in a topological space (X, 7 ) is connected if it is connected as a topological subspace of (X, 7 ) (with the induced topology) .
It follows from this definition and Definition 1 that the property of a set of being connected is independent of the ambient space. More precisely, if (X, 7X ) and (Y, 7y ) are topological spaces containing E and inducing the same topology on E, then E is connected or not connected simultaneously in both X and Y.
Example 1. Let E = {x E JRI x =f. 0}. The set E_ = {x E E l x < 0} is nonempty, not equal to E, and at the same time open-closed in E (as is E+ = { x E JRI x > 0}), if E is regarded as a topological space with the topology induced by the standard topology of R Thus, as our intuition suggests, E is not connected. Proposition. (Connected subsets of R ) A
C
nonempty set E lR is connected if and only if for any x and belonging to E, the inequalities x y imply that y E E . z
<
< z
Thus, the only connected subsets o f the line are intervals (finite or infi nite) : open, half-open, and closed.
Proof. N e c e s s i t y. Let E be a connected subset of JR, and let the triple of points a, b, c be such that a E E, b E E, but c tJ_ E, even though a < c < b. Setting A = {x E El x < c } , B = {x E El x > c } , we see that a E A, b E B, that is, A =f. 0 , B =f. 0 , and An B = 0 . Moreover E = A U B, and both sets A and B are open in E. This contradicts the connectedness of E. S u f f i c i en c y. Let E be a subspace of lR having the property that to gether with any pair of points a and b belonging to it, every point between them in the closed interval [a, b] also belongs to E. We shall show that E is connected. Suppose that A is an open-closed subset of E with A =f. 0 and B = E \ A =f. 0. Let a E A and b E B. For definiteness we shall assume that a < b. (We certainly have a =f. b, since A n B = 0 . ) Consider the point 4 That is, sets that are simultaneously open and closed.
20
9 *Continuous Mappings (General Theory) E
3
c 1 = sup { A n [a , b]}. Since A a ::; c 1 ::; b E B, we have c 1 E. Since A is closed in E, we conclude that c 1 E A. Considering now the point c 2 = inf { B n [c 1 , b]} we conclude similarly, since B is closed, that c 2 E B. Thus a ::; c 1 c 2 ::; b, since c 1 E A, c 2 E B, and A n B = 0 . But it now follows from the definition of c 1 and c 2 and the relation E = A B that no point of the open interval h , c 2 [ can belong to E. This contradicts the original property of E. Thus the set E cannot have a subset A with these properties, and that proves that E is connected. D U
<
9.4. 1 Problems and Exercises 1. a) Verify that if A is an open-closed subset of (X, r ) , then B = X \ A is also such a set. b) Show that in terms of the ambient space the property of connectedness of a set can be expressed as follows: A subset E of a topological space (X, T ) is connected if and only if there is no pair of open (or closed) subsets G'x , G'Jc that are disjoint and such that E n G'x # 0, E n G'Jc # 0, and E c G'x U G'fc . 2.
Show the following: a) The union of connected subspaces having a common point is connected. b) The intersection of connected subspaces is not always connected. c) The closure of a connected subspace is connected.
3. One can regard the group G L ( n) of nonsingular n x n matrices with real entries as 2 an open subset in the product space lRn , if each element of the matrix is associated with a copy of the set lR of real numbers. Is the space GL(n) connected?
4. A topological space is locally connected if each of its points has a connected neighborhood. a) Show that a locally connected space may fail to be connected. b) The set E in JR 2 consists of the graph of the function x 1--t sin � (for x =I 0) plus the closed interval { ( x , y ) E 1R2 I x = 0 1\ I Y I :S 1} on the y-axis. The set E is endowed with the topology induced from JR 2 • Show that the resulting topological space is connected but not locally connected.
5. In Subsect. 7.2.2 we defined a connected subset of lRn as a set E C lRn any two of whose points can be joined by a path whose support lies in E. In contrast to the definition of topological connectedness introduced in the present section, the concept we considered in Chapt. 7 is usually called path connectedness or arcwise connectedness. Verify the following: a) A path-connected subset of lRn is connected. b) Not every connected subset of lRn with n > 1 is path connected. (See Prob lem 4.) c) Every connected open subset of lRn is path connected.
9.5 Complete Metric Spaces
21
9 . 5 Complete Metric Spaces In this section we shall be discussing only metric spaces, more precisely, a class of such spaces that plays an important role in various areas of analysis. 9.5. 1 Basic Definitions and Examples
By analogy with the concepts that we already know from our study of the space JR. n , we introduce the concepts of fundamental ( Cauchy ) sequences and convergent sequences of points of an arbitrary metric space.
Definition 1 . A sequence {x n ; n E N} of points of a metric space (X, d) is a fundamental or Cauchy sequence if for every c > 0 there exists N E N such that d(xm , X n ) < c for any indices m,n E N larger than N.
{x n ; n E N} of points of a metric space (X, d) converges to the point a E X and a is its limit if n-too lim d(a, X n ) 0. A sequence that has a limit will be called convergent, as before.
Definition 2. A sequence
=
We now give the basic definition.
Definition 3. A metric space of its points is convergent.
(X, d) is complete if every Cauchy sequence
Example 1. The set JR. of real numbers with the standard metric is a com plete metric space, as follows from the Cauchy criterion for convergence of a numerical sequence. We remark that, since every convergent sequence of points in a metric space is obviously a Cauchy sequence, the definition of a complete metric space essentially amounts to simply postulating the Cauchy convergence cri terion for it.
Example 2. If the number 0, for example, is removed from the set JR., the remaining set JR. \ 0 will not be a complete space in the standard metric. Indeed, the sequence X n = 1 /n, n E N, is a Cauchy sequence of points of this set, but has no limit in JR. \ 0.
The space JR. n with any of its standard metrics is complete, as was explained in Subsect. 7.2.1.
Example 3.
Example 4.
Consider the set C[a, b] of real-valued continuous functions on a closed interval [a, b] C JR., with the metric (9.9) d(f, g) a::5x::5b max l f(x) - g(x)l ( see Sect. 9. 1 , Example 7). We shall show that the metric space C[a, b] =
complete.
IS
22
9 *Continuous Mappings (General Theory)
Proof.
Let
is,
Un (x) : n E N} be a Cauchy sequence of functions in C[a, b], that
'Vc;
E N ( ( N n N) x E [a, b] ( l fm (x) - fn (x) l c:) ) . (9. 10) For each fixed value of x E [a, b], as one can see from (9.10) , the numerical sequence Un (x); n E N} is a Cauchy sequence and hence has a limit f(x) by >
0 3N
E N 'Vm E N ====}
'Vn 'V
m
>
1\
>
====;.
<
the Cauchy convergence criterion. Thus f(x) := lim --+
n oo fn (x) , x E [a, b] . We shall verify that the function f(x) i s continuous o n [a, b], f E C[a, b]. It follows from (9.10 ) and (9. 1 1 ) that the inequality
l f(x) - fn (x) l :::; E 'Vx E [a, b]
(9. 1 1 ) that is, ( 9.12 )
holds for n > N. We fix the point x E [a, b] and verify that the function f is continuous at this point. Suppose the increment h is such that (x + h ) E [a, b]. The identity
f(x + h) - f(x) = f(x + h) - fn (x + h) + fn (x + h) - fn (x) + fn (x) - f(x) implies the inequality
l f(x + h) - f(x) l :::; l f(x + h ) - fn (x + h) l + l fn (x + h) - fn (x) l + l fn (x) - f(x) l .
( 9.13 )
By virtue of ( 9.12 ) the first and last terms on the right-hand side of this last inequality do not exceed c; if n > N. Fixing n > N, we obtain a function fn E C[a, b], and then choosing 6 = 6(c:) such that l fn (x + h) - fn (x) l < c; for l hl < 6, we find that l f(x + h) - f(x) l < 3c: if l h l < 6. But this means that the function f is continuous at the point x. Since x was an arbitrary point of the closed interval [a, b] , we have shown that f E C[a, b] . D Thus the space C[a, b] with the metric ( 9.9 ) is a complete metric space. This is a very important fact, one that is widely used in analysis.
Example 5.
If instead of the metric ( 9.9 ) we consider the integral metric b
d(f, g) = j l f - g l (x) dx
( 9. 14 )
a
on the same set
C[a, b] , the resulting metric space is no longer complete.
9.5 Complete Metric Spaces
23
Proof. For the sake of notational simplicity, we shall assume [a, b] = [ - 1, 1] and consider, for example, the sequence {fn E C[ - 1, 1]; n E N} of functions defined as follows: - 1 if - 1 :::; x :::; - 1 In ,
{
( See Fig. 9.2. )
fn (x) = nx 1,
:
if if
<
-1
1 ln
Fig.
<
- 1 In x 1 In , 1ln :::; x :::; 1 .
1
x
9.2.
It follows immediately from properties of the integral that this sequence is a Cauchy sequence in the sense of the metric (9.14) in C[ - 1, 1] . At the same time, it has no limit in C[ - 1, 1]. For if a continuous function f E C[ - 1, 1] were the limit of this sequence in the sense of metric (9.14), then f would have to be constant on the interval - 1 :::; x < 0 and equal to - 1 while at the same time it would have to be constant and equal to 1 on the interval 0 < x :::; 1, which is incompatible with the continuity of f at the point x = 0. D
Example 6. It is slightly more difficult to show that even the set R[a, b] of real-valued Riemann-integrable functions defined on the closed interval [a, b] is not complete in the sense of the metric 9.14.5 We shall show this, using the Lebesgue criterion for Riemann integrability of a function. Proof. We take [a, b] to be the closed interval [0, 1], and we shall construct a Cantor set on it that is not a set of measure zero. Let L1 E] O , 113[. We remove from the interval [0, 1] the middle piece of it of length .:1. More precisely, we remove the .:1 1 2-neighborhood of the midpoint of the closed interval [0, 1]. 5
In regard to the metric (9. 14) on R[a, b] see the remark to Example 9 in Sect. 9. 1 .
24
9 *Continuous Mappings (General Theory)
On each of the two remaining intervals, we remove the middle piece of length 1/3. On each of the four remaining closed intervals we remove the middle piece of length L1 · 1/3 2 , and so forth. The length of the intervals removed in this process is L1 + L1 · 2/3 + L1 · 4/3 2 + · · · + L1 · (2/3) n + · · · = 3..:1. Since 0 < L1 < 1/3, we have 1 - 3..:1 > 0, and, as one can verify, it follows from this that the (Cantor) set K remaining on the closed interval [0, 1] does not have measure zero in the sense of Lebesgue. Now consider the following sequence: {fn E R[O, 1 ] ; n E N}. Let fn be a function equal to 1 everywhere on [0, 1] except at the points of the intervals removed at the first n steps, where it is set equal to zero. It is easy to verify that this sequence is a Cauchy sequence in the sense of the metric (9.14). If some function f E R[O, 1] were the limit of this sequence, then f would have to be equal to the characteristic function of the set K at almost every point of the interval [0, 1] . Then f would have discontinuities at all points of the set K. But , since K does not have measure 0, one could conclude from the Lebesgue criterion that f � R[O , 1] . Hence R[a, b] with the metric (9.14) is not a complete metric space. D Ll ·
9 . 5 . 2 The Completion of a Metric Space
Example 7. Let us return again to the real line and consider the set Ql of rational numbers with the metric induced by the standard metric on R It is clear that a sequence of rational numbers converging to v'2 in is a Cauchy sequence, but does not have a limit in Ql, that is, Ql is not a complete space with this metric. However, Ql happens to be a subspace of the complete metric space JR., which it is natural to regard as the completion of Ql. Note that the set Q C could also be regarded as a subset of the complete metric space JR.2 , but it does not seem reasonable to call lR. 2 the completion of Q.
JR.
JR.
Definition 4. The smallest complete metric space containing a given metric
space
(X, d) is the completion of (X, d).
This intuitively acceptable definition requires at least two clarifications: what is meant by the "smallest" space, and does it exist? We shall soon be able to answer both of these questions; in the meantime we adopt the following more formal definition.
(X, d) is a subspace of the metric space (Y, d) and the set X Y is everywhere dense in Y, the space (Y, d) is called a completion of the metric space (X, d). Definition 6 . We say that the metric space (X 1 , d i ) i s isometric t o the metric space (X2 , d2 ) if there exists a bijective mapping f : X 1 X2 such that d2 ( f(a), f(b) ) = d 1 (a, b) for any points a and b in X1 . (The mapping f : xl x2 is called an isometry in that case. ) Definition 5 . If a metric space C
--+
--+
9.5 Complete Metric Spaces
25
It is clear that this relation is reflexive, symmetric, and transitive, that is, it is an equivalence relation between metric spaces. In studying the properties of metric spaces we study not the individual space, but the properties of all spaces isometric to it. For that reason one may regard isometric spaces as identical.
Example 8. Two congruent figures in the plane are isometric as metric spaces, so that in studying the metric properties of figures we abstract completely, for example, from the location of a figure in the plane, identifying all congruent figures. By adopting the convention of identifying isometric spaces, one can show that if the completion of a metric space exists at all, it is unique. As a preliminary, we verify the following statement. Lemma. The following inequality holds for any quadruple of points a, b, v of the metric space (X, d) : (9. 15) l d(a, b) - d(u, v ) l :::; d(a , ) + d(b, v ) . Proof By the triangle inequality d(a, b) :::; d(a, ) + d( , v ) + d(b, v) .
u,
u
u
u
By the symmetry of the points, this relation implies (9. 1 5) . 0 We now prove uniqueness of the completion.
If the metric spaces (Yt , d 1 ) and (Y2 , d2 ) are completions of the same space (X, d), then they are isometric. Proof. We construct an isometry f : Y1 -+ Y2 follows. For x E X we set f(x) = x. Then d2 (/(x l ), f(x2 ) ) = d ( f(x l ) , f(x2 ) ) = d(xt , x 2 ) = d1 (x 1 . x2 ) for x 1 , x 2 E X. If Y 1 E Y1 \ X, then Y l is a limit point for X, since X is everywhere dense in Y1 . Let {x n ; n E N} be a sequence of points of X converging to y1 in the sense of the metric d 1 . This sequence is a Cauchy sequence in the sense of the metric d 1 . But since the metrics d1 and d2 are both equal to d on X, this sequence is also a Cauchy sequence in ( Y2 , d2 ) . The latter space is complete, and hence this sequence has a limit y2 E Y2 . It can be verified in the standard manner that this limit is unique. We now set f(y1 ) = Y2 . Since any point Y2 E Y2 \ X, just like any point Y1 E Y1 \ X, is the limit of a Cauchy sequence of points in X, the mapping f : Y1 -+ Y2 so Proposition 1 .
as
constructed is surjective. We now verify that
(9. 16) for any pair of points
y � , y� of Y1 .
26
9 *Continuous Mappings (General Theory)
If y� and y� belong to X, this equality is obvious. In the general case we take two sequences { x�; E N} and { x�; E N} converging to y� and y� respectively. It follows from inequality (9 . 1 5 ) that
n
n
d 1 (y� , yn = nlim -+oo d 1 (x�, x� ) ,
or, what is the same,
(9.17) f( YD and
By construction these same sequences converge to y� y� = f(y�) respectively in the space (Y2 , d2 ). Hence d2 (y�, y�) = nlim (9.18) -+oo d(x � , x� ) . Comparing relations (9.17 ) and (9.18), we obtain Eq. (9.16). This equality then simultaneously establishes that the mapping f : Y1 -+ Y2 is injective and hence completes the proof that f is an isometry. 0 In Definition 5 of the completion (Y, d) of a metric space (X, d) we required that (X, d) be a subspace of (Y, d) that is everywhere dense in (Y, d). Under
the identification of isometric spaces one could now broaden the idea of a completion and adopt the following definition. Definition 5'. A metric space (Y, dy) is a completion of the metric space (X, dx) if there is a dense subspace of (Y, dy) isometric to (X, dx ). We now prove the existence of a completion.
Proposition 2.
Proof.
Every metric space has a completion.
If the initial space itself is complete, then it is its own completion. We have already essentially demonstrated the idea for constructing the completion of an incomplete metric space (X, dx) when we proved Proposi tion 1. Consider the set of Cauchy sequences i n the space (X, dx ) . Two such sequences { x�; E N} and { x�; E N} are called equivalent or confinal if dx(x�, x�) 0 as It is easy to see that confinality really is an equivalence relation. We shall denote the set of equivalence classes of Cauchy sequences by S. We introduce a metric in S by the following rule. If s ' and s" are elements of S, and { x�; E N} and { x�; E N} are sequences from the classes s ' and s" respectively, we set
-+
n n n -+ oo. n
n
( 9.19 ) d(s', s" ) = nlim -+oo dx (x�, x � ) . It follows from inequality (9 . 1 5 ) that this definition is unambiguous: the
limit written on the right exists (by the Cauchy criterion for a numeri cal sequence) and is independent of the choice of the individual sequences { x�; E N} and { x�; E N} from the classes s' and s"
n
n
27
9.5 Complete Metric Spaces
The function d( s' , s") satisfies all the axioms of a metric. The resulting metric space (S, d) is the required completion of the space (X, dx ) . Indeed, (X, dx) is isometric to the subspace (Sx , d) of the space (S, d) consisting of the equivalence classes of fundamental sequences that contain constant sequences {x n = x E X; E N}. It is natural to identify such a class s E S with the point x E X. The mapping f : (X, dx) -+ ( Sx , d) is obviously an isometry. It remains to be verified that (Sx , d) is everywhere dense in (S, d) and that (S, d) is a complete metric space. We first verify that (Sx, d) is dense in (S, d). Let s be an arbitrary element of S and {x n ; E N} a Cauchy sequence in (X, dx) belonging to the class s E S. Taking �n = f(x n ), E N, we obtain a sequence {�n ; E N} of points of (Sx, d) that has precisely the element s E S as its limit , as one can see from (9.19). We now prove that the space (S, d) is complete. Let {s n ; E N} be an arbitrary Cauchy sequence in the space (S, d). For each E N we choose an element �n in (Sx, d) such that d(s n , �n ) < 1 / . Then the sequence {�n ; E N}, like the sequence {s n ; E N}, is a Cauchy sequence. But in that case the sequence {x n = f - 1 (�n ); E N} will also be a Cauchy sequence. The sequence {x n ; E N} defines an element s E S, to which the given sequence {sn ; E N} converges by virtue of relation (9.19). 0
n
n
n
n
n
nn
n
n
n n
n
Remark 1 . Now that Propositions 1 and 2 have been proved, it becomes un derstandable that the completion of a metric space in the sense of Definition 51 is indeed the smallest complete space containing (up to isometry) the given metric space. In this way we have justified the original Definition 4 and made it precise.
Remark 2. The construction of the set JR. of real numbers, starting from the set Q of rational numbers could have been carried out exactly as in the construction of the completion of a metric space, which was done in full generality above. That is exactly how the transition from Q to JR. was carried out by Cantor. Remark 3. In Example 6 we showed that the space R[a, b] of Riemann integrable functions is not complete in the natural integral metric. Its com pletion is the important space £[a, b] of Lebesgue-integrable functions. 9.5.3 Problems and Exercises 1. a ) Prove the following n e s t e d b a l l l e m m a. Let (X, d) be a metric space and B(x 1 , r 1 ) :J · · · :J B(x n , rn ) :J · · · a nested sequence of closed balls in X whose radii tend to zero. The space (X, d) is complete if and only if for every such sequence there exists a unique point belonging to all the balls of the sequence.
28
9 *Continuous Mappings (General Theory)
b) Show that if the condition r n -+ 0 as n -+ oo is omitted from the lemma stated above, the intersection of a nested sequence of balls may be empty, even in a complete space. 2. a) A set E C X of a metric space (X, d) is nowhere dense in X if it is not dense in any ball, that is, if for every ball B(x, r) there is a second ball B(x 1 , r1 ) C B(x, r) containing no points of the set E . A set E is of first category in X if it can be represented as a countable union of nowhere dense sets. A set that is not of first category is of second category in X . Show that a complete metric space i s a set o f second category (in itself) . b) Show that if a function f E c< = > [a, b] is such that Vx E [a, b] 3n E N Vm > n u<m> (x) = 0) , then the function f is a polynomial.
9 . 6 Continuous Mappings of Topological Spaces From the point of view of analysis, the present section and the one following contain the most important results in the present chapter. The basic concepts and propositions discussed here form a natural, some times verbatim extension to the case of mappings of arbitrary topological or metric spaces, of concepts and propositions that are already well known to us in . In the process, not only the statement but also the proofs of many facts turn out to be identical with those already considered; in such cases the proofs are naturally omitted with a reference to the corresponding propositions that were discussed in detail earlier. 9 . 6 . 1 The Limit of a Mapping a. The Basic Definition and Special Cases of it
f : X -+ Y be a mapping of the set X with a fixed base { B} in X into a topological space Y . The point A E Y is the limit of the mapping f : X -+ Y over the base B, and we write lim f ( x ) = A, if for every neighborhood V(A) of A in Y there exists an element B E B of the base B whose image under the mapping f is contained in V(A) . Definition 1 . Let
B=
t3
In logical symbols Definition 1 has the form lim f ( x ) = A := 'v'V(A) c Y 3B E B t3
c
( f(B) V(A)) .
We shall most often encounter the case in which X , like Y , is a topological space and B is the base of neighborhoods or deleted neighborhoods of some point a E X. Retaining our earlier notation a for the base of deleted 0 neighborhoods { U (a)} of the point a, we can specialize Definition 1 for this base: lim f ( ) = A := 'v'V(A) C Y 3 U(a) C X ( f ( U(a) ) C V(A) ) .
x -+
x -+ a
x
9.6 Continuous Mappings of Topological Spaces
29
If (X, dx) and (Y, dy) are metric spaces, this last definition can be re stated in e-8 language: lim
x -ta
f(x) = A := Ve
In other words, lim
x -+a
>
0
:38
>
0 Vx E X
(o < dx (a, x) < 8 ===* dy ( A, f(x) ) < c: ) .
f(x) = A � xlim dy ( A, f(x) ) = 0 . --+ a
Thus we see that, having the concept of a neighborhood, one can define the concept of the limit of a mapping f : X -+ Y into a topological or metric space Y just as was done in the case Y = lR or, more generally, Y = R n . b. Properties of the Limit of a Mapping We now make some remarks on the general properties of the limit . We first note that the uniqueness of the limit obtained earlier no longer holds when Y is not a Hausdorff space. But if Y is a Hausdorff space, then the limit is unique and the proof does not differ at all from the one given in the special cases Y = lR or Y = Rn . Next, if f : X -+ Y is a mapping into a metric space, it makes sense to speak of the boundedness of the mapping (meaning the boundedness of the set f( X ) in Y), and of ultimate boundedness of a mapping with respect to the base B in X (meaning that there exists an element B of B on which f is bounded) . It follows from the definition of a limit that if a mapping f : X -+ Y of a set X with base B into a metric space Y has a limit over the base B, then it is ultimately bounded over that base. c. Questions Involving the Existence of the Limit of a Mapping
Let Y be a set with base By and g : Y -+ Z a mapping of Y into a topological space Z having a limit over the base By . Let X be a set with base Bx and f : X -+ Y a mapping of X into Y such that for every element By E By there exists an element Bx E Bx whose image is contained in By, that is, f ( Bx) By . Under these hypotheses the composition g o f : X -+ Z of the mappings f and g is defined and has a limit over the base Bx, and lim g o f(x) = lim g(y) . Bv Bx For the proof see Theorem 5 of Sect . 3.2.
Proposition 1. (Limit of a composition of mappings. )
C
Another important proposition on the existence of the limit is the Cauchy criterion, to which we now turn. This time we will be discussing a mapping f : X -+ Y into a metric space, and in fact a complete metric space. In the case of a mapping f : X -+ Y of the set X into a metric space (Y, d) it is natural to adopt the following definition.
30
9 *Continuous Mappings (General Theory)
Definition 2. The
the quantity
oscillation of the mapping f : X -+ Y on a set E d ( f(x i ), f(x 2 ) ) . w(f, E) = x 1 sup ,x E E
C
X is
2
The following proposition holds.
Proposition 2 . ( Cauchy criterion for existence of the limit of a mapping. )
Let X be a set with a base B, and let f : X -+ Y be a mapping of X into a complete metric space ( Y, d) . A necessary and sufficient condition for the mapping f to have a limit over the base B is that for every c 0 there exist an element B in B on which the oscillation of the mapping is less than c . >
More briefly: >
3 lim f(x) -{==:} Vc 0 3B E B (w(f, B) < c) . 8
For the proof see Theorem 4 of Sect. 3.2. It is useful to remark that the completeness of the space Y is needed only in the implication from the right-hand side to the left-hand side. Moreover, if Y is not a complete space, it is usually this implication that breaks down. 9 . 6 . 2 Continuous Mappings a. Basic Definitions
Definition 3. A mapping f : X -+ Y of a topological space (X, Tx ) into a topological space ( Y, Ty ) is continuous at a point a E X if for every neigh borhood V ( f(a) ) C Y of the point f(a) E Y there exists a neighborhood U(a) C X of the point a E X whose image f ( U(a) ) is contained in V ( f(a) ) .
Thus,
f : X -+ Y
is continuous at a E X := = VV ( f(a)) 3U(a)
c
( f ( U(a) ) V ( f(a))) . In the case when X and Y are metric spaces (X, d ) and ( Y, dy ) , Defini
tion 3 can of course be stated in c-r5 language:
x
f : X -+ Y is continuous at a E X := dy (! (a), f(x) ) < c) . = Vc 0 3r5 0 Vx E X ( d (a , x) < r5 Definition 4. The mapping f : X -+ Y is continuous if it is continuous at each point x E X . >
>
x
===}
The set o f continuous mappings from X into Y will b e denoted C(X, Y ) .
9.6 Continuous Mappings of Topological Spaces
Theorem 1 . (Criterion for continuity. ) A
31
mapping f : X -+ Y of a topolog ical space (X, Tx) into a topological space (Y, Ty) is continuous if and only if the pre-image of every open (resp. closed) subset of Y is open (resp. closed) in X. Proof. Since the pre-image o f a complement i s the complement o f the pre image, it suffices to prove the assertions for open sets. We first show that if f E C(X, Y) and Gy E Ty, then Gx f - 1 (Gy) belongs to Tx. If G x 0 , it is immediate that the pre-image is open. If G x # 0 and a E G x, then by definition of continuity of the mapping f at the point a, for the neighborhood Gy of the point f(a) there exists a neighborhood Ux(a) of a E X such that f ( Ux (a) ) Gy. Hence Ux(a) Gx f - 1 (Gy ) . U Ux (a), we conclude that G x is open, that is, G x E Tx. Since Gx aE G x We now prove that if the pre-image of every open set in Y is open in X, then f E C(X, Y). But , taking any point a E X and any neighbor hood Vy ( f(a) ) of its image f(a) in Y, we discover that the set Ux(a) f - 1 ( Vy ( f(a) ) ) is an open neighborhood of a E X, whose image is contained in Vy (! (a) ) . Consequently we have verified the definition of continuity of the mapping f : X -+ Y at an arbitrary point a E X. 0 Definition 5 . A bijective mapping f : X -+ Y of one topological space (X, Tx) onto another (Y, Ty) is a homeomorphism if both the mapping itself and the inverse mapping f - 1 Y -+ X are continuous. =
=
c
=
C
=
=
:
Definition 6 . Topological spaces that admit homeomorphisms onto one an other are said to be homeomorphic.
As Theorem 1 shows, under a homeomorphism f : X -+ Y of the topolog ical space (X, Tx) onto (Y, Ty) the systems of open sets Tx and Ty correspond to each other in the sense that G x E Tx {:} f ( G x) = Gy E Ty. Thus, from the point of view of their topological properties homeomorphic spaces are absolutely identical. Consequently, homeomorphism is the same kind of equivalence relation in the set of all topological spaces as, for example, isometry is in the set of metric spaces. b. Local Properties of Continuous Mappings We now exhibit the lo cal properties of continuous mappings. They follow immediately from the corresponding properties of the limit .
Let (X, Tx), (Y, Ty) and (Z, Tz) be topological spaces. If the mapping Y -+ Z is continuous at a point b E Y and the mapping f : X -+ Y is continuous at a point a E X for which f(a) b, then the composition of these mappings g f X -+ Z is continuous at a E X. Proposition 3. (Continuity of a composition of continuous mappings. )
o
:
=
g :
This follows from the definition of continuity of a mapping and Proposi tion 1 .
32
9 *Continuous Mappings (General Theory)
Proposition 4.
( Boundedness of a mapping in a neighborhood of a point
If a mapping f X ---+ Y of a topological space (X, ) into a metric space (Y, d) is continuous at a point a E X , then it is bounded in some neighborhood of that point.
of continuity. )
:
r
This proposition follows from the ultimate boundedness ( over a base ) of a mapping that has a limit. Before stating the next proposition on properties of continuous mappings, we recall that for mappings into or we defined the quantity
JR. JR.n
w(f; a) r-+ lim0 w ( f , B(a , r) ) to be the oscillation of f at the point a. Since both the concept of the oscil lation of a mapping on a set and the concept of a ball B(a, r) make sense in any metric space, the definition of the oscillation w(f, a) of the mapping f at the point a also makes sense for a mapping f X ---+ Y of a metric space (X, dx) into a metric space (Y, dy ). Proposition 5 . A mapping f X ---+ Y of a metric space (X, dx) into a metric space (Y, dy) is continuous at the point a E X if and only ifw(f, a) = 0 . :=
:
:
This proposition follows immediately from the definition o f continuity of a mapping at a point. c. G lobal Properties of Continuous Mappings We now discuss some of the important global properties of continuous mappings.
The image of a compact set under a continuous mapping is compact. Proof. Let f : K ---+ Y be a continuous mapping of the compact space ( K, ) into a topological space (Y, y ) , and let { cy. , E A} be a covering of f(K) by sets that are open in Y. By Theorem 1 , the sets { Gx f - 1 ( GY.) , E A} form an open covering of K. Extracting a finite covering 0�1 , , G�n , we find a finite covering G':1 , , G':n of f(K) Y. Thus f(K) is compact in Y. D Corollary. A continuous real-valued function f : K ---+ JR. on a compact set assumes its maximal value at some point of the compact set (and also its minimal value, at some point). Proof. Indeed, f(K) is a compact set in JR., that is, it is closed and bounded. This means that inf f(K) E f(K) and sup f(K) E f(K). D In particular, if K is a closed interval [a, b] JR., we again obtain the Theorem 2 .
T
•
•
a
•
=
C
C
•
•
•
a
TK
classical theorem of Weierstrass. Cantor ' s theorem on uniform continuity carries over verbatim to map pings that are continuous on compact sets. Before stating it , we must give a necessary definition.
9.6 Continuous Mappings of Topological Spaces
33
Definition 7. A mapping f : X ---+ Y of a metric space (X, dx) into a metric space (Y, dy) is uniformly continuous if for every E: > 0 there exists 8 > 0 such that the oscillation w(f, E) of f on each set E C X of diameter less than 8 is less than E: . Theorem 3. ( Uniform continuity. ) A
continuous mapping f : K ---+ Y of a compact metric space K into a metric space (Y, dy) is uniformly continuous. In particular, if K is a closed interval in lR and Y = JR, we again have the classical theorem of Cantor, the proof of which given in Subsect . 4.2.2. carries over with almost no changes to this general case. Let us now consider continuous mappings of connected spaces.
The image of a connected topological space under a continuous mapping is connected. Proof. Let f : X ---+ Y be a continuous mapping of a connected topological space (X, Tx) onto a topological space (Y, Ty ) . Let Ey be an open-closed subset of Y. By Theorem 1, the pre-image Ex = f - 1 (Ey) of the set Ey is open-closed in X. By the connectedness of X, either Ex = 0 or Ex = X. But this means that either Ey = 0 or Ey = Y = f(X). 0 Corollary. If a function f X ---+ lR is continuous on a connected topological space (X, T) and assumes values f(a) = A E lR and f(b) = B E JR, then for any number C between A and B there exists a point c E X at which f ( c ) = C. Proof. Indeed, by Theorem 4 f(X) i s a connected set i n JR. But the only connected subsets of lR are intervals ( see the Proposition in Sect . 9.4). Thus the point C belongs to f(X) along with A and B. 0 In particular, if X is a closed interval, we again have the classical Theorem 4.
:
intermediate-value theorem for a continuous real-valued function. 9.6.3 Problems and Exercises
1. a) If the mapping f : X -+ Y is continuous, will the images of open (or closed) sets in X be open (or closed) in Y? b) If the image, as well as the inverse image, of an open set under the mapping f : X -+ Y is open, does it necessarily follow that f is a homeomorphism? c) If the mapping f : X -+ Y is continuous and bijective, is it necessarily a homeomorphism? d) Is a mapping satisfying b) and c) simultaneously a homeomorphism? 2. Show the following. a) Every continuous bijective mapping of a compact space into a Hausdorff space is a homeomorphism. b) Without the requirement that the range be a Hausdorff space, the preceding statement is in general not true.
34
9 *Continuous Mappings (General Theory)
3. Determine whether the following subsets of IR.n are (pairwise) homeomorphic as topological spaces: a line, an open interval on the line, a closed interval on the line; a sphere; a torus.
4. A topological space (X, r ) is arcwise connected or path connected if any two of its points can be joined by a path lying in X. More precisely, this means that for any points A and B in X there exists a continuous mapping f : I -+ X of a closed interval [a, b] C JR. into X such that f( a ) = A and f(b) = B. a) Show that every path connected space is connected. b) Show that every convex set in IR.n is path connected. c) Verify that every connected open subset of IR.n is path connected. d) Show that a sphere S(a, r) is path connected in IR.n , but that it may fail to be connected in another metric space, endowed with a completely different topology. e) Verify that in a topological space it is impossible to join an interior point of a set to an exterior point without intersecting the boundary of the set.
9 . 7 The Contraction Mapping Principle Here we shall establish a principle that, despite its simplicity, turns out to be an effective way of proving many existence theorems. Definition 1 . A point
f (a)
= a.
a
E
X is a fixed point of a mapping f : X -+ X if
f : X -+ X of a metric space (X, d) into itself contraction if there exists a number q, 0 q 1 , such that the
Definition 2 . A mapping
is called a inequality
holds for any points
x 1 and x2 in X.
<
<
(9.20)
Theorem. Picard6�Banach7 fixed-point principle. A
( ) contraction mapping f : X -+ X of a complete metric space (X, d) into itself has a unique fixed point a . Moreover, for any point x0 E X the recursively defined sequence x0, x 1 = f( x o ) , . . . , Xn+l f( xn ), . . . converges to a . The rate of convergence is given by the estimate =
(9.21) 6
Ch. E. Picard ( 1856�1941 ) � French mathematician who obtained many impor tant results in the theory of differential equations and analytic function theory. 7 S. Banach ( 1892� 1945) � Polish mathematician, one of the founders of functional analysis.
9. 7 The Contraction Mapping Principle
35
Proof. We shall take an arbitrary point x0 E X and show that the sequence xo, x 1 = f ( xo ) , . . . , Xn+l = f ( xn ) , . . . is a Cauchy sequence. The mapping f is a contraction, so that by Eq.
(9.20)
and
From this one can see that the sequence x 0 , x 1 , . . . , X n , . . . is indeed a Cauchy sequence. The space (X, d) is complete, so that this sequence has a limit lim X n =
n -+cx:>
a E X.
It is clear from the definition of a contraction mapping that a contraction is always continuous, and therefore
-+cx:> X n ) = f(a) . -+ oo f( xn ) = f ( nlim n -+ oo Xn+l = nlim
a = lim
Thus a is a fixed point of the mapping f. The mapping f cannot have a second fixed point , since the relations a i f(ai ), i = 1, 2, imply, when we take account of (9.20), that
=
which is possible only if d( a 1 , a 2 ) = 0, that is, a 1 = a2 . Next , by passing to the limit as k -+ oo in the relation
we find that The following proposition supplements this theorem.
Proposition. ( Stability of the fixed point . ) Let (X, d) be a complete metric space and (D, ) a topological space that will play the role of a parameter space in what follows. Suppose to each value of the parameter t E D there corresponds a contrac tion mapping ft X -+ X of the space X into itself and that the following conditions hold. a ) The family Ut ; t E D} is uniformly contracting, that is, there exists q, 0 q 1, such that each mapping ft is a q-contraction. T
:
<
<
36
9 *Continuous Mappings (General Theory) -t
b) For each x E X the mapping ft(x) : Q X is continuous as a function of t at some point to E il, that is tlim -- Ho ft (x) = fto (x ) . Then the solution a(t) E X of the equation x = ft(x) depends continuously on t at the point t0, that is, tlim -Ho a(t) = a(t0) . Proof. As was shown in the proof of the theorem, the solution a(t) of the equation x = ft(x) can be obtained as the limit of the sequence {x n + l = ft(x n ); n = 0, 1, . . . } starting from any point Xo E X. Let xo = a(t0) a(to) ) . fto (Taking account of the estimate (9.21) and condition a) , we obtain d ( a(t), a( to) ) = d(a(t), xo ) :::; 1-d(x , xo) = -1-d(ft ( a(to) ) , fto ( a(to) )) . :::; 1-q l 1-q By condition b) , the last term in this relation tends to zero as t t 0 . Thus -t
it has been proved that
tlim -Ho a(t) = a( to) . D -Ho d (a(t), a( to) ) = 0 , that is, tlim
Example 1 . As an important example of the application of the contraction mapping principle we shall prove, following Picard, an existence theorem for the solution of the differential equation y'(x) = f ( x, y(x) ) satisfying an initial condition y(xo) = Yo · If the function f E C(JR2 , JR) is such that
where M is a constant, then, for any initial condition y(xo) = Yo , (9.22) there exists a neighborhood U(xo) of xo E lR and a unique function y = y(x) defined in U(x0) satisfying the equation y' = f ( x, y) (9.23) and the initial condition (9.22) . Proof. Equation ( 9.23 ) and the condition (9.22 ) can be jointly written as a
single relation
X
y(x) = Yo + j f ( t, y(t) ) dt .
xo
(
9.24 )
37
9. 7 The Contraction Mapping Principle
Denoting the right-hand side of this equality by A(y), we find that A : C(V(x0), lR) -+ C(V(x0), lR) is a mapping of the set of continuous functions defined on a neighborhood V (x0) of x0 into itself. Regarding C(V(x0), lR ) as a metric space with the uniform metric (see formula (9.6) from Sect . 9.1), we find that
d(Ay1 ,Ayz) =
II1ax
x E V ( xo )
X
X
I J!(t, yl (t)) dt - j f (t,yz(t)) dt l ::; xo X
xo
I J M I Y1 (t) - Yz(t) l dtl :S M i x - xol d(yl , Yz) . If we assume that l x - x0 I ::; Ju then the inequality :S II1ax
x E V ( xo )
xo
2
,
is fulfilled on the corresponding closed interval I, where max Thus we have a contraction mapping xEI
I Y 1 ( x) - Yz ( x) I ·
d(y1 , y ) 2
A : C(I, JR) C(I, JR) of the complete metric space ( C(I, JR), d) (see Example 4 of Sect . 9.5) into itself, which by the contraction mapping principle must have a unique fixed point y = Ay. But this means that the function in C(I, JR) just found is the unique function defined on I x0 and satisfying Eq. (9.24). 0 -t
3
Example 2. As an illustration of what was just said, we shall seek a solution of the familiar equation
with the initial condition principle. In this case
(9.22)
y' y =
on the basis of the contraction mapping X
Ay = Yo + J y(t) dt , 1. and the principle is applicable at least for l x - xo I Starting from the initial approximation y(x) 0, we construct succes sively the sequence O, y 1 = A(O), . . . , Yn + l (t) A(yn (t)), . . . of approxima tions xo
::; q
=
=
<
38
9 *Continuous Mappings (General Theory)
Yo , Yo ( l + (x - xo)) , 2 Yo ( l + (x - xo) + !(x - x0) ) , Yn +l ( t )
=
Yo ( l + (x - xo) + � (x - xo) 2 + · · · + � (x - x0) n ) ,
from which it is already clear that
y(x) = yo ex - xo .
The fixed-point principle stated in the theorem above also goes by the name of the contraction mapping principle. It arose as a generalization of Pi card ' s proof of the existence theorem for a solution of the differential equation ( 9.23), which was discussed in Example 1 . The contraction mapping principle was stated in full generality by Banach. Example 3. Newton 's method of finding a root of the equation f(x) = 0. Sup pose a real-valued function that is convex and has a positive derivative on a closed interval [a, ,8] assumes values of opposite signs at the endpoints of the interval. Then there is a unique point a in the interval at which f(a) = 0. In addition to the elementary method of finding the point a by successive bisection of the interval, there also exist more sophisticated and rapid methods of finding it , using the properties of the function f. Thus, in the present case, one may use the following method, proposed by Newton and called Newton 's method or the method of tangents. Take an arbitrary point xo E [a, ,8] and write the equation y f(xo) + J'(xo)(x - xo) of the tangent to the graph of the function at the point ( x0, f(x0)) . We then find l the point x 1 = x0 - [f'(xo) ] - f(x0) where the tangent intersects the x-axis ( Fig. 9.3). We take x 1 as the first approximation of the root a and repeat this operation, replacing x0 by x 1 . In this way we obtain a sequence
=
·
(9. 2 5)
of points that , as one can verify, will tend monotonically to a in the present case. In particular, if f(x) = x k - a, that is, when we are seeking {'l'a, where a > 0, the recurrence relation ( 9.25 ) has the form
X n +l = X n -
x� - a
� ,
kx n which for k = 2 becomes the familiar expression X n +l =
� (Xn + x:J .
9. 7 The Contraction Mapping Principle
Fig.
The method
method.
39
9.3.
(9. 25) for forming the sequence { Xn } is called Newton 's
If instead of the sequence recurrence relation
(9.25) we consider the sequence obtained by the
Xn+ 1 = Xn - [f (xo ) l - 1 · f(xn ) ,
( 9.26 )
x0. x A(x) = x - [f'(xa)r 1 · f(x) .
( 9.27)
1
we speak of the modified Newton 's method. 8 The modification amounts to computing the derivative once and for all at the point Consider the mapping H
By Lagrange ' s theorem
where � is a point lying between Thus, if the conditions
x 1 and x2 .
( 9.28 )
c
A(I) I
[f'(xo ) ] - 1 f'(x)l
and
( 9.29 ) · I :::; q < 1 , hold on some closed interval I C JR, then the mapping A I ---+ I defined by relation ( 9.27 ) is a contraction of this closed interval. Then by the general principle it has a unique fixed point on the interval. But, as can be seem from (9. 27), the condition A(a) = a is equivalent to = 0. Hence, when conditions ( 9.28 ) and ( 9.29 ) hold for a function j, the mod ified Newton ' s method ( 9.26 ) leads to the required solution = a of the equation = 0 by the contraction mapping principle. :
f(a)
8
f (x)
x
In functional analysis it has numerous applications and is called the Newton Kantorovich method. L.V. Kantorovich (1912-1986) - eminent Soviet mathemati cian, whose research in mathematical economics earned him the Nobel Prize.
40
9 *Continuous Mappings ( General Theory )
9. 7. 1 Problems and Exercises 1. Show that condition (9.20) in the contraction mapping principle cannot be re placed by the weaker condition 2. a ) Prove that if a mapping f : X -+ X of a complete metric space (X, d) into itself is such that some iteration of it f n : X -+ X is a contraction, then f has a unique fixed point. b ) Verify that the mapping A : C(I, R) -+ C(I, R) in Example 2 is such that for any closed interval I C R some iteration A n of the mapping A is a contraction. c ) Deduce from b ) that the local solution y = y0e x - x o found in Example 2 is actually a solution of the original equation on the entire real line.
3. a ) Show that in the case of a function on (a: , ,8] that is convex and has a positive derivative and assumes values of opposite signs at the endpoints, Newton's method really does give a sequence {x n } that converges to the point a E (o:, ,B] at which f(a) = 0. b ) Estimate the rate of convergence of the sequence (9.25) to the point a.
1 0 *Different ial Calculus
from a more General Point of View
1 0 . 1 Normed Vector Spaces Differentiation is the process of finding the best local linear approximation of a function. For that reason any reasonably general theory of differentiation must be based on elementary ideas connected with linear functions. From the course in algebra the reader is well acquainted with the concept of a vector space, as well as linear dependence and independence of systems of vectors, bases and dimension of a vector space, vector subspaces, and so forth. In the present section we shall present vector spaces with a norm, or as they are described, normed vector spaces, which are widely used in analysis. We begin, however, with some examples of vector spaces. 10. 1 . 1 Some Examples of Vector Spaces in Analysis
JRn
Example 1.
en
The real vector space and the complex vector space are classical examples of vector spaces of dimension n over the fields of real and complex numbers respectively.
Example 2. In analysis, besides the spaces JRn and en exhibited in Example 1, we encounter 1 n the space closest to them, which is the space f of sequences
or complex numbers. The vector-space operations xin=£, (asx in, . . JR. ,nxand, . . e. ) nof, arerealcarried out coordinatewise. One peculiarity of this space, when compared with JRn or e n is that any finite subsystem of the countable system of vectors { x i = (O, . . . , O,x i = 1, 0, . . . ) , i E N} is linearly independent , that is, f is an infinite-dimensional vector space ( of countable
dimension in the present case ) . The set of finite sequences ( all of whose terms are zero from some point on ) is a vector subspace f of the space £, also infinite-dimensional. 0
Example 3. Let F[a, b] be the set of numerical-valued ( real- or complex valued ) functions defined on the closed interval [a, b]. This set is a vector space over the corresponding number field with respect to the operations of addition of functions and multiplication of a function by a number.
10 *Differential Calculus from a General Viewpoint
42
{0
The set of functions of the form
e,.(x) =
, if
1,
if
X E [a, b] X E [a, b]
and and
x =/=x=
T , T
is a continuously indexed system of linearly independent vectors in F[a, b]. The set C[a, b] of continuous functions is obviously a subspace of the space F[a, b] just constructed.
Example 4. If X1 and X2 are two vector spaces over the same field, there is a natural way of introducing a vector-space structure into their direct product X1 x X2 , namely by carrying out the vector-space operations on elements X = (xi. X 2 ) E X1 X X2 coordinatewise. Similarly one can introduce a vector-space structure into the direct prod uct X 1 x x Xn of any finite set of vector spaces. This is completely anal ogous to the cases of Rn and e n . ·
·
·
10. 1 . 2 Norms in Vector Spaces
We begin with the basic definition.
X be a vector space over the field of real or complex numbers. A function I I : X --+ R assigning to each vector x E X a real number x ll ll is called a norm in the vector space X if it satisfies the following three conditions: a) ll x ll = 0 {::} x = 0 (nondegeneracy) ; b ) 11..\ x ll = l..\l ll x ll (homogeneity) ; c ) ll x 1 + x 2 ll :=::; ll x i !I + ll x 2 ll (the triangle inequality) .
Definition 1. Let
Definition 2 . A vector space with a norm defined on it is called a
vector space.
Definition 3. The value of the norm at a vector is called the
vector.
normed
norm of that
The norm of a vector is always nonnegative and, as can be seen by a) , equals zero only for the zero vector.
Proof. Indeed, by c) , taking account of a) and b) , we obtain for every x E X, 0 = I lOll = ll x + ( -x) ll :=::; ll x ll + II - x ll = ll x ll + 1 - 1 l ll x ll = 2 ll x ll D ·
By induction, condition c) implies the following general inequality.
(10.1)
43
10. 1 Normed Vector Spaces
and taking account of b) , one can easily deduce from c) the following useful inequality.
(10.2)
Every normed vector space has a natural metric
(10.3)
x2 )
The fact that the function d( x 1 , just defined satisfies the axioms for a metric follows immediately from the properties of the norm. Because of the vector-space structure in X the metric d in X has two additional special properties: that is, the metric is translation-invariant , and that is, it is homogeneous. Definition 4. If a normed vector space is complete as a metric space with
the natural metric
Banach space. Example 5. If for
p
(10.3),
2:
it is called a
complete normed vector space
or
1 we set (10.4)
(xi, . . . , xn ) n
E JR , it follows from Minkowski ' s inequality that we for x = obtain a norm on JRn . The space JRn endowed with this norm will be denoted JR� . One can verify that
(10.5) and that as p
l x i P -+ max { lx 1 1 , . . . , l xn l} -+ +oo. Thus, it is natural to set l x l oo := max { l x 1 l , . . . , l xn l} .
It then follows from
(10.6) (10.7)
(10.4) and (10.5) that
l x l oo l x i P l x l t ::=:;
::=:;
::=:;
n ll x ll oo for P
2: 1
·
(10.8)
It is clear from this inequality, as in fact it is from the very definition of the norm in Eq. (10.4), that JR� is a complete normed vector space.
l xi P
44
10 *Differential Calculus from a General Viewpoint
Example 6. X = X1 x
The preceding example can be usefully generalized as follows. If x Xn is the direct product of normed vector spaces, one can introduce the norm of a vector x (x 1 , . . . , xn ) in the direct product by setting ·
·
·
=
(10.9) where ll x i ll i s the norm o f the vector Xi E Xi . Naturally, inequalities (10.8) remain valid in this case as well. From now on, when the direct product of normed spaces is considered, unless the contrary is explicitly stated, it is assumed that the norm is defined in accordance with formula (10.9) (including the case p +oo) .
=
Example 7. Let p � 1. We denote by fp the set of sequences x = (x l , . . . , xn , . . . ) of real or complex numbers such that the series I:00= l xn l p nl converges, and for x E fp we set (10.10)
Using Minkowski ' s inequality, one can easily see that fp is a normed vector space with respect to the standard vector-space operations and the norm (10.10). This is an infinite-dimensional space with respect to which R� is a vector subspace of finite dimension. All the inequalities (10.8) except the last are valid for the norm (10.10). I t i s not difficult t o verify that fp i s a Banach space.
Example 8. In the vector space C[a, b] of numerical-valued functions that are continuous on the closed interval [a, b], one usually considers the following norm: (10.11) 11!11 max l f(x) l .
:=
x E [a ,b]
We leave the verification of the norm axioms t o the reader. We remark that this norm generates a metric on C[a, b] that is already familiar to us (see Sect. 9.5), and we know that the metric space that thereby arises is complete. Thus the vector space C[a, b] with the norm (10.11) is a Banach space.
Example 9.
One can also introduce another norm in b
)
1
C[a, b]
II J II P := (/ IJI P (x) dx -;; , p � 1 , a
which becomes (10.11) as p -+ +oo. It is easy to see (for example, Sect . 9.5) that the space norm ( 10.12) is not complete for 1 :S p < +oo.
(10.12) C[a, b] with the
10.1 Normed Vector Spaces
45
10. 1 . 3 Inner Products in Vector Spaces
An important class of normed spaces is formed by the spaces with an inner product. They are a direct generalization of Euclidean spaces. We recall their definition. Definition 5. We say that a
Hermitian form is defined in a vector space X ) ( , ) : X x X --+ C
( over the field of complex numbers if there exists a mapping
having the following properties: = a) = b) + = c) + where are vectors in X and
(x1( AX,x1 ,X2 ) ) (x.A(x1,x 2 ,x 1) , 2 ), 2 (x1x 1 , x x, x2 , x3 ) (x1, x3 ) (x2 , x3A),E C. 2 3
It follows from a ) , b ) , and c ) , for example, that
(x1, .Ax2 ) = (.Ax2 ,x 1) = .A(x2 ,x 1) X (x2 ,x 1) X(xbx2 ) ; (x1, x2 + x3 ) = (x2 + x3 , x 1) (x2 , x 1) + (x3 , x 1) (x1, x2 ) + (x1, x3 ) ; (x, x) (x, x) , that is, (x, x) is a real number. =
=
=
=
=
A Hermitian form is called nonnegative if d) � 0 and nondegenerate if e) = 0 <=? = 0. If X is a vector space over the field of real numbers, one must of course consider a real-valued form In this case a ) can be replaced by = which means that the form is symmetric with respect to its vector arguments and An example of such a form is the dot product familiar from analytic geometry for vectors in three-dimensional Euclidean space. In connection with this analogy we make the following definition.
(x,x) (x, x)
x
2 ). (x1,x2 ) (x2 ,x 1), x 1 (x1,x x2 .
Definition 6. A nondegenerate nonnegative Hermitian form in a vector space is called an inner product in the space.
Example 1 0. An inner product of vectors (y 1 , . . . , yn ) in JR.n can be defined by setting x n (x, y) L=l xi yi ' i n and in e by setting n :=
i yi . x (x, y) := L i=l
(10.13) (10.14)
10 *Differential Calculus from a General Viewpoint
46
Example 1 1 .
In £ 2 the inner product of the vectors x and y can be defined as 00
(x, y ) := L x i y i , i= l
The series in this expression converges absolutely since
Example 1 2.
00
00
00
i= l
i= l
i= l
An inner product can be defined in
C[a, b] by the formula
b
(!, g) := J ( ! · g)(x) dx .
(10.15)
a
It follows easily from properties of the integral that all the requirements for an inner product are satisfied in this case. The following important inequality, known as the
inequality, holds for the inner product : l (x , y ) l 2 :::; (x, x)
·
( y, y ) ,
Cauchy-Bunyakovskii (10.16)
where equality holds if and only if the vectors x and y are collinear.
Proof. Indeed, let a = (x, x), b = (x, y ) , and and c 2 0. If c > 0, the inequalities
with
A = - � imply
or
c = ( y, y ) . By hypothesis a 2 0
O <- a - -bbc - -bbc + -bbc
0 :::; ac - bb = ac - I W , (10.17) which is the same as ( 10.16). The case a 0 can be handled similarly. If a = c = 0, then, setting A = -b in ( 10.17), we find 0 :::; - bb - bb = -2 I W , that is, b = 0, and (10.16) is again true. If x and y are not collinear, then 0 < (x + Ay, x + A Y ) and consequently inequality (10.16) is a strict inequality in this case. But if x and y are collinear, >
it becomes equality as one can easily verify. 0
10.1 Normed Vector Spaces
47
A vector space with an inner product has a natural norm:
ll x ll := v'<.X:X)
and metric
(10.18)
d(x, y) := ll x - Yll ·
Using the Cauchy-Bunyakovskii inequality, we verify that if (x, y ) is a nondegenerate nonnegative Hermitian form, then formula (10.18) does indeed define a norm. Proof. In fact , since the form Next ,
ll x ll = v (x, x ) = 0 � X = 0 ,
(x, y) is nondegenerate.
II Axl l = V (Ax, Ax ) = JA X (x, x ) = I A I V<.X:X} = I A I II x ll ·
We verify finally that the triangle inequality holds:
ll x + Yll :::; ll x ll + IIYII · Thus, we need to show that
J (x + y, x + y )
:::; V<.X:X} + JT.Y:Y) ,
or, after we square and cancel, that
(x, y ) + (y, x ) :::; 2vf (x, x ) · (y, y ) .
But
(x, y ) + (y, x ) = (x, y ) + (x, y ) = 2 Re (x, y ) :::; 2 l (x, y ) l ,
and the inequality to be proved now follows immediately from the Cauchy Bunyakovskii inequality (10.16). 0 In conclusion we note that finite-dimensional vector spaces with an inner product are usually called Euclidean or Hermitian (unitary) spaces according as the field of scalars is � or C respectively. If a normed vector space is infinite dimensional, it is called a Hilbert space if it is complete in the metric induced by the natural norm and a pre-Hilbert space otherwise.
48
10 *Differential Calculus from a General Viewpoint
10. 1 .4 Problems and Exercises 1. a) Show that if a translation-invariant homogeneous metric d(x 1 , x 2 ) is defined in a vector space X, then X can be normed by setting llxll = d(O, x) . b) Verify that the norm in a vector space X is a continuous function with respect to the topology induced by the natural metric (10.3) . c) Prove that if X is a finite-dimensional vector space and llxll and llxll ' are two norms on X, then one can find positive numbers M , N such that
M l l xll
<S;
llxll'
<S;
N ll xll
( 10.19)
for any vector x E X. d) Using the example of the norms llxll 1 and llxll = in the space £, verify that the preceding inequality generally does not hold in infinite-dimensional spaces. a) Prove inequality (10.5). b) Verify relation (10.6) . c) Show that as p -+ +oo the quantity II J II P defined by formula (10.12) tends to the quantity 11!11 given by formula (10. 1 1 ) . 2.
a ) Verify that the normed space £p considered i n Example 7 is complete. b) Show that the subspace of lp consisting of finite sequences (ending in zeros) is not a Banach space.
3.
4. a) Verify that relations (10. 1 1 ) and (10.12) define a norm in the space C[a, b] and convince yourself that a complete normed space is obtained in one of these cases but not in the other. b) Does formula (10.12) define a norm in the space R[a, b] of Riemann-integrable functions? c) What factorization (identification) must one make in R[a, b] so that the quantity defined by (10.12) will be a norm in the resulting vector space? 5. a) Verify that formulas (10. 13)-(10.15) do indeed define an inner product in the corresponding vector spaces. b) Is the form defined by formula (10.15) an inner product in the space R[a, b] of Riemann-integrable functions? c) Which functions in R[a, b] must be identified so that the answer to part b) will be positive in the quotient space of equivalence classes?
Using the Cauchy-Bunyakovskii inequality, find the greatest lower bound of the values of the product (1/ J) (x) dx ) on the set of continuous J(x) dx real-valued functions that do not vanish on the closed interval [a, b] . 6.
(!
) (!
10.2 Linear and Multilinear Transformations
49
10.2 Linear and Multilinear Transformations 10. 2 . 1 Definitions and Examples
We begin by recalling the basic definition.
Definition 1 . If X and Y are vector spaces over the same field ( in our case, either lR or C) , a mapping A : X ---+ Y is linear if the equalities
A(x 1 + x 2 ) = A( x i ) + A(x 2 ) , A(.-\x) = .-\A(x) hold for any vectors x, x 1 , x 2 in X and any number .-\ in the field of scalars. For a linear transformation A : X ---+ Y we often write Ax instead of A(x). Definition 2 . A mapping A Xn ---+ y of the direct product of the vector spaces X 1 , . . . , Xn into the vector space Y is multilinear ( n-linear) if the mapping y = A(x 1 , . . . , x n ) is linear with respect to each variable for :
xl X . . . X
all fixed values of the other variables.
: xl X . . . X
The set of n-linear mappings A Xn ---+ y will be denoted £(X1 , . . . , Xn ; Y). In particular for n 1 we obtain the set .C(X; Y ) of linear mappings from X1 = X into Y. For n = 2 a multilinear mapping is called bilinear, for n = 3, trilinear, and so forth. One should not confuse an n-linear mapping A E .C ( X1 , . . . , Xn ; Y ) with a linear mapping A E .C(X; Y) of the vector space X = X 1 x x Xn ( in this connection see Examples 9- 11 below ) . If Y = lR or Y = C, linear and multilinear mappings are usually called linear or multilinear functionals. When Y is an arbitrary vector space, a linear mapping A : X ---+ Y is usually called a linear transformation from X into Y, and a linear operator in the special case when X = Y. =
·
·
·
Let us consider some examples of linear mappings.
Example 1 .
Let £ be the vector space of finite numerical sequences. We define 0 a transformation A : £ ---+ £ as follows:
0 0
Example 2. We define the functional A : C[a, b] ---+ lR by the relation A(!) := f(xo) , where f E C ( [a, b], lR ) and x0 is a fixed point of the closed interval [a, b].
10 *Differential Calculus from a General Viewpoint
50
Example 3.
A : C([a, b], �) --+ � by the relation
We define the functional
J f(x) dx . b
A(f)
:=
a
Example 4 . We define the transformation the formula X A (f )
where
:=
A C([a, b], �) :
--+
C([a, b], �) by
J f(t) dt , a
x is a point ranging over the closed interval [a, b].
All of these transformations are obviously linear. Let us now consider some familiar examples of multilinear mappings.
Example 5. The usual product (x , . . . , n ) f-t · . . . n of n real numbers is a typical example of an n-linear functional A .C( �, . . . , �; �) .
1 X
X1 E
The inner product (x 1 , x 2 ) � space over the field � is a bilinear function.
Example 6.
A
·
X
----
n
(x 1 , x 2 )
in a Euclidean vector
Example 7. The cross product (x 1 , x 2 ) � [x 1 , x 2 ] of vectors in three dimensional Euclidean space E 3 is a bilinear transformation, that is, A
E
.C(E3 ' E3 ; E3 ). Example 8. If X is a finite-dimensional vector space over the field � {e 1 , . . . , e n } is a basis in X, and x x i e i is the coordinate representation of' the vector x E X, then, setting xl xn =
A(x., . . . , X n ) we obtain an n-linear function
�
det
( � �) .
.........
Xn
...
A x n --+ �:
Xn
As a useful supplement to the examples just given, we investigate in ad dition the structure of the linear mappings of a product of vector spaces into a product of vector spaces.
Example 9. Let = xl X . . . X be the vector space that is the direct product of the spaces X 1 , and let A : --+ Y be a linear mapping of into a vector space Y . Representing every vector x = (x1 , in the form
X
X
X . . . , Xm , m
X
. . . , Xm ) E X
51 10.2 Linear and Multilinear Transformations ) ( x = (x 1 , . . . ,xm = 2 = x 1 , 0, . . . , 0) + (0, X , 0, . . . , 0) + · · · + (0, . . . , 0, Xm ) ( 10.20)
and setting
( 10.21)
Xi E xi' = {1 ' . . . ' m} ' we observe that the mappings Ai : xi ---+ y are
for i linear and that
( 10.22)
Since the mapping X . . . X is obviously linear for any linear mappings Y, we have shown that formula ( 10.22) gives the general form of any linear mapping A X . . • X Y).
AiA: X: Xi ---+= x1
Xm( ---+ y E C X = x1
Xm ;
Example 1 0. Starting from the definition o f the direct product Y Y1 x · · · x Yn of the vector spaces Y1 , . . . , Yn and the definition of a linear mapping A : X ---+ Y, one can easily see that any linear mapping A :(X ---+ Y = Y1 ) · · Yn ) has the form x Ax = A 1 x, . . . , An x = ( 1 , . . . , Yn E Y, where
Ai : X ---+
Yi
Example 1 1 .
mapping
X •
>--+
X y
y
are linear mappings.
Combining Examples
9 and 10,
we conclude that any linear X
X . . . X
X
A : x1 Xm = X ---+ y = y1 Yn of the direct product X = X 1 x · · · x Xm of vector spaces into another direct product Y = Y1 x · · · x Yn has the form (�1 ) ( A " A ,� ) ( �1 ) . = Ax ' = · ( 10.23) ...... . Yn An1 · · · Anm Xm where A ij : Xj ---+ are linear mappings. In particular, if X 1 = X2 = · · · = Xm = R and Y1 = Y2 = · · · = Yn = R, then A ij : X1 ---+ are the linear mappings R x ai j X E R, each of which is given by a single number aij · Thus in this case relation (10.23) becomes the familiar numerical notation for a linear mapping A : R m ---+ R n . .
•
.
y
Yi
3
Yi
>--+
10.2.2 The Norm of a Transformation
X . . . X be a multilinear transformation mapping the direct product of the normed vector spaces into a normed space Y.
Definition 3. Let
A : x1
Xn ---+ y
X 1 , . . . , Xn
52
10 *Differential Calculus from a General Viewpoint
The quantity
(10.24) where the supremum is taken over all sets X I , . . . , X n of nonzero vectors in the spaces X I , . . . , Xn , is called the n o rm of the multilinear transformation
A.
On the right-hand side of Eq. (10.24) we have denoted the norm of a vector by the symbol 1 · 1 subscripted by the symbol for the normed vector space to which the vector belongs, rather than the usual symbol · for the norm of a vector. From now on we shall adhere to this notation for the norm of a vector; and, where no confusion can arise, we shall omit the symbol for the vector space, taking for granted that the norm ( absolute value ) of a vector is always computed in the space to which it belongs. In this way we hope to introduce for the time being some distinction in the notation for the norm of a vector and the norm of a linear or multilinear transformation acting on a normed vector space. Using the properties of the norm of a vector and the properties of a multilinear transformation, one can rewrite formula (10.24) as follows:
x
I I
(10.25)
ei,
where the last supremum extends over all sets . . . , e n of unit vectors in the spaces XI , . . . , Xn respectively ( that is, = 1, i = 1, . . . , n ) . In particular, for a linear transformation : X � Y, from (10.24) and (10.25) we obtain = sup (10.26) = sup
I AI
x;60
I Ax l IXl
l eAi I
lei= I
I Ae l .
It follows from Definition 3 for the norm of a multilinear transformation then the inequality A that if I A I < oo ,
iE i
(10.27)
holds for any vectors X X , i = 1, . . . , n . In particular, for a linear transformation we obtain
I Ax l I A I I x l . :s
(10.28)
In addition, it follows from Definition 3 that if the norm of a multilinear transformation is finite, it is the greatest lower bound of all numbers M for which the inequality holds for all values of X
i E Xi , i = 1,
(10.29) . . . , n.
10.2 Linear and Multilinear Transformations
A : x1
53
Xn
X X --* y is if there exists M such that inequality (10.29) holds for all values of in the spaces respectively.
Definition 4. A multilinear transformation
bounded
x 1 , . . . , Xn
E JR. X
1 , . . . , Xn
0
0
0
Thus the bounded transformations are precisely those that have a finite norm. On the basis of relation (10.26) one can easily understand the geometric meaning of the norm of a linear transformation in the familiar case --* In this case the unit sphere in maps under the transformation into some ellipsoid in whose center is at the origin. Hence the norm of in this case is simply the largest of the semiaxes of the ellipsoid. On the other hand, one can also interpret the norm of a linear transforma tion as the least upper bound of the coefficients of dilation of vectors under the mapping, as can be seen from the first equality in (10.26). It is not difficult to prove that for mappings of finite-dimensional spaces the norm of a multilinear transformation is always finite, and hence in par ticular the norm of a linear transformation is always finite. This is no longer true in the case of infinite-dimensional spaces, as can be seen from the first of the following examples. Let us compute the norms of the transformations considered in Exam ples 1 -8.
JRn .
JRn
JRm
A : JRAm A
Example 1'. If we regard f0 as a subspace of the normed space fp , in which the vector e n = (0, . . . , 0, 1, 0, . . . ) has unit norm, then, since Ae n = n e n , it n-1 is clear that I A I = oo. Example 2' . If I f I = amax x ::;b l f (x ) I ::; 1, then l A/ I = l f (xo ) I ::; 1, and l A/ I = 1 ::; if f ( xo ) = 1, so that I A I = 1. '-v--'
We remark that if we introduce, for example, the integral norm
b 1!1 = j l f l (x) dx a on the same vector space C ( [a , b],JR), the result of computing I A l may change considerably. Indeed, set [a , b] = [ 0 , 1] and x0 = 1. The integral norm of the function fn = x n on [0, 1] is obviously n� 1 , while A fn = Ax n = x n l x =1 = 1. It follows that I A I = oo i n this case. Throughout what follows, unless the contrary is explicitly stated, the space C([a , b], JR.) is assumed to have the norm defined by the maximum of the absolute value of the function on the closed interval [a , b].
10 *Differential Calculus from a General Viewpoint
54
Example 3'. If l f l I AJ I
=
max
a :'O x :'O b
l f(x) l :S 1, then
I J f(x) dx l :S J IJI (x) d J 1 d b
=
b
b
x :S
a
a
x=b-a
.
a
f(x) 1, we obtain I A1 I = b - a, and therefore II A II b - a . Example 4 ' . If l f l amax l f(x) l :S 1, then :'O x :'O b =
But for
max
a :'O x :'O b
But for
X
=
I J f(t) dt l :S a
max
1
x
a :'O x :'O b a
l f(t) 1, we obtain =
max
a :'O x :'O b
and therefore in this example
=
( x - a) b - a . IJI (t) dt :S amax :'O x :'O b
X
J 1 dt a
=
=
b-a,
II A ll = b - a .
Example 5 ' . We obtain immediately from Definition 3 that II A II = 1 in this
case.
Example 6 1 . By the Cauchy-Bunyakovskii inequality and if XI
=
xz , this inequality becomes equality. Hence
Example 7'. We know that
II A II = 1.
where rp is the angle between the vectors XI and xz , and therefore II A II :S 1. At the same time, i f the vectors XI and x 2 are orthogonal, then sin rp = 1 . Thus II A II = 1.
Example 8 1 • If we assume that the vectors lie in a Euclidean space of dimen sion n , we note that A x i , . . . , Xn = det x i , . . . , X n is the volume of the parallelepiped spanned by the vectors XI , . . . , Xn , and this volume is maxi mal if the vectors XI , . . . , Xn are made pairwise orthogonal while keeping their lengths constant. Thus, det x i , . . . , xn :S l x i · . . . · x n , equality holding for orthogonal vectors. Hence in this case II A II = 1.
(
l (
)
(
)l
)
i
l l
10.2 Linear and Multilinear Transformations
55
Let us now estimate the norms of the operators studied in Examples 9� 11. We shall assume that in the direct product X = X1 x · · · x Xm of the normed spaces xl ' . . . ' Xm the norm of the vector X = (x l ' . . . ' X m ) is introduced in accordance with the convention in Sect . 10.1 ( Example 6).
Example 9' . Defining a linear transformation Xm = X A : xl X . . . X
--+
y '
as has been shown, is equivalent to defining the m linear transformations Ai : Xi --+ Y given by the relations Ai xi = A((O, . . . , O, x i , O , . . . , O)) , i = 1, . . . , m . When this is done, formula (10.22) holds, by virtue of which
Thus we have shown that
m
II A II ::; 2:::: II Ai ll . i= l On the other hand, since
I Ai xil = I A ( (o, . . . , O, xi , o, . . . , o) ) l ::; ::; II A l i i (0, . . . , 0, Xi , 0, . . . , 0) l x = II A II I xi l x, , we can conclude that the estimate
II Ai ll ::; II A II also holds for all
i = 1,
. . . , m.
Example 10' . Taking account of the norm introduce in Y this case we immediately obtain the two-sided estimates
= Y1 x
· · ·
x Yn , in
n
II Ai ll ::; II A II ::; 2:::: II Ai ll . i= l Example 11 1 • Taking account of the results of Examples 9 and 10, one can
conclude that
m n
II Aij ll ::; II A II ::; 2:::: 2:::: II AiJ II . i= l j = l
10 *Differential Calculus from a General Viewpoint
56
1 0 . 2 . 3 The Space of Continuous Transformations
From now on we shall not be interested in all linear or multilinear transfor mations, only continuous ones. In this connection it is useful to keep in mind the following proposition. X . . . X
--t
For a multilinear tmnsformation A : x1 Xn y mapping a product of normed spaces X1 , . . . , Xn into a normed space Y the following conditions are equivalent: a ) A has a finite norm, b) A is a bounded tmnsformation, ) A is a continuous tmnsformation, d) A is continuous at the point (0, . . . , 0) E X1 x · · · x Xn . Proof. We prove a closed chain of implications a ) b) ) d) a ) . It is obvious from relation (10.27) that a ) b). Let us verify that b) ) that is, that (10.29) implies that the operator A is continuous. Indeed, taking account of the multilinearity of A, we can Proposition 1.
c
::::}
::::} c ,
write that
::::}
::::} c
::::}
::::}
A(x 1 + h 1 , x2 + h2 , . . . , Xn + hn ) - A(x 1 , x2 , . . . , xn ) = = A( h l , X 2 , . . . , Xn ) + · · · + A(x l , X 2 , . . . , Xn - 1 , hn ) = + A(h 1 , h 2 , X 3 , . . . , X n ) + · · · + A(x 1 , . . . , X n - 2 , h n - 1 , h n ) + (10.29) we now obtain the estimate I A(x l + h 1 , X 2 + h2 , . . . , Xn + hn ) - A(x 1 , X 2 , . . . , Xn ) :S :S M (lh 1 l · l x 2 l · · · · · l x n l + · · · + l x 1 l · l x 2 l · · · · · l x n - l l · lh n l +
From
A is continuous at each point (x 1 , . . . , Xn ) E In particular, if (x 1 , . . . , X n ) = (0, . . . , 0) we obtain d ) from c ) . It remains to be shown that d) a ) . Given e: 0 we find 8 8(e:) 0 such that I A(x 1 , . . . , x n ) l e: when max{ l x 1 l , . . . , l xn l } 8. Then for any set e 1 , . . . , e n of unit vectors we obtain I A(e 1 , . . . , e n ) I 81n I A(8e 1 , . . . , 8en ) l 8n , that is, II A II 8"n D from which it follows that
x1
X . . . X
Xn . >
<
::::} >
=
=
<
< 00 .
<
<
€
10.2 Linear and Multilinear Transformations
57
We have seen above ( Example 1) that not every linear transformation has a finite norm, that is, a linear transformation is not always continuous. We have also pointed out that continuity can fail for a linear transformation only when the transformation is defined on an infinite-dimensional space. From here on .C(X � , . . . , Xn ; Y) will denote the set of c o n t i n u o u s mul tilinear transformations mapping the direct product of the normed vector spaces X1 , . . . , Xn into the normed vector space Y. In particular, .C(X; Y) is the set of continuous linear transformations from X into Y. In the set .C(X 1 , . . . , Xn ; Y) we introduce a natural vector-space structure: and
(.AA)(x 1 , . . . , Xn ) := .AA(x 1 , . . . , X n ) . It is obvious that if A , B E .C(X 1 , . . . , Xn ; Y), then (A + B ) E .C(X1 , . . . , Xn ; Y) and (.AA) E .C(Xb · · · , Xn ; Y). Thus .C(X1 , . . . , Xn ; Y) can be regarded a vector space. Proposition 2 . The norm of a multilinear tmnsformation is a norm in the vector space .C(X1 , . . . , Xn ; Y) of continuous multilinear tmnsformations. Proof. We observe first of all that by Proposition 1 the nonnegative number I A ll oo is defined for every transformation A E .C(X1 , . . . , Xn ; Y). Inequality (10.27) shows that II A II = 0 {::} A = 0 . as
<
Next, by definition of the norm of a multilinear transformation
Finally, if A and B are elements of the space
.C(X1 , . . . , Xn ; Y), then
58
10 *Differential Calculus from a General Viewpoint
From now on when we use the symbol .C ( X 1 , . . . , Xn ; Y ) we shall have in mind the vector space of continuous n-linear transformations normed by this transformation norm. In particular .C ( X, Y ) is the normed space of continuous linear transformations from X into Y . We now prove the following useful supplement t o Proposition 2. E .C ( X; Y ) and Supplement . If X, Y, and Z are normed spaces and B E .C ( Y; Z), then � 0
li B l I B I I I . A A B .A B Proof.l i BIndeed, l I ( l l ) l I Il I l l A = sup oxA x � sup xl iAxB II I l l i B II I I Il = sup AXx = · A . o
x;t'O
x ;i O
x;t'O
o
If Y is a complete normed space, then .C ( X1 , . . . , Xn ; Y ) is also a complete normed space. Proof. We shall carry out the proof for the space .C ( X; Y ) of continuous linear
Proposition 3 .
transformations. The general case, as will be clear from the reasoning below, differs only in requiring a more cumbersome notation. Let be a Cauchy sequence in .C ( X; Y ) . Since for any E X we have
x
A 1 , A2 , . . . , An . . .
A 1 x, A2 x, . . . , An x, . . .
it is clear that for any E X the sequence is a Cauchy sequence in Y . Since Y is complete, it has a limit in Y , which we denote by
Ax.Thus, We shall show that The linearity of
x :
lim A n x . Ax : = n--too
A follows X Y is a continuous linear transformation. A from the relations ) ) ( ( + A 2 x 2 = lim >. 1 A n x 1 + A 2 A n x 2 lim A >' xl l n n---t oo n--+oo A lim A xl + A2 lim A X2 . n n--+oo n I Next , forI any fixed E > 0 and sufficiently1 n--+oo l large l l values of E N we have Am - An < E, and therefoIre Amx - An x � c x at each vector x E X . Letting tend to infinity in this last relation and l of a vector, using the continuity of the norm l l wel obtain Ax - An x � c x . ---+
=
m, n
m
I
I
59
10.2 Linear and Multilinear Transformations
( ) Thus A - A n c: , and sinceI AI = AI n +l A - A n , we conclude that A An + c: . I l l Consequently, we have shown that A E C(X; Y) and A - A n ( 0 as n that is, A = lim A n in the sense of the norm of the space C X; Y) . n--+oo :s:;
:s:;
--7
--7 oo ,
D
In conclusion, we make one special remark relating to the space of mul tilinear transformations, which we shall need when studying higher-order differentials.
E { 1, . . . , n } there is a bijection between the C (Xt, . . . , Xm ; C ( Xm + l , . . . , Xn ; Y)) and C (Xt, . . . , Xn ; Y)
Proposition 4.
spaces
For each
m
that preserves the vector-space structure and the norm. Proof. We shall( exhibit this isomorphism. (C XLetm+l!B, . .E. , CXnX; t,Y).. . . ,Xm ; C ( Xm+l , . . . ,Xn ; Y )) , that is, !B (x 1 , . . . ,xm ) We set
E
(10.30)
Then
I I !B =
I (l I l!B x l , · · · ,l Xm )l sup xt · · · · · xm ( I ) ) ( l l j !B xl, .l . . , Xm xm+bl . · . , Xn sup l l xm+l l · · ·l · · xn sup xt · · · · · xm ( lj A xt,l . . . ,xl n )l i I I = sup xt · · · · · xn = A .
X 1 , . . . 1Xm. x, ,t: o
X rn + l , . . . , Xn Xj ,t'O
x t , . . . ,xn X k #O
We leave to the reader the verification that relation isomorphism of these vector spaces. D
(10.30) defines an
the space ) ( X2; . we. . ; Cfind( Xnthat (C Xt;4 nCtimes, Y)) · · · ; ( is isomorphic to the space C X 1 , . . . , Xn ; Y) of n-linear transformations. Applying Proposition
10 *Differential Calculus from a General Viewpoint
60
10.2.4 Problems and Exercises 1 . a) Prove that if A : X --+ Y is a linear transformation from the normed space X into the normed space Y and X is finite-dimensional, then A is a continuous operator. b) Prove the proposition analogous to that stated in a) for a multilinear oper ator. 2. Two normed vector spaces are isomorphic if there exists an isomorphism between them (as vector spaces) that is continuous together with its inverse transformation. a) Show that normed vector spaces of the same finite dimension are isomorphic. b) Show that for the infinite-dimensional case assertion a) is generally no longer true. c) Introduce two norms in the space c ( [a, b] , lR ) in such a way that the identity mapping of C ( [a, b] , lR ) is not a continuous mapping of the two resulting normed spaces. 3. Show that if a multilinear transformation of n-dimensional Euclidean space is continuous at some point, then it is continuous everywhere. 4. Let A : En --+ En be a linear transformation of n-dimensional Euclidean space and A : En --+ En the adjoint to this transformation. Show the following. a) All the eigenvalues of the operator A · A* : En --+ En are nonnegative. b) If >. 1 :S · · · :S A n are the eigenvalues of the operator A · A* , then I All = A. c) If the operator A has an inverse A- 1 : En --+ En , then II A- 1 11 = );-,- . d) If (a; ) is the matrix of the operator A : En --+ En in some basis, then the estimates •
n
L(a; )2 j=1
:S
II A II
:S
n
L (a; )2
i ti = l
:S
v'nii A II
hold. 5 . Let IP'[x] be the vector space of polynomials in the variable x with real coefficients. We define the norm of the vector P E IP'[x] by the formula
IPI =
1
J P2 (x) dx. 0
a) Is the operator D : IP'[x] -+ IP'[x] given by differentiation (D ( P(x) ) : = P'(x)) continuous in the resulting space? b) Find the norm of the operator F : IP'[x] --+ IP'[x] of multiplication by x, which acts according to the rule F ( P(x) ) = x · P(x) .
6.
A ll
Using the example of projection operators in IR 2 , show that the inequality liB o :S II B II · II A II may be a strict inequality.
10.3 The Differential of a Mapping
61
10.3 The Differential of a Mapping 10.3 . 1 Mappings Differentiable at a Point Definition 1 . Let X and Y be normed spaces. A mapping f : E -+ Y of a set E C X into Y is differentiable at an interior point x E E if there exists a continuous linear transformation L(x) : X -+ Y such that
(10.31) f(x + h) - f(x) L(x)h + a(x; h) , where a(x; h ) = o(h) as h -+ 0, x + h E E. 1 Definition 2 . The function L( x) E .C( X ; Y) that is linear with respect to h and satisfies relation (10.31) is called the differential, the tangent mapping, or the derivative of the mapping f : E -+ Y at the point x. As before, we shall denote L(x) by df(x), Df(x), or f'(x). =
We thus see that the general definition of differentiability of a mapping at a point is a nearly verbatim repetition of the one already familiar to us from Sect . 8.2, where it was considered in the case X = � m , Y �n . For that reason, from now on we shall allow ourselves to use such concepts introduced there as increment of a function, increment of the argument, and tangent space at a point without repeating the explanations, preserving the corresponding notation. We shall, however, verify the following proposition in general form. =
Proposition 1 . If a mapping f : E -+ Y is differentiable at an interior point x of a set E X, its differential L(x) at that point is uniquely determined. Proof. Thus we are verifying the uniqueness of the differential. Let L 1 (x) and L 2 (x) be linear mappings satisfying relation ( 10.31 ) , that is a 1 (x; h) , f(x + h) - f(x) - L 1 (x)h (10.32) f(x + h) - f(x) - L 2 (x)h a 2 (x; h) , where a i (x; h) o(h) as h -+ 0, x + h E E, i 1 , 2. Then, setting L(x) = L 2 (x) - L 1 (x) and a(x; h) a 2 (x; h) - a 1 (x; h ) and C
=
=
=
subtracting the second equality in ( 10.32) from the first , we obtain
L(x)h a(x; h) . =
Here L(x) is a mapping that is linear with respect to h, and a(x; h) = o(h) as h -+ 0, x + h E E. Taking an auxiliary numerical parameter .>., we can now
1 The notation "a ( x; h ) = o ( h ) as h -+ 0, x + h E E" , of course, means that lim
h-+0, x+ h E E
l a ( x; h ) I Y · l h i :X 1
=
0.
62
10 *Differential Calculus from a General Viewpoint
write
l i a(x; Ah) l lhl 0 as A 0 I L(x)h l I L(x)(Ah) IAI I Ahl Thus L(x)h 0 for any h =1- 0 (we recall that x is an interior point of E). Since L(x)O 0, we have shown that L 1 (x)h L 2 (x)h for every value o f h . =
---+
=
=
=
---+
•
=
D
If E is an open subset of X and f : E ---+ Y is a mapping that is dif ferentiable at each point x E E, that is, differentiable on E , by the unique ness of the differential of a mapping at a point , which was just proved, a function E 3 x H f'(x) E .C(X; Y) arises on the set E, which we denote f' : E ---+ .C(X; Y). This mapping is called the derivative of f, or the deriva tive mapping relative to the original mapping f : E ---+ Y. The value f'(x) of this function at an individual point x E E is the continuous linear transfor mation f'(x) E .C(X; Y) that is the differential or derivative of the function f at the particular point x E E. We note that by the requirement of c o n t i n u i t y of the linear mapping L(x) Eq. (10.31) implies that a mapping that is differentiable at a point is necessarily continuous at that point . The converse is of course not true, as we have seen in the case of numerical functions. We now make one more important remark.
Remark. If the condition for differentiability of the mapping f at some point a is written as f(x) - f(a) L(x)(x - a) + a( a; x) , where a( a; x) ( x - a) as x ---+ a, it becomes clear that Definition 1 actually applies to a mapping f : A ---+ B of any affine spaces (A, X) and (B, Y) whose vector spaces X and Y are normed. Such affine spaces, called normed affine spaces, are frequently encountered, so that it is useful to keep this remark in =
=
o
mind when using the differential calculus.
Everything that follows, unless specifically stated otherwise, applies equally to both normed vector spaces and normed affine spaces, and we use the notation for vector spaces only for the sake of simplicity. 1 0 . 3 . 2 The General Rules for Differentiation
The following general properties of the operation of differentiation follow from Definition 1 . In the statements below X , Y, and Z are normed spaces and U and V open sets in X and Y respectively.
63
10.3 The Differential of a Mapping
If the mappings fi : U -+ Y, i = 1 , 2, are differentiable at a point x E U, a linear combination of them (A.t/1 + >..2 /2 ) : U -+ Y is also differentiable at x, and
a. Linearity of Differentiation
Thus the differential of a linear combination of mappings is the corre sponding linear combination of their differentials. b. Differentiation of a Composition of Mappings ( Chain Rule )
If the mapping f : U -+ V is differentiable at a point x E U X, and the mapping g : V -+ Z is differentiable at f ( x) = y E V Y, then the composition o f of these mappings is differentiable at x, and (g o J)'(x) = g' ( f(x) ) o f'(x) . C
C
g
Thus, the differential of a composition is the composition of the differen tials.
Let f : U -+ Y be a mapping that is continuous at x E U X and has an inverse f - 1 : V -+ X that is defined in a neighborhood of y = f ( x) and continuous at that point. If the mapping f is differentiable at x and 1 its tangent mapping f' ( x) E .C ( X; Y ) has a continuous inverse [f'(x) r E .C ( Y; X ) , then the mapping f - 1 is differentiable at y = f ( x) and u - 1 ] ' (f(x)) = [f'(x) r 1 . c. Differentiation of the Inverse of a Mapping C
Thus, the differential of an inverse mapping is the linear mapping inverse to the differential of the original mapping at the corresponding point. We omit the proofs of a, b, and c, since they are analogous to the proofs given in Sect. 8.3 for the case X = Y=
rn:.m ,
rn:.n .
10.3.3 Some Examples
Example 1 . If f : U -+ Y is a constant mapping of a neighborhood U = U(x) X of the point x, that is, f(U) = y0 E Y , then f'(x) = 0 E .C ( X ; Y ) . Proof. Indeed, in this case it is obvious that f(x + h) - f(x) - Oh = Yo - Yo - 0 = 0 = o (h) . D Example 2. If the mapping f : X -+ Y is a continuous linear mapping of a normed vector space X into a normed vector space Y , then f' ( x) = f E .C ( X; Y ) at any point x E A. c
64
10 *Differential Calculus from a General Viewpoint
Proof. Indeed, f(x + h) - f(x) - fh = fx + fh - fx - fh = 0
.D We remark that strictly speaking f' (x) E .C(TXx ; TYf ( x ) ) here and h is a vector of the tangent space TXx . But parallel translation of a vector to any point x E X is defined in a vector space, and this allows us to identify the tangent space T Xx with the vector space X itself. (Similarly, in the case of an affine space (A, X) the space T Aa of vectors "attached" to the point a E A can be identified with the vector space X of the given affine space. ) Consequently, after choosing a basis i n X, we can extend i t t o all the tangent spaces T Xx . This means that if, for example, X = IR.m , Y = IR.n , and the mapping f E .C(IR. m ; IR. n ) is given by the matrix ( ai ) , then at every point x E IR. m the tangent mapping f ' (x) TIR.:- ---+ TIR.�f< x ) will be given by the same matrix. In particular, for a linear mapping x J-t ax = y from JR. to JR. with x E JR. and h E TIR.x rv JR. , we obtain the corresponding mapping TIR.x h � a h E TJR. f ( x ) · :
3
Taking account of these conventions, we can provisionally state the result of Example 2 as follows: The mapping f ' : X ---+ Y that is the derivative of a linear mapping f : X ---+ Y of normed spaces is constant, and f' (x) = f at each point x E X.
Example 3. From the chain rule for differentiating a composition of mappings and the result of Example 2 one can conclude that if f : U ---+ Y is a mapping of a neighborhood U = U(x) c X of the point x E X and is differentiable at x , while E .C(Y; Z) , then
A
(A o f) ' (x) = A o J' (x) .
For numerical functions, when = Z = JR., this is simply the familiar possibility of moving a constant factor outside the differentiation sign.
Y
Example 4. Suppose once again that U = U(x) is a neighborhood of the point x in a normed space X, and let J : U ---+
Y = Y1
X · · · X
Yn
be a mapping of U into the direct product of the normed spaces Y1 . . . . , Y . Defining such a mapping is equivalent to defining the n mappings U ---+ Yi , i = 1 , . . . , n , connected with f by the relation X f-t f (X ) =
Y = ( Yl , · . . , Yn ) = ( fi (X ) , . . . , fn ( X)) ,
which holds at every point of U.
/in
10.3 The Differential of a Mapping
65
If we now take account of the fact that in formula ( 10.31) we have
(f x + h) - f (x ) = (h (x (+ h) - h (x ) , . . . , fn )(X + h) - fn (x ) ) , L( (x ) h) = ( L 1((x ) h), . . . , Ln ((x ) h ) ), o: x ; h = o: l x ; h , . . . ,o:n x ; h ,
then, referring to the results of Example 6 of Sect. 10. 1 and Example 10 of Sect. 10.2, we can conclude that the mapping f is differentiable at if and only if all of its components U Yi are differentiable at i 1 , . . . , n; and when the mapping is differentiable, we have the equality
x : -+ = x, fi f ( ) ( ) ( )) ! ' X = (!{ X , . . . , �� X . ( Example 5. Now let A E £ X1 , . . . , Xn ; Y), that is, A is a continuous n-linear transformation from the product X 1 x · · · x Xn of the normed vector spaces X1 , . . . , Xn into the normed vector space Y. We shall prove that the mapping
A : xl X . . . X Xn = X -+
y
is differentiable and find its differential.
Proo(f. Using) the( multilinearity of A, we find that ) ( A x + h( - A x =) A xl( + h 12, . . . ,Xn )+ hn ) - A ((x 1 , . . . ,xn ) = ) = A xl,( . . . , X2n + A h l , X) , . . . , Xn +( · · · + A xl,2 . . . , Xn - l ), hn + + A h l , h , X3 , . . . , Xn + · · · + A xl, . . . , Xn - , hn - l , hn + X =l Xl 1 x · l· · l x Xn satisfies the inequalities l l n , :::; L I l xi x, x x i=l xi x, and the norm A of � � l satisfies � ) l I I I 6A 1is finite and I (the6 transformation A , · · · , n :::; A X · · · X n ' we can conclude that A (x + h) - A (x ) =( A (xl2 +. h 1 , . .). ,Xn + hn )( - A (xl, . . . ,xn ) )= ( ) =(A) h l , X , . . , Xn + · · · + A xl, . . . , Xn - l , hn + o: x ; h , ) ( where o: x ; h = o h as h -+ 0. Since the norm in
:::;
66
10 *Differential Calculus from a General Viewpoint
But the transformation
is a continuous transformation (because A is continuous) that is linear in h = (h 1 , . . . , h n ) · Thus we have established that A ' (x)h = A ' (x 1 , . . . , X n ) (h 1 , . . . , h n ) = = A(h 1 , X 2 , . . . , X n ) or, more briefly,
+ · · · + A(x 1 , . . . , Xn - 1 , hn ) ,
In particular, if: a) x 1 · . . . · X n is the product of n numerical variables, then d(x 1 . . . · X n ) = dx 1 · X 2 · . . . · X n ·
b) (x 1 , x 2 ) is the inner product in
+ · · · + X 1 · . . . · Xn- 1 · dxn ;
E3 , then
c) [x 1 , x 2 ] is the vector cross product in
E3 , then
d) (x 1 , x 2 , x 3 ) is the scalar triple product in
E 3 , then
e) det (x 1 , . . . , xn ) is the determinant of the matrix formed from the co ordinates of n vectors x 1 , . . . , X n in an n-dimensional vector space X with a fixed basis, then
Example 6. Let U be the subset of .C(X; Y) consisting of the continuous linear transformations A : X ---+ Y having continuous inverse transformations A - 1 : Y ---+ X (belonging to .C(Y; X ) ) . Consider the mapping U 3 A H A - 1 E .C(Y; X) ,
which assigns to each transformation A E U its inverse A - 1 E .C(Y; X) . Proposition 2 proved below makes it possible to determine whether this mapping is differentiable.
10.3 The Differential of a Mapping
1 space and A E U, then for any h 67E lIf Xl is I a complete 1 - 1 , the transformation A + h also belongs to .C(X; Y) such that h AU and the following relation holds: Proposition 2.
<
( 10.33)
Proof. Since
( 1 0.34) ) ) ( ( it suffices to find the operator E+A- 1 h - 1 inverse to E +A - 1 h E .C(X; X ) , where E i s the identity mapping l l itself. I exI ::;oIfofXtheintol supplement 1 h . Taking account Let L1 A 1 ::; 2 of 1 · h , so that toby1 Proposition Sect. 10.2, we can observe that L1 the assumptions A made with respect to the operator h we may assume that .1 q 1. We now verify that ( E .1 ) - 1 E + L1 + ,12 + . . . + ,1n + . . . , ( 10.35) where (the series on side is formed from the linear operators ) Ethe.C(X;right-hand ,1n SinceL1 X. .i s aL1complete X). normed ::; i t follows from Proposition I l ::;vector I I space, 3 of Sect. 10.2 that the space .C(X; X) is also complete. It then follows im l l relation L1n L1 n qn and the convergence of the mediately from the series :z= qn for q 1 that the series ( 10.35) formed from the vectors in n=O that space converges. The direct verification that ( E + L1 + L12 + · ) ( E - L1 ) ( E + L1 + ,12 + . . · ) _ ( L1 + ,12 + ,13 + . . · ) E and ( E - L1 ) ( E + L1 + L12 + ) ( E + L1 + ,12 + . . ) ( L1 + ,12 + ,13 + . . ) E ) shows that we have indeed found ( E - L1 - 1 . :=
<
=
_
=
o
·
o
00
<
·· =
=
=
··· =
=
·
_
· =
It is worth remarking that the freedom in carrying out arithmetic op erations on series (rearranging the terms! ) in this case is guaranteed by the absolute convergence (convergence in norm) of the series under consideration. Comparing relations ( 0.34) and ( 10.35) , we conclude that
(A + h) - 1 A - 1 1- A - 1 hA - 1 + (A - 1 h) 2 A - 1 - . . . ) + ( - 1 ) n (A - 1 h n A - 1 + ( 1 0.36) =
··
·
···
10 *Differential Calculus from a General Viewpoint
68
Since
I ll i 1 ) I �( - A - 1 h n A - 1 1 :S � AI- 1 h n1 A1 -l 1 :S � l 3 l l 1 h2, 3 2 :S A - 1 h f q m = :� q m=O
Eq. ( 10.33) follows in particular from ( 10.36) .
D
Returning now to Example 6, we can say that when the space X is com plete the mapping � under consideration is necessarily differen tiable, and
A
A- 1
this means that if A is a nonsingular square matrix and - 1Inisparticular, its inverse, then under a perturbation of the matrix A by a( matrix) h Awhose elements are close to zero, we can write the inverse matrix A + h - 1 in first approximation in the following form:
More precise formulas can obviously be obtained starting from Eq. (10.36) .
Example 7.
ping
Let
X be a complete normed vector space. The important mapexp : .C(X; X) ---* .C(X; X)
is defined as follows:
1 n 1 1 2 ·· · -A expA : = E + -A A +··· + + + 1! 2! n!
( 10.37)
(X; X). 1 -\ATheE l.C series in ( 10.37) converges, since .C(X; X) is a complete space and An :S II A'rn. , while the numerical series n=Of II Aifn. converges. It is not difficult to verify that exp ( A + h ) = expA + L (A) h + o ( h) as h ---* (10.38) where L (A ) h = h + 12.1 ( Ah + hA ) + 13.1 (A2 h + AhA + hA2 ) + . . . )+··· 2 2 1 ( n- 1 n n n 1 · · h h · hA hA · · · A A A + A + + + + I i I :::; I () n! exp A = eii A II , that is, L (A) E .C (.C(X; X), .C(X; X) ) . and L A if
�
oo ,
10.3 The Differential of a Mapping
69
Thus, the mapping .C(X; X) 3 A H exp A E .C (X; X) is differentiable at every value of A. We remark that if the operators A and h commute, that is, Ah = hA, then, as one can see from the expression for L(A)h, in this case we have L(A)h = (exp A)h. In particular, for X = � or X = C, instead of ( 10.38) we again obtain exp(A + h) = exp A + (exp A)h + o(h) as h -+ 0 .
( 10.39)
Example 8 . We shall attempt to give a mathematical description of the in stantaneous angular velocity of a rigid body with a fixed point o (a top) . Consider an orthonormal frame at the point o rigidly attached to the body. It is clear that the position of the body is completely characterized by the position of this orthoframe, and the triple of instantaneous velocities of the vectors of the frame obviously give a complete characteriza tion of the instantaneous angular velocity of the body. The position of the frame itself at time t can be given by an orthogonal matrix ( a1, ) i, j = 1 , 2 , 3 composed of the coordinates of the vectors with re spect to some fixed orthonormal frame in space. Thus, the motion of the top corresponds to a mapping t H O(t) from � (the time axis) into the group 80(3) of special orthogonal 3 x 3 matrices. Consequently, the angular veloc ity of the body, which we have agreed to describe by the triple is the matrix O(t) = : (w{)(t) = (a{)(t), which is the derivative of the matrix O(t) = (a{)(t) with respect to time. Since O(t) is an orthogonal matrix, the relation
{ e1, e2, e3}
{ e l ' e2' e3}
{ e1, e2, e3}
e1 , e2, e3
{ eb e2, e3},
( 10.40) O(t)O*(t) = E holds at any time t, where O*(t) is the transpose of O(t) and E is the identity matrix. We remark that the product A · B of matrices is a bilinear function of A and B, and the derivative of the transposed matrix is obviously the transpose
of the derivative of the original matrix. Differentiating ( 1 0.40) and taking account of these things, we find that
6(t)O * (t) + O(t)O*(t) = o or
O(t) = - O(t)O*(t)O(t) ,
( 10.41)
since O*(t)O(t) = E. In particular, if we assume that the frame coincides with the spatial frame of reference at time t, then O(t) = E, and it follows from ( 10.41) that ( 1 0.42) 6(t) = -6 * (t) ,
{ eb e2 , e3}
10 *Differential Calculus from a General Viewpoint
70
(wi)
that is, the matrix O(t) = : D(t) = of coordinates of the vectors } in the basis { el ' e2 ' e3 } turns out to be skew-symmetric: {
e l ' e2 ' e3
D(t) =
( wwl
i �
w3
Thus the instantaneous angular velocity of a top is actually characterized by three independent parameters, as follows in our line of reasoning from rela tion ( 10.40) and is natural from the physical point of view, since the position of the frame { e3 } , and hence the position of the body itself, can be described by three independent parameters ( in mechanics these parameters may be, for example, the Euler angles ) . + in the tangent If we associate with each vector w = + space at the point o a right-handed rotation of space with angular velocity l w l about the axis defined by this vector, it is not difficult to conclude from these results that at each instant of time t the body has an instantaneous angular velocity and that the velocity at that time can be adequately described by the instantaneous angular velocity vector w(t) ( see Problem 5 below ) .
e1 , e2 ,
w 1 e1 w2 e2 w3 e3
10.3.4 The Partial Derivatives of a Mapping
Let u = U(a) be a neighborhood of the point a E = xl X . . . X the direct product of the normed spaces and let f : --+ a mapping of into the normed space V. In this case
U
Y =
U XYm bein
X X1 , . . . , Xm , f( x ) = f( xi , . . . , x m ) ,
and hence, if we fix all the variables but k E { 1 , . . . , m} \ i, we obtain a function
( 10.43)
Xi in (10.43) by setting Xk
defined in some neighborhood
=
ak for
( 10.44)
Ui of ai in X. to the original mapping (10.43) the mapping 'P i : Ui Y is calledRelative the partial mapping with respect to the variable Xi at a E X. If the mapping ( 10.44) is differentiable at Xi ai , its deriva tive at that point is called the partial derivative or partial differential off at a with respect to the variable xi. Definition 3. --+
=
Definition 4.
We usually denote this partial derivative by one of the symbols
8d(a) , Dd(a) , 88xif (a) ' f�; (a) .
10.3 The Differential of a Mapping
71
In accordance with these definitions Dd(a) E .C(Xi ; Y). More precisely, Dd(a) E C(TXi (ai ); TY(f(a) )) . The differential df(a) of the mapping (10.43) at the point a ( if f is dif
ferentiable at that point ) is often called the in this situation in order to distinguish it from the partial differentials with respect to the individual variables. We have already encountered all these concepts in the case of real-valued functions of m real variables, so that we shall not give a detailed discussion of them. We remark only that by repeating our earlier reasoning, taking account of Example 9 in Sect. 9.2, one can prove easily that the following proposition holds in general.
total differential
If the mapping (10.43) is differentiable at the point a = (a l , . . . ' am ) E xl X . . . X Xm = X, it has partial derivatives with respect to
Proposition 3 .
each arevariable and the total differential and the partial differen tials relatedat bythatthepoint, equation ( 10.45) df(a)h 81 /(a)h l + · · · + 8m f( a )hm , where h (h 1 , . . . , hm ) E TX1 (a 1 ) X · · · X TXm (am ) = TX(a) . =
=
We have already shown by the example of numerical functions that the existence of partial derivatives does not in general guarantee the differentia bility of the function ( 10.43) . 10.3.5 Problems and Exercises 1. a) Let A E .C(X; X) be a nilpotent opemtor, that is, there exists k E N such that A k = 0. Show that the operator (E - A) has an inverse in this case and that (E - A) -1 = E + A + · · · + A k - 1 . b) Let D : lP'[x] -+ lP'[x] be the operator of differentiation on the vector space lP'[x] of polynomials. Remarking that D is a nilpotent operator, write the operator exp(aD) , where a E JR., and show that exp(aD) ( P(x) ) = P(x + a) =: Ta ( P(x) ) . c) Write the matrices of the operators D : 1P'n [x] -+ 1P'n [x] and Ta : 1P'n [x] -+ 1P'n [x] from part b) in the basis e; = (�':__�; , , 1 :S i :S n, in the space 1P'n [x] of real polynomials of degree n in one variable. 2. a) If A, B E .C(X; X) and ::JB - 1 E .C(X; X), then exp(B - 1 AB) = B - 1 (exp A)B. b) If AB = BA, then exp(A + B) = exp A · exp B. c) Verify that exp O = E and that exp A always has an inverse, namely (exp A) - 1 = exp( -A) . 3. Let A E .C(X; X). Consider the mapping I{) A : JR. -+ .C(X; X) defined by the correspondence JR. 3 t o-+ exp(tA) E .C(X; X). Show the following. a) The mapping i{) A is continuous. b) ipA is a homomorphism of JR. as an additive group into the multiplicative group of invertible operators in .C(X; X ) .
10 *Differential Calculus from a General Viewpoint
72
Verify the following. a) If A1 , . . . , A n are the eigenvalues of the operator A E .C(Cn; en) , then exp A l , . . , exp A n are the eigenvalues of exp A. b) det (exp A) = exp(tr A) , where tr A is the trace of the operator A E 4.
.
.C(Cn , cn ) .
c ) I f A E .C( JR n , JR n ) , then det(exp A) > 0 . d) I f A * i s the transpose o f the matrix A E .C(Cn , cn) and A i s the matrix whose elements are the complex conjugates of those of A, then (exp A)* = exp A* and exp A = exp A. e) The matrix �1 �1 is not of the form exp A for any 2 x 2 matrix A. 5. We recall that a set endowed with both a group structure and a topology is called a topological group or continuous group if the group operation is continuous. If there is a sense in which the group operation is even analytic, the topological group is called a Lie group. 2 A Lie algebra is a vector space X with an anticommutative bilinear operation [ , ] : X x X -+ X satisfying the Jacobi identity: [[a, b] , c] + [[b, c] , a] + [[c, a] , b] = 0 for any vectors a, b, c E X . Lie groups and algebras are closely connected with each other, and the mapping exp plays an important role in establishing this connection (see Problem 1 above). An example of a Lie algebra is the oriented Euclidean space E 3 with the opera tion of the vector cross product. For the time being we shall denote this Lie algebra by LA 1 . a) Show that the real 3 x 3 skew-symmetric matrices form a Lie algebra (which we denote LA 2 ) if the product of the matrices A and B is defined as [A, B] = AB - BA. b) Show that the correspondence
(
)
is an isomorphism of the Lie algebras LA 2 and LA 1 . c) Verify that if the skew-symmetric matrix f? and the vector w correspond to each other as shown in b) , then the equality Dr = [w, r] holds for any vector r E E 3 , and the relation PnP - 1 +-t Pw holds for any matrix P E 80(3) . d) Verify that if JR. 3 t r+ O ( t ) E S0(3) is a smooth mapping, then the matrix n ( t ) = o- 1 ( t ) 6 ( t ) is skew-symmetric. e) Show that if r ( t ) is the radius vector of a point of a rotating top and f? ( t ) is the matrix (0- 1 0) (t) found in d), then r(t) = ( f?r ) ( t ) . f) Let r and w be two vectors attached at the origin of E3 . Suppose a right handed frame has been chosen in E 3 , and that the space undergoes a right-handed rotation with angular velocity lwl about the axis defined by w. Show that r(t) = [w, r ( t )] in this case. 2 For the precise definition of a Lie group and the corresponding reference see Problem 8 in Sect. 15.2.
10.4 The Finite-increment (Mean-value) Theorem
73
g) Summarize the results of d), e) , and f) and exhibit the instantaneous angular velocity of the rotating top discussed in Example 8. h) Using the result of c) , verify that the velocity vector w is independent of the choice of the fixed orthoframe in E3, that is, it is independent of the coordinate system. Let r = r(s) = ( x 1 (s), x 2 (s), x3(s) ) be the parametric equations of a smooth curve in E3, the parameter being arc length along the curve (the natu ral parametrization of the curve) . a) Show that the vector e 1 (s) = * (s) tangent t o the curve h as unit length. b) The vector � (s) = � (s) is orthogonal to e 1 . Let e2(s) be the unit vector formed from � (s). The coefficient k(s) in the equality � (s) = k(s)e 2 (s) is called the curvature of the curve at the corresponding point. c) By constructing the vector e 3 ( s) = [e 1 ( s), e 2 ( s)) we obtain a frame { e 1 , e 2 , e 3 } at each point, called the Prenet frame 3 or companion trihedral of the curve. Verify the following Frenet formulas: 6.
� (s) Ts (s) � (s)
k(s)e 2 (s) , x(s)e 3 (s) , - x(s)e 2 (s) . Explain the geometric meaning of the coefficient x( s) called the torsion of the - k(s)e 1 (s)
curve at the corresponding point.
10.4 The Finite-increment Theorem and some Examples of its Use 10.4. 1 The Finite-increment Theorem
In our study of numerical functions of several variables in Subsect. 5.3 . 1 we proved the finite-increment theorem for them and discussed in detail various aspects of this important theorem of analysis. In the present section the finite increment theorem will be proved in its general form. So that its meaning will be fully obvious, we advise the reader to recall the discussion in that subsection and also to pay attention to the geometric meaning of the norm of a linear operator (see Subsect. 10.2.2) .
Let f : U -+ Y be a continuous mapping of an open set U of a normed space X into a normed space Y. If the closed interval [x, x + h] = { � E X I � = x + ()h, 0 ::::; () ::::; 1 } is contained in U and the mapping f is differentiable at all points of the open interval ]x, x + h[= { � E X I � = x + ()h, 0 () 1 } , then the following
Theorem 1 . (The finite-increment theorem) .
<
3 J. F. Frenet (1816�1900) � French mathematician.
<
74
10 *Differential Calculus from a General Viewpoint
estimate holds: l
' (f x + h) - f (x ) I Y :S sup l f (.; ) I .C(x; Y J i h l x . (10.46) ' .;) l l the inequality ' ' Proof. We remarkl first of all that ( x" - x l (f x" ) - f (x ) l if wesupcouldl f prove (10.4 7) ' ' in which the supremum extends over the whole interval [x , x "] , for every closed interval [x , x "] c] x, x + h[ , then, using the continuity of f and the norm together with the fact 1that 1 !' (0 1 !' (.; ) 1 sup :S sup ' , we would obtain inequality (10.46) in the limit as x ---+ x and x " ---+ x + h . Thus, it suffices to prove that l ( ) ( )l l l (10.48) l ' ( f x )+l h - f x :S M h , where M = sup f x + O h and the function f is assumed differentiable o:s:e:s:t on the entire closed interval [x , x + h] . I J( ) ( ) 1 l ( ) ( ) l l computation I J(The )very (simple ) x3 - f xl r :S 2 l x3 -l f2 x2 +I f x2( l - f x2r l :Sl 2 I )l :S M x 3 - x + M x - XI = M xa - x + x - XI = = M x:l - xi I , which uses only the triangle inequality and the properties of a closed interval, shows that if an inequality of the form (10.48) holds on the portions [x i, x 2 ] and [ x 2 ,x 3 ] of the closed interval [ x i,x 3 ], then it also holds on [x i,x 3 ]. Hence, if estimate (10.48) fails for the closed interval [x , x + h] , then by successive bisections, one can obtain a sequence of closed intervals [ a k, b k] h[ contracting to some point x 0 E [x , x + h] such that (10.48) fails on ]x,eachx +interval b [ a Since k, k] · x0 E [ak, bk], consideration of the closed intervals [closed ak, x0]intervals and [x o , b k] enables us to assume that we have found a sequence of of the form [x 0, x0 + h k] [x , x + h] , where h k ---+ 0 as k ---+ e E] x , x+h [
:S
e E [x' , x " j
e E [x' , x " ]
e E ] x ,x + h [
C
C
on which
oo
(10.49)
If we prove (10.48) with replaced by c: , where c: is any positive number, we will still obtain (10.48) as c: ---+ 0, and hence we can also replace (10.49) by
M
M+
(10.49')
75
10.4 The Finite-increment (Mean-value) Theorem
and we can now show that this is incompatible with the assumption that is differentiable at Indeed, by the assumption that f is differentiable,
x0( . l l ' ( (l ( )l ) ) ) ' / f xo + hk - f xo f xo:::; h1 k +( o )hl kl :::;l (l l ) :::; ) l l xo hk + o hk (M + e hk
f
=
The finite-increment theorem has the following useful, purely technical corollary. Corollary. If E .C(X; Y), that is, is a continuous linear mapping of the normed space X into the normed space Y and f : U -+ Y is a mapping
A A satisfying thel hypotheses theorem, then l ( ) of( the) finite-increment ll l f x + h - f x - Ah :::; sup II!'(�) - A h . Proof. For the proof it suffices( to apply ( the )finite-increment theorem to the mapping ) t F t f x + th - Ath of the unit interval [ 0(, 1] lR into Y,( since ' ( f x + h) - f (x ) - Ah , 'F( 1 ) - F(O) F O )I ' J(O )xI +:::; Ol h )' (h - AOh ) for 0l l l ll 1 , F :::; f x + h - A hl ,l l sup II! '( � ) - A h . sup II F'(O) II Remark. As can be seen from the proof of Theorem 1 , in its hypotheses there is no need to require that f be differentiable as a mapping f : U -+ Y; it suffices that its restriction to the closed interval [x , x + h] be a continuous ]mapping x,x + h[ .of that interval and differentiable at the points of the open interval � E] x , x + h [
t-t
c
=
=
=
O
<
�E]x,x+h[
(}
<
D
This remark applies equally to the corollary of the finite-increment theo rem just proved. 10.4.2 Some Applications of the Finite-increment Theorem a. Continuously Differentiable Mappings Let
( ) 10. 50
f : U -+ Y be a mapping of an open subset U of a normed vector space X into a normed space Y. If f is differentiable at each point x E U, then, assigning to the
76
10 *Differential Calculus from a General Viewpoint
.C(X; Y) tangent to f at that point, we obtain (10.51) f ' : U .C(X; Y) . Since the space .C( X; Y) of continuous linear transformations from X into Y is, as we know, a normed space (with the transformation norm) , it makes sense to speak of the continuity of the mapping (10.51). Definition. When the derivative mapping (10.51) is continuous in U, the mapping (10.50), in complete agreement with our earlier terminology, will be said to be continuously differentiable. As before, the set of continuously differentiable mappings of type (10.50) will be denoted by the symbol C ( l l (U, Y), or more briefly, CC 1 l (U), if it is point x the mapping f' (x) E the derivative mapping
-t
clear from the context what the range of the mapping is. Thus, by definition fE
c C l l (U, Y)
{:}
f' E
C ( U, .C(X; Y) ) .
Let us see what continuous differentiability of a mapping means in differ ent particular cases.
Example 1 . Consider the familiar situation when X = Y = JR, and hence f : U -t lR is a real-valued function of a real argument. Since any linear mapping A E .C(JR; JR) reduces to multiplication by some number a E JR, that is, Ah = ah and obviously I I A II = l a l , we find that f ' (x)h = a(x)h, where a(x) is the numerical derivative of the function f at the point x. Next , since ( f ' (x + <5 )
- J ' (x) ) h = J ' (x + <5 ) h - J ' (x) h = = a(x + <5 ) h - a(x)h = ( a(x + <5 ) - a (x) ) h ,
(
10.52 )
it follows that
I I J ' (x + <5 ) - f ' (x) ll = la(x + <5 ) - a(x) l and hence in this case continuous differentiability of the mapping f is equiv alent to the concept of a continuously differentiable numerical function (of class cC l l (U, JR)) studied earlier.
Example 2. This time suppose that X is the direct product X1 x · · · x Xn of normed spaces. In this case the mapping (10.50) is a function f(x) = f(x 1 , . . . , x m ) of m variables X i E Xi , i = 1, . . . , m, with values in Y. If the mapping f is differentiable at x E U, its differential df(x) at that point is an element of the space .C ( X1 x · · · x Xm = X; Y). The action of df(x) on a vector h = (h 1 , . . . , h m ) , by formula (10.45), can be represented as df(x)h = 8l f(x) h l
+
· · · + 8m f(x)h m ,
10.4 The Finite-increment (Mean-value) Theorem 77 ) ( where 8d x : Xi Y, i = 1, are the partial derivatives of the mapping f at the point x under consideration. Next , (d ( ) d ( ) ) m (8 ( ) ( + <5 ) - 8d x h i . (10.53) f x + <5) - f x h L x d i=l -r
. . . , m,
=
But by the properties of the standard norm in the direct product of normed spaces ( see Example 6 in Subsect. 10.1.2) and the definition of the norm of a transformation, we find that
ld ( l l ( d ( )l l ) ( 8d x + <5) - 8d x ccx,; Y } :::; l f x + <5) - f x lccx; Y } :::; m 8 ( ) ccx,; Y } . (10.54) ( 8 + ) x x <5 d d L :::; i=l Thus in this case the differentiable mapping (10.50) is continuously differen tiable in U if and only if all its partial derivatives are continuous in U. In particular, if CX) = !R m and Y = IR, we again obtain the familiar concept of a continuously differentiable ( ) numerical function of real variables ( a function of class C l U,IR , where U !R m ). m
c
Remark. It is worth noting that in writing (10.52) and (10.53) we have made essential use of the canonical identification TXx "' X , which makes it possible to compare or identify vectors lying in different tangent spaces. We shall now show that continuously differentiable mappings satisfy a Lipschitz condition.
C) I K is a convex compact set in a normed space X and f ( l K E C Y ) , where Y is also a normed space, then the mapping f : K Y , fsatisfies a Lipschitz condition on K, that is, there exists a constant M > 0 such that the inequality
Proposition 1 .
-7
(10.55)
holds for any points x 1 , x' 2 E K . Proof. By hypothesis f : K
-r .C ( X; Y ) is a continuous mapping of the compact set into the metric space .C ( X; Y ) . Since the norm is a continuous function on a normed space with its natural metric, the mapping H being the composition of continuous functions, is itself a continuous mapping of the compact set into R But such a mapping is necessarily bounded. Let M be a constant such that :::; M at any point E Since is convex, for any two points x 1 E and x 2 E the entire interval [x 1 , x 2 ] is contained in Applying the finite-increment theorem to that interval, we immediately obtain relation (10.55). 0
K
K
K.
l '( ) l Kf x
K
l '( ) l x fx , x K. K
78
10 *Differential Calculus from a General Viewpoint
Under of' Proposition 1 there exists a non ( ) the hypotheses to 0 as 8 negative function w l8 tending ) lhl (f xl +l h) - f (x ) - f --+(x )+h0l ::;suchw (8that (10.56) at any point x E K for h 8 if x + h E K . ' l l ' theorem) we 'can) lwrite ll Proof. Byl the corollary ) ) to) the finite-increment
Proposition 2 .
<
f ( x + h - f (x - f ( x h ::; sup' f ( x' + Oh - f ( x h and, setting (w 8) sup l f ( x2 ) - f (xl ) l , ' ) ( we obtain (10.56) in view of the uniform continuity of the function x f x , which is continuous on the compact set K . 0<0< 1
=
X l , X2 E K l x 1 - x2 l < 6
D
H
b. A Sufficient Condition for Differentiability We shall now show that by using the general finite-increment theorem, we can obtain a general suffi cient condition for differentiability of a mapping in terms of its partial deriva tives.
U be a neighborhood of the point x in a normed space X is the direct product of the normed spaces x1 X . . . X xX1 ,X and XletXmfLet, which U Y be a mapping of U into a normed space Y. If the m mapping f has partial derivatives with respect to all its variables in U, then it is differentiable at the point x if the partial derivatives are all continuous at that point. =
Theorem 2 . • . .
:
--+
Proof. To simplify the writing we carry out the proof for the case m = verify immediately that the mapping
2. We
h ( h 1 , h2 ) , x. f (x + h) -( f (x ) - Lh2 2 ) ( 2 ) ( ) 2 ( ) 2 f x( 1 + h 1 , X 2+ h 2 )- f x( 1 , X 2 - oif2 ) x h 1 -( o f 2x) h f x 1 + h 1 , X + h -( f x 21 , X +2 ) h -( oi f 2x) 1 , X82 h(1 + 2 ) 2 + f x1 , x + h - f x1 , x - f x1 , x h , l ( corollary to Theorem l by the 1 we obtain ) ) ) ) ( ( ( f x 1 + h 1 , X2l 8+ h(2 - f x 1 , X22 - oif2 ) x 18, X2( h 1 -2 o) l2 fl xl1 , X2 h2 ::; d x 1l + !01( h 1 ,x + h -) d x(1 ,x ) hl 1l +l ::; sup + sup 82 x 1 ,x2 + 02 h2 - 82 f x 1 ,x2 h2 . (10.57)
which is linear in = is the total differential of f at Making the elementary transformations =
= =
=
0 < 01 < 1
0 < 02 < 1
10.4 The Finite-increment ( Mean-value ) Theorem
79
Since max{ l h 1 l , l h 2 } :::; l h l , it follows obviously from the continuity of the partial derivatives fhf and chf at the point X = ( x 1 , X 2 ) that the right-hand side of inequality (10.57) is o(h) as h = (h 1 , h 2 ) --+ 0. D Corollary. A mapping f U --+ Y of an open subset U of the normed space X = X1 x x Xm into a normed space Y is continuously differentiable if and only if all the partial derivatives of the mapping f are continuous. Proof. We have shown in Example 2 that when the mapping f U --+ Y is · · ·
:
:
differentiable, it is continuously differentiable if and only if its partial deriva tives are continuous. We now see that if the partial derivatives are continuous, then the map ping f is automatically differentiable, and hence (by Example 2) also contin uously differentiable. D 10.4.3 Problems and Exercises
1. Let f : I --+ Y be a continuous mapping of the closed interval I = [0, 1] C lR into a normed space Y and g : I --+ lR a continuous real-valued function on I. Show that if f and g are differentiable in the open interval ] O, 1 [ and the relation I f' ( t) I S g' ( t) holds at points of this interval, then the inequality l f(1) - f(O) I S g(1) - g(O) also holds.
a) Let f : I --+ Y be a continuously differentiable mapping of the closed interval C lR into a normed space Y. It defines a smooth path in Y. Define the length of that path. b) Recall the geometric meaning of the norm of the tangent mapping and give an upper bound for the length of the path considered in a) . c ) Give a geometric interpretation of the finite-increment theorem. 2.
I = [0, 1]
3. Let f : U --+ Y be a continuous mapping of a neighborhood U of the point a in a normed space X into a normed space Y. Show that if f is differentiable in U \ a and f'(x) has a limit L E .C ( X; Y ) as x --+ a, then the mapping f is differentiable at a and J ' (a) = L.
4. a ) Let U be an open convex subset of a normed space X and f : U --+ Y a mapping of U into a normed space Y. Show that if J ' (x) = 0 on U, then the mapping f is constant. b ) Generalize the assertion of a ) to the case of an arbitrary domain U ( that is, when U is an open connected subset of X ) . c ) The partial derivative � of a smooth function f : D --+ lR defined in a domain D c JR 2 of the xy-plane is identically zero. Is it true that f is then independent of y in this domain? For which domains D is this true?
80
10 *Differential Calculus from a General Viewpoint
1 0 . 5 Higher-order Derivatives 1 0 . 5 . 1 Definition of the nth Differential
U be an open set in a normed space X and f:U Y (10.58) a mapping of U into a normed space Y. If the mapping (10.58) is differentiable in U, then the derivative of f , given by f' : U .C(X; Y) , (10.59) is defined in U. The space .C(X; Y) =: Y1 is a normed space relative to which the mapping (10.59) has the form (10.58), that is, f' : U Y1 , and it makes sense to speak of differentiability for it. If the mapping ( 10.59) is differentiable, its derivative ( ! ')' : U .C(X; YI ) = .C(X; .C(X; Y)) is called the second derivative or second differential of f and denoted f" or J C 2 ) . In general, we adopt the following inductive definition. Definition 1 . The derivative of order n E Nor n th differential of the map ping (10.58) at the point x E U is the mapping tangent to the derivative of f of order n - 1 at that point. If the derivative of order k E N at the point x E U is denoted f ( k l (x), Definition 1 means that (10.60) Thus, if J C n l (x) is defined, then f ( n ) (x) E .C(X; Yn ) = .C(X; .C(X; Yn - I )) = · . . . . . = .C(X; .C(X; . . . ; .C(X; Y)) . . . ) . Consequently, by Proposition 4 of Sect. 10.2, f ( n ) (x ) , the differential of order n of the mapping (10.58) at the point x can be interpreted as an element of the space .C(X, . . . , X; Y) of continuous n-linear transformations. '-v-' n We note once again that the tangent mapping f'(x) : TXx TYJ ( x ) is Let
-t
-'t
-'t
-'t
factors
-'t
a mapping of tangent spaces, each of which, because of the affine or vector space structure of the spaces being mapped, we have identified with the corresponding vector space and said on that basis that f'(x) E .C(X; Y). It is this device of regarding elements f'(x 1 ) E .C(TXx 1 ; TYJ ( x , ) ) and
81
10.5 Higher-order Derivatives
j'(x2 ) E £ (TXx 2 , TYf (x 2 ) ) , which lie in different spaces, as vectors in the same space C(X; Y) that provides the basis for defining higher-order dif ferentials of mappings of normed vector spaces. In the case of an affine or vector space there is a natural connection between vectors in the different tangent spaces corresponding to different points of the original space. In the final analysis, it is this connection that makes it possible to speak of the continuous differentiability of both the mapping ( 10.58) and its higher-order differentials. 10.5.2 Derivative with Respect to a Vector and Computation of the Values of the nth Differential
When we are making the abstract Definition 1 specific, the concept of the derivative with respect to a vector may be used to advantage. This concept is introduced for the general mapping (10.58) just as was done earlier in the case X = lRm , Y = R Definition 2 . If
X and Y are normed vector spaces over the field
derivative of the mapping (10.58) with respect to the vector h E TXx the point x E U is defined as the limit f(x + t h) - J(x) ' Dh f ( x) := lim IR 3 t -+ 0 t
JR, the X at
rv
provided this limit exists. It can be verified immediately that and that if the mapping f is differentiable at the point x E derivative at that point with respect to every vector; moreover Dh f ( x) =
f' ( x)h ,
and, by the linearity of the tangent mapping,
U,
(10.61) it has a (
10.62 )
(
10.63 )
It can also be seen from Definition 2 that the value Dh f ( x) of the deriva tive of the mapping f : U -+ Y with respect to a vector is an element of the vector space TYJ (x) Y , and that if is a continuous linear transformation from Y to a normed space Z, then rv
L
(10.64) We shall now try to give an interpretation to the value j ( n ) (h 1 , . . . , h n ) of the nth differential of the mapping f at the point x on the set (h 1 , . . . , h n ) of vectors h; E TXx "' X, i = 1, . . . , n.
10 *Differential Calculus from a General Viewpoint
82
1. In this case, by formula (10.62) f'(x)(h) = f'(x)h = Dh f(x) . We now consider the case = 2. Since f< 2 l (x) E .C(X; .C(X ; Y)), fixing a vector h 1 E X , we assign a linear transformation (J< 2 l (x)h l ) E .C(X; Y) to it by the rule h 1 f ( 2 ) (x )h 1 . Then, after computing the value of this operator at the vector h 2 E X, we obtain an element of Y: (10.65) We begin with
n
=
n
f-t
But
f < 2 l (x)h = (f')'(x)h = Dhf '(x) ,
and therefore
(10.66)
If A E .C(X; Y) and h E X, this pairing with Ah can be regarded not only as a mapping h f-t Ah from X into Y, but as a mapping A f-t Ah from .C(X; Y) into Y, the latter mapping being linear, just like the former. Comparing relations (10.62), (10.64), and (10.66), we can write Thus we finally obtain
Similarly, one can show that the relation
f ( n ) (x)(h 1 , . . . , hn ) := ( . . . (f ( n ) (x)h 1 ) . . . hn ) = Dh 1 Dh 2 . . Dh n f(x) (10.67) holds for any E N, the differentiation with respect to the vectors being carried out sequentially, starting with differentiation with respect to h n and ending with differentiation with respect to h 1 . n
•
1 0 . 5 . 3 Symmetry of the Higher-order Differentials
In connection with formula (10.67), which is perfectly adequate for compu tation as it now stands, the question naturally arises: To what extent does the result of the computation depend on the order of differentiation?
If the form f ( n ) (x) is defined at the point x for the mapping ( 10.58), it is symmetric with respect to any pair of its arguments.
Proposition.
10.5 Higher-order Derivatives
83
Proof. The main element in the proof is to verify that the proposition holds in the case n = 2. Let h 1 and h 2 be two arbitrary fixed vectors in the space T Since U is open in X, the following auxiliary function of t is defined for all values of t E lR sufficiently close to zero:
Xx "' X.
We consider also the following auxiliary function:
g(v) = f(x + t (h 1 + v)) - f(x + tv) , which is certainly defined for vectors v that are collinear with the vector h 2 and such that l v l :S: l h 2 l · We observe that
We further observe that , since the function f : U --+ Y has a second differential f" ( x) at the point x E U, it must be differentiable at least in some neighborhood of x. We shall assume that the parameter t is sufficiently small that the arguments on the right-hand side of the equality that defines Ft (h 1 , h2 ) lie in that neighborhood. We now make use of these observations and the corollary of the mean value theorem in the following computations:
1 Ft (h 1 , h 2 ) - t 2 j" (x)(h 1 , h 2 ) l = = l g(h 2 ) - g(O) - t 2 j"(x)(h 1 , h 2 ) l :S: :S: sup ll g'(fhh 2 ) - t 2 J" (x)h i ! i l h 2 l = 0<112 < 1 = sup II (J '(x + t(h 1 + fhh 2 )) - f'(x + t 8 2 h 2 ))t - t 2 f" (x)hi !i l h 2 1 . 0<112 < 1 By definition of the derivative mapping we can write that
and as t --+ 0. Taking this relation into account, one can continue the preceding computation, finding after cancellation that as t --+
0. But this equality means that
84
10 *Differential Calculus from a General Viewpoint
Since it is obvious that Ft(h 1 , h 2 ) = Ft(h 2 , h l ), it follows from this relation that j"(x)(h 1 , h 2 ) = j"(x)(h 2 , h l ). One can now complete the proof of the proposition by induction, repeating verbatim what was said in the proof that the values of the mixed partial derivatives are independent of the order of differentiation. D Thus we have shown that the nth differential of the mapping the point x E U is a symmetric n-linear transformation
(10.58) at
f ( n l (x) E C(TXx, . . . , TXx; TYJ ( x J ) "" C(X, . . . , X; Y) whose value on the set (h 1 , . . . , hn ) of vectors h i E TXx = X, i = 1, . . . , n, can be computed by formula (10.67). If X is a finite-dimensional space having a basis { e 1 , , e k } and hJ = hj e i is the expansion of the vector hJ , j = 1, . . . , n, with respect to that basis, then by the multilinearity of f ( n l (x) we can write f ( n) ( X )(h 1 , . . . , hn ) - J ( n ) ( X )(hi1t e,. 1 , . . . , hnin e,n. ) -- J (n)(X)( e,. 1 , . . . , e,. n )hi1t . . . . . hnin . Using our earlier notation Oi1 ... in f(x) for De 1 De n f(x), we find finally that f ( n ) ( X )(h 1 ' ' hn ) - U2 1 ... 2 n J( X )hi11 hnin ' •
_
.
.
_
•
· · ·
- "' ·
·
•
•
•
•
•
where as usual summation extends over the repeated indices on the right hand side within their range of variation, that is, from 1 to k. Let us agree to use the following abbreviation:
( 10.68) In particular, if we are discussing a finite-dimensional space X and h = hi e i , then f (n) ( X )hn - u2l"'2. n J(X)hi t , . . . . hin ' _
"' ·
which is already very familiar to us from the theory of numerical functions of several variables. 1 0 . 5 .4 Some Remarks
In connection with the notation ( 10.68 ) consider the following example, which is quite useful and will be used in the next section.
, xn )
Example. Let A E £ (X1 , . . . , X ; Y), that is, y = A(x 1 , . . . is a contin uous n-linear transformation from the product of the normed vector spaces X1 , . . . , X into the normed vector space Y. It was shown in Example 5 of Sect . 10.4 that A is a differentiable mapping A : x1 X X X ---+ y and
n
n
0
0
0
n
10.5 Higher-order Derivatives
85
A'(x 1 , . . . , Xn )(h 1 , · · · , hn ) = = A(h 1 , X2 , . . . , Xn ) + · · · + A(x l , . . . , Xn - 1 , hn ) . Thus, if X1 = · · · = Xn = X and A is symmetric, then A'(x, . . . , x)(h, . . . , h) = nA(x, . . . , x, h) =: (nAxn - 1 )h . n- 1 Hence, if we consider the function F : X --+ Y defined by the condition X x F (x) = A(x, . . . , x) =: Axn , �
3
H
it turns out to be differentiable and
F'(x)h = (nAxn - 1 )h , that is, in this case
F'(x) = nAxn - 1 , where Ax n - 1 := A(x, . . . , x, ·). n- 1 In particular, if the mapping (10.58) has a differential f ( n l (x) at a point x E U , then the function F (h) = f ( n l (x)hn is differentiable, and (10.69) �
To conclude our discussion of the concept of an nth-order derivative, it is useful to add the remark that if the original function (10.58) is defined on a set U in a space X that is the direct product of normed spaces X 1 , . . . , Xm , one can speak of the first-order partial derivatives &If(x), . . . , &m f(x) of f with respect to the variables x; E X;, i = 1, . . . , m , and the higher-order partial derivatives &; 1 .. · i n f ( x). On the basis of Theorem 2 of Sect . 10.4, we obtain by induction in this case that if all the partial derivatives 0;1 . . . ; n f(x) of a mapping f : U --+ Y are continuous at a point X E X = x1 X . . . X Xm , then the mapping f has
an n-th order differential f ( n l (x) at that point. If we also take account of the result of Example 2 from the same sec tion, we can conclude that the mapping U x f ( n l (x) E C(X, . . . , X Y) is continuous if and only if all the nth-order partial derivativesn U x &; , ... ; f(x) E C(X; "" . . , X; Y) of the original mapping f : U --+ Y are con tinuous ( or, what is the same, the partial derivatives of all orders up to n inclusive are continuous ) . The class of mappings (10.58) having continuous derivatives up to order n inclusive in U is denoted c< n l (U, Y), or, where no confusion can arise, by the briefer symbol c< n l ( U) or even c< n l . 3
n
n;
H
; '--v---' factors 3 H
10 *Differential Calculus from a General Viewpoint
86
In particular, if X = X1 x · · · x Xn , the conclusion reached above can be written in abbreviated form as
where
C , as always, denotes the corresponding set of continuous functions.
1 0 . 5 . 5 Problems and Exercises 1.
Carry out the proof of Eq. ( 10.64) in full.
2.
Give the details at the end of the proof that f ( n) (x) is symmetric.
3 . a ) Show that if the functions Dh, Dh2 f and Dh2 Dh, f are defined and continuous at a point x E U for a pair of vectors h 1 , h 2 and the mapping ( 10.58) in the domain U, then the equality Dh, Dh2 f(x) = Dh2 Dh 1 f(x) holds. b ) Show using the example of a numerical function f(x, y) that, although the continuity of the mixed partial derivatives ::JY and ::Jx implies by a) that they are equal at this point, it does not in general imply that the second differential of the function exists at the point. c ) Show that, although the2 existence of f ( 2) (x, y) , guarantees that the mixed partial derivatives ::JY and �Jx exist and are equal, it does not in general guar antee that they are continuous at that point. 4. Let A E .C (X, . . . , X ; Y) where A is a symmetric n-linear transformation. Find the successive derivatives of the function x >--+ Ax n : = A ( x, . . . , x) up to order n + 1 inclusive.
1 0 . 6 Taylor's Formula and the Study of Extrema 10.6. 1 Taylor's Formula for Mappings Theorem 1 . If a mapping f U ---+ Y from a neighborhood U = U(x) of a point x in a normed space X into a normed space Y has derivatives up to order n - 1 inclusive in U and has an n-th order derivative f ( n l (x) at the point x, then 1 f ( n ) (x)hn + o(ihi n ) f(x + h) = f(x) + f'(x)h + + 1 (10.70) n. as h ---+ 0. Equality (10. 70) i s one of the varieties o f Taylor ' s formula, written here :
· · ·
for rather general classes of mappings.
10.6 Taylor's Formula and the Study of Extrema
Proof.
We prove Taylor ' s formula by induction. For = 1 it is true by definition of f'(x). 1) E N. Assume formula (10.70) is true for some Then by the mean-value theorem, formula (10.69) of Sect. induction hypothesis, we obtain
n
(n -
87
10.5, and the
l f(x + h ) - (f(x) + f'(x)h + · · · + �! f (n ) (x) hn ) l :::; :::; sup l f'(x + Bh) - ( f'(x) + f"(x)( Bh) + . . . 0 <8< 1 · · · + (n -1 1) ! f ( n ) (x)(Bh) n - 1 ) l lhl = o (IBhl n - 1 ) lhl = o (lhl n )
as
-t
h 0.
D
We shall not take the time here to discuss other versions of Taylor ' s for mula, which are sometimes quite useful. They were discussed earlier in detail for numerical functions. At this point we leave it to the reader to derive them ( see, for example, Problem 1 below ) . 10.6.2 Methods of Studying Interior Extrema
Using Taylor ' s formula, we shall exhibit necessary conditions and also suffi cient conditions for an interior local extremum of real-valued functions defined on an open subset of a normed space. As we shall see, these conditions are analogous to the differential conditions already known to us for an extremum of a real-valued function of a real variable. -t
Let f : U lR be a real-valued function defined on an open set U in a normed space X and having continuous derivatives up to order k - 1 2: 1 inclusive in a neighborhood of a point x E U and a derivative x itself. f ( k l(x) of order k at the point If f'(x) = 0, . . . , f( k - 1 l (x) = 0 and f ( k ) (x) =J 0, then for x to be an extremum of the function f it is: n e c e s s a ry that k be even and that the form f ( k ) (x) h k be semidefinite,4 and s uffi c i e n t that the values of the form f ( k ) (x)h k on the unit sphere lhl = 1 be bounded away from zero; moreover, x is a local minimum if the inequalities f ( k ) (x)hk 2: 8 > 0 hold on that sphere, and a local maximum if Theorem 2 .
4 This means that the form f ( k ) (x)h k cannot take on values of opposite signs, although it may vanish for some values h #- 0. The equality J ( i l (x) = 0, as usual, is understood to mean that J ( i l (x)h = 0 for every vector h.
10 *Differential Calculus from a General Viewpoint
88
Proof. For the proof we consider the Taylor expansion (10. 70) of f in a neigh borhood of The assumptions enable us to write
x.
a(h) -+ h -+ f ( k l(x)h� f:- 0. t ho f(x + tho) - f(x) = k1! f ( k )(x)(tho) k + a(tho) i tho l k = = (�! f ( k )(x)h� ± a(tho) l h o l k )t k and the expression in the outer parentheses has the same sign as f ( k l(x)h�. For x to be an extremum it is necessary for the left-hand side ( and hence also the right-hand side ) of this last equality to be of constant sign when t changes sign. But this is possible only if k is even. This reasoning shows that if x is an extremum, then the sign of the differ ence f (x + th o) - f (x) is the same as that of f ( k l(x)h� for sufficiently small in that case there cannot be two vectors h o, h 1 at which the form t;f ( khence l(x) assumes values with opposite signs. We now turn to the proof of the sufficiency conditions. For definiteness we consider the case when f ( k )(x)h k 8 > 0 for l h l = 1. Then f (x + h) - f (x) = k1! f ( k )(x)hk + a(h) i h l k = k = (�!f ( k l (x) ( � ) + a(h) ) i h i k ( �! 8 + a(h)) l h l k , ll and, since a(h) -+ 0 as h -+ 0, the last term in this inequality is positive for all vectors h f:- 0 sufficiently close to zero. Thus, for all such vectors h, f (x + h) - f (x) > 0 , that is, x is a strict local minimum. The sufficient condition for a strict local maximum is verified simil a(h) j( k l(x)
where 0 as 0. is a real-valued function, and We first prove the necessary conditions. Since f:- 0 on which f:- 0, there exists a vector Then for values of the real parameter sufficiently close to zero,
:2:
:2:
iarly. D
S(x,
Remark 1 . If the space X is finite-dimensional, the unit sphere 1) with center at E X, being a closed bounded subset of X, is compact. Then the continuous function = ( a k-form) has both a maximal and a minimal value on 1). If these values are of opposite sign, then does not have an extremum at If they are both of the same sign, then, as was shown in Theorem 2, there is an extremum. In the latter case, a
x
f
J( k )(x)hk aS(x, i l· · iJ(x)hi 1 · hik x. •
•
•
•
10.6 Taylor's Formula and the Study of Extrema
89
sufficient condition for an extremum can obviously be stated as the equivalent requirement that the form J ( k ) (x)h k be either positive- or negative-definite. It was this form of the condition that we encountered in studying real valued functions on JR n .
Remark 2. As we have seen in the example of functions f : JRn -+ JR , the semi-definiteness of the form J ( k ) h k exhibited in the necessary conditions for an extremum is not a sufficient criterion for an extremum.
Remark 3. In practice, when studying extrema of differentiable functions one normally uses only the first or second differentials. If the uniqueness and type of extremum are obvious from the meaning of the problem being studied, one can restrict attention to the first differential when seeking an extremum, simply finding the point x where f'(x) = 0. 10.6.3 Some Examples
Example 1 . Let L E C ( l l (JR 3 , JR) and f E C ( l l ( [a, b], JR ) . In other words, 1 2 ( u , u , u3 ) L( u l , u2 , u3 ) is a continuously differentiable real-valued func tion defined in JR 3 and x f(x) a smooth real-valued function defined on the closed interval [a, b] JR. H
c
Consider the function
H
F : c< 1 l ( [a, b] , JR ) -+ lR
(10.71)
defined by the relation
J b
c< 1 l ( [a, bj , lR) f F(f) = L(x, f(x), J'(x) ) dx E lR . 3
H
a
(10.72)
Thus, ( 10.71) is a real-valued functional defined on the set of functions f E C(1) ( [a, b] , JR) .
The basic variational principles connected with motion are known in physics and mechanics. According to these principles, the actual motions are distinguished among all the conceivable motions in that they proceed along trajectories along which certain functionals have an extremum. Ques tions connected with the extrema of functionals are central in optimal control theory. Thus, finding and studying the extrema of functionals is a problem of intrinsic importance, and the theory associated with it is the subject of a large area of analysis the calculus of variations. We have already done a few things to make the transition from the analysis of the extrema of numer ical functions to the problem of finding and studying extrema of functionals seem natural to the reader. However, we shall not go deeply into the special problems of variational calculus, but rather use the example of the functional �
90
10 *Differential Calculus from a General Viewpoint
(10. 72) to illustrate only the general ideas of differentiation and study of local extrema considered above. We shall show that the functional (10.72) is a differentiable mapping and find its differential. We remark that the function (10.72) can be regarded as the composition of the mappings (10.73 )
F1 : c( l l ( [a , b], JR) ---+ C ( [a,' b],) JR) ( () () defined by the formula ) ) ( ( F1 f x = L x, f x f x followed by the mapping d ) ) ( ( C ( [a, b], lR ) g F2 g = J g x x E lR . 3
(10.74 )
b
H
(
a
10.75 )
F2 { ( f ) ( ) ( f ( ) f' ( ) ) F( 1 ) ( f ( ) f' ( ) ) ' ( ) F h x = 82 L x, x , x h x + 83 L x, x x h x ( 10.76) for h E C ( l ) ( [a , b], lR ) . Indeed, by the corollary to the mean-value theorem, we can write in the By properties of the integral, the mapping is obviously linear and continuous, so that its differentiability is clear. We shall show that the mapping is also differentiable, and that
present case
L3 ( I( ( ) ( ) ) 2 3 8 i - i= 1 iL u 1) ,u ,u L1) l 0 <11sup< 1 8)1 L u + (8L1)) l - I81 L1 u , 82 L ( u + 8-:1 - 82l L (u( , 83 L ( u) + 8-:1 ( -) l83 L u I · L1l 8iL u + Bu - 8iL u · t=l,max2 ,3 L1i , (10.77) 3 0max <11 < 31 ( i � , 2 , 1 ) ( l l 2 3) 2 where{ I u = l u '1 l u} u:� andI = -:1 1 ,::1 ,::1 . If wei now recall that the i norm f ccl) of the function f in c ( l l ( [a , b], lR' ) is max f e, f c ( where f e is' the maximum absolute2 value( of ) , theu3 =function (x ) , 1 on the closed interval [a, b]), then, setting u = u = x, x f f ) ) ( ( ,::1 1 = 0, ,::12 = h x , and -:13 = h x , we obtain from8inequality ) , i = taking (u 1 , u2 , u(10.77), 3 account of the uniform continuity of the functions 1, 2, 3, iL ' ' ' f f f ( ) of(JR)3 , fthat I ( subsets on bounded ( ) ( )) ( ( ) ( )) max L x, x + h x , ' x + h x - L x, x ', x ' - l - 82 L (x, f (x ) , f (x)) h (x ) - 83 L (X, f (x( l) fl (x )))h (x )l =l = o h ccl) as h cc') 0 . :S:
:S:
:S:
'
'
L1
'
'
a :C: x :C: b
---+
10.6 Taylor's Formula and the Study of Extrema
91
But this means that Eq. (10.76) holds. By the chain rule for differentiating a composite function, we now conclude that the functional (10.72) is indeed differentiable, and
J b
F' (f)h = ( (82 L(x, f(x)f' (x) )h(x) + o3 L (x, f(x), f' (x)) )h' (x) ) dx . (10. 78) We often consider the restriction of the functional (10.72) to the affine space consisting of the functions f E C( l ) ( [a, b], IR ) that assume fixed values f(a) A f(b) = B at the endpoints of the closed interval [a, b]. In this case, the functions h in the tangent space rc? ) must have the value zero at the endpoints of the closed interval [a, b]. Taking this fact into account , we may integrate by parts in (10.78) and bring it into the form a
=
F'(f)h
,
J (82 L(x, f(x), f'(x)) - � 83 L (x, f(x)J'(x)) )h(x) dx , (10.79) b
=
d
a
of course under the assumption that L and f belong to the corresponding class In particular, if f is an extremum (extremal) of such a functional, then by Theorem 2 we have F'(f)h = 0 for every function h E ( [a, b] , IR ) such that h(a) = h(b) = 0. From this and relation ( 10.79) one can easily conclude (see Problem 3 below) that the function f must satisfy the equation
C(2) .
C( l )
82 L(x, f(x), f'(x)) - d� 03 L(x, J(x), f'(x)) = 0 .
(10.80)
This is a frequently-encountered form of the equation known in the cal culus of variations as the Euler-Lagrange equation. Let us now consider some specific examples.
Example 2. The shortest-path problem.
Among all the curves in a plane joining two fixed points, find the curve that has minimal length. The answer in this case is obvious, and it rather serves as a check on the formal computations we will be doing later. We shall assume that a fixed Cartesian coordinate system has been chosen in the plane, in which the two points are, for example, (0, 0) and ( 1, 0). We confine ourselves to just the curves that are the graphs of functions f E ( [0, 1], IR ) assuming the value zero at both ends of the closed interval [0, 1]. The length of such a curve
C( l )
F(f) =
1
J \/1 0
+
(f ' ) 2 (x) dx
(10.81)
92
10 *Differential Calculus from a General Viewpoint
depends on the function f and is a functional of the type considered in Example 1. In this case the function has the form
L
(10.80) for an extremal here reduces to �dx ( f'(x)1 2 ) -- o ' J1 + (/ ) (x)
and therefore the necessary condition the equation
from which it follows that
f'(x) (10.82) -V-;=1::::+==:::: ('=:=/1=':=;)2;:(x:::::::;:: ) :;::: const on the closed interval [0 , 1] . Since the function v' l�u2 is not constant on any interval, Eq. (10.82) is possible only if f'(x) const on [a, b]. Thus a smooth extremal of this problem must be a linear function whose graph passes through the points (0, 0) and ( 1 , 0). It follows that f(x) 0, and we arrive at the closed interval =
=
=
of the line joining the two given points.
Example 3. The brachistochrone problem. The classical brachistochrone problem, posed by Johann Bernoulli I in 1696 , was to find the shape of a track along which a point mass would pass from a prescribed point Po to another fixed point P1 at a lower level under
the action of gravity in the shortest time. We neglect friction, of course. In addition, we shall assume that the trivial case in which both points lie on the same vertical line is excluded. In the vertical plane passing through the points Po and P1 we introduce a rectangular coordinate system such that Po is at the origin, the x-axis is directed vertically downward, and the point P1 has positive coordinates (x 1 , yt). We shall find the shape of the track among the graphs of smooth functions defined on the closed interval [0, x 1 ] and satisfying the condition f(O) = 0 , f(xt) = Yl · At the moment we shall not take time to discuss this by no means uncontroversial assumption (see Problem 4 below) . If the particle began its descent from the point Po with zero velocity, the law of variation of its velocity in these coordinates can be written as V=
�
(10.83)
Recalling that the differential of the arc length is computed by the formula ds
= J(dx) 2 + (dy) 2 = J1 + ( !' ) 2 (x) dx ,
we find the time of descent
(10.84)
10.6 Taylor's Formula and the Study of Extrema Xl
2 F(f) = vk J 1 + (f') (x) dx
(10.85)
X
0
93
along the trajectory defined by the graph of the function y = f ( x) on the closed interval [0, x 1 ] . For the functional (10.85)
and therefore the condition the equation
_i_dx (
(10.80) for an extremum reduces in this case to
f' (x )
Jx( 1 + (f' ) 2 (x)
) =O'
from which it follows that
f'(x) = c,.fi ' y/ 1 + (f') 2 (x)
(10.86)
where c is a nonzero constant, since the points are not both on the same vertical line. Taking account of (10.84), we can rewrite (10.86) in the form dy = c,.fi · ds
(10.87)
However, from the geometric point of view dx = cos ip , ds
dy . ip = sm ' ds
(10.88)
where ip is the angle between the tangent to the trajectory and the positive x-axis. By comparing Eq. (10.87) with the second equation in (10.88), we find x=
1
2
c
. 2 ip . sm
(10.89)
(10.88) and (10.89) that ( sin2 ip ) = 2 sin2 ip dx dy dy . dx = tan ip = tan ip � =
But it follows from dip
dx dip
from which we find
dip
dip
c2
c2
'
(10.90)
94
10 *Differential Calculus from a General Viewpoint Setting 2/c2 = : a and 2cp = : t, we write relations ( 10.89) and ( 10.90) as
x
a(1 - cos t) , = a(t - sin t) + b =
y
( 10.91)
.
Since a =f. 0, it follows that x = 0 only for t = 2k7r, k E Z. It follows from the form of the function ( 10.91) that we may assume without loss of generality that the parameter value t = 0 corresponds to the point Po = (0, 0) . In this case Eq. ( 1 0.90) implies b = 0, and we arrive at the simpler form
x
y
a ( 1 - cos t) , = a(t - sin t)
=
( 10.92)
for the parametric definition of this curve. Thus the brachistochrone is a cycloid having a cusp at the initial point Po where the tangent is vertical. The constant a, which is a scaling coefficient, must be chosen so that the curve ( 10.92) also passes through the point P1 . Such a choice, as one can see by sketching the curve ( 10.92 ) , is by no means always unique, and this shows that the necessary condition ( 10.80) for an extremum is in general not sufficient. However, from physical considerations it is clear which of the possible values of the parameter a should be preferred (and this, of course, can be confirmed by direct computation) . 10.6.4 Problems and Exercises 1. Let f : U ---+ Y be a mapping of class c
g(t) = f(x + t h ) - ( f(x) + J ' (x) (t h ) + · · + ·
C
is defined on the closed interval [0 , 1] ]0, 1 [, and that the estimate
�� J
lR and differentiable on the open interval
holds for every t E ] O, 1 [. b ) Show that l g(1) - g(O) I :S: ( n � l ) ! M i h l n + l . c ) Prove the following version of Taylor's formula:
l f (x + h) - (f(x) + J' (x)h +
·
·
·
+
�/ n) (x)hn ) I :::; ( :1 ) ! l h l n + l .
d ) What can be said about the mapping f f ( n + ll (x) 0 in U? =
:
n
U
---+
Y if it is known that
10.6 Taylor's Formula and the Study of Extrema
95
2. a) If a symmetric n-linear operator A is such that Axn = 0 for every vector x E X, then A(x 1 , . . . , x n ) 0, that is, A equals zero on every set X 1 , . . . , x n of vectors in X. b) If a mapping f U Y has an nth-order differential f ( n) (x) at a point x E U and satisfies the condition =
:
---+
= 0, 1, . . . , n, are i-linear operators, and a ( h ) 0 as h 0, then L i = f ( i l (x) , i = 0, 1 , . . . , n. c) Show that the existence of the expansion for f given in the preceding problem does not in general imply the existence of the n-th order differential J < n l ( x) (for n > 1 ) for the function at the point x. d) Prove that the mapping .C(X; Y) 3 A A- 1 E .C(X; Y) is infinitely differentiable in its domain of definition, and that ( A- 1 ) ( n ) ( A ) ( h 1 , . . . , h n ) = A- 1 h n A- 1 . ( - 1 ) n A - 1 h 1 A - 1 h2 3. a) Let rp E C ( [a, b] , lR ) . Show that if the condition ---+
where L i , i
---+
>-+
·
. . .
·
b
J rp(x)h(x) dx = 0 a
holds for every function h E C ( 2 ) ( [a, b] , lR ) such that h ( a) = h ( b) = 0, then rp(x) = 0 on [a, b] . b) Derive the Euler-Lagrange equation (10.80 ) as a necessary condition for an extremum of the functional ( 10 . 7 2) restricted to the set of functions f E c <2 l ( [a, b] , 1R) assuming prescribed values at the endpoints of the closed interval [a, b] . Find the shape y = f(x) , a :::; x :::; b, of a meridian of the surface of revolution (about the x-axis) having minimal area among all surfaces of revolution having circles of prescribed radius ra and rb as their sections by the planes x = a and x = b respectively. 4.
5. a) The function L in the brachistochrone problem does not satisfy the conditions of Example 1, so that we cannot justify a direct application of the results of Example 1 in this case. Show by repeating the derivation of formula (10. 79) with necessary modifications that this equation and Eq. (10.80 ) remain valid in this case. b) Does the equation of the brachistochrone change if the particle starts from the point Po with a nonzero initial velocity (the motion is frictionless in a closed pipe)? c) Show that if P is an arbitrary point of the brachistochrone corresponding to the pair of points P0 , H , the arc of that brachistochrone from Po to P is the brachistochrone of the pair P0 , P. d) The assumption that the brachistochrone corresponding to a pair of points Po , P1 can be written as y = f(x) , is not always justified, as was revealed by the
96
10 *Differential Calculus from a General Viewpoint
final formulas ( 10.92) . Show by using the result of c) that the derivation of (10.92) can be carried out without any such assumption as to the global structure of the brachistochrone. e) Locate a point P1 such that the brachistochrone corresponding to the pair of points Po , P1 in the coordinate system introduced in Example 3 cannot be written in the form y = f(x) . f) Locate a point P1 such that the brachistochrone corresponding to the pair of points Po , P1 in the coordinate system introduced in Example 3) has the form l y = f(x) , and f rf. C ( ) ( [a, b] , lR ) . Thus it turns out that in this case the functional ( 10.85) we are interested in has a greatest lower bound on the set C ( l ) ( [a, b] , lR ) , but not a minimum. g) Show that the brachistochrone of a pair of points Po , P1 of space is a smooth curve. Let us measure the distance d(Po , PI ) of the point Po of space from the point P1 in a homogeneous gravitational field by the time required for a point mass to move from one point to the other along the brachistochrone corresponding to the points. a) Find the distance from the point Po to a fixed vertical line, measured in this sense. b) Find the asymptotic behavior of the function d( Po , H ) as the point H 1s raised along a vertical line, approaching the height of the point Po . c ) Determine whether the function d(Po , PI ) is a metric. 6.
1 0 . 7 The General Implicit Function Theorem In this concluding section of the chapter we shall illustrate practically all of the machinery we have developed by studying an implicitly defined func tion. The reader already has some idea of the content of the implicit theo rem, its place in analysis, and its applications from Chap. 8. For that rea son, we shall not go into detail here with preliminary explanations of the essence of the matter preceding the formalism. We note only that this time the implicitly defined function will be constructed by an entirely different method, one that relies on the contraction mapping principle. This method is often used in analysis and is quite useful because of its computational efficiency. Theorem. Let X, Y, and Z be normed spaces (for example, IR.m , IR. n , and JR k ) , Y being a complete space. Let W = {(x, y) E X x Y l l x - xol < a 1\ IY Yo I < ,8} be a neighborhood of the point (xo, Yo) in the product X x Y of the spaces X and Y. Suppose that the mapping F : W -+ Z satisfies the following conditions: 1 . F(xo, Yo) = 0; 2. F(x, y) is continuous at (xo, Yo);
10.7 The General Implicit Function Theorem
97
3. F'(x, y) is defined in W and continuous at (xo , Yo); 4. F�(xo , Yo) is an invertible5 transformation. Then there exists a neighborhood U = U(xo) of xo E X, a neighborhood V = V(yo ) of yo E Y, and a mapping f : U -t V such that: 1 ' . u X v W; 2'. ( F(x, y) = 0 in U X V) {::} (y = f(x), where x E U and f(x) E V) ; 3'. Yo = f(xo); 4'. f is continuous at x0 . In essence, this theorem asserts that if the linear mapping F� is invertible at a point (hypothesis 4), then in a neighborhood of this point the relation F(x, y) = 0 is equivalent to the functional dependence y = f(x) (conclu sion 2'). Proof. 1 ° To simplify the notation and obviously with no loss of generality, we may assume that x0 = 0, y0 = 0, and consequently W = {(x, y) E X x Y l l x l < a IYI < ,B } . 2° The main role in the proof is played by the auxiliary family of functions gx ( Y ) := y - ( Fy (O, 0) ) - 1 F(x, y) , (10.93) which depend on the parameter x E S, l x l < a , and are defined on the set {y E Y I I YI < ,B } . Let us discuss formula (10.93). We first determine whether the mappings gx are unambiguously defined and where their values lie. The mapping F is defined for (x, y) E W, and its value F(x, y) at the pair (x, y) lies in Z. The partial derivative F� (x, y) at any point (x, y) E W, as we know, is a continuous linear mapping from Y into Z. By hypothesis 4 the mapping F�(O , 0) : Y -t Z has a continuous inverse F�(O, ( 0) ) - 1 : Z -t Y. Hence the composition ( F� ( O , 0) ) - 1 F(x, y) really is defined, and its values lie in Y. Thus, for any x in the a-neighborhood Bx( O, a ) := {x E X l l x l < a } of �he point 0 E X, the function gx is a mapping gx : By( O , ,B) -t Y from the ,B-neighborhood By (O, ,B) := {y E Y I IYI < ,B } of the point 0 E Y into Y. The connection of the mappings (10.93) with the problem of solving the equation F(x, y) = 0 for y obviously consists of the following: the point Yx is a fixed point of gx if and only if F(x, Yx ) = 0. c
A
1
·
·
Let us state this important observation firmly:
g x ( Yx ) = Y x
"¢===>
F(x, Yx ) = 0 .
(10.94) Thus, finding and studying the implicitly defined function y = Yx = f(x) reduces to finding the fixed points of the mappings ( 10.93) and studying the way in which they depend on the parameter x. 5
That is, 3 [F� (xo , YoW 1 E £(Z; Y) .
98
10 *Differential Calculus from a General Viewpoint
3 ° We shall show that there exists a positive number "( < min {a, ,6} such that for each x E X satisfying the condition l x l < "( < a, the mapping 9x : By (O, "f) -+ Y of the ball By(O, "f) : = {y E Y I IYI < "( < ,6} into Y is a contraction with a coefficient of contraction that does not exceed, say 1/2. Indeed, for each fixed x E Bx (O, a) the mapping 9x : By(O, ,6) -+ Y is differentiable, as follows from hypothesis 3 and the theorem on differentiation of a composite mapping. Moreover,
g� (y) = ey - ( F� (0, 0) ) - 1 · ( F� (x, y) ) = = (F� (0, 0) ) - 1 (F� (O, O) - F� (x, y)) . (10.95) By the continuity of F� (x, y) at the point (0, 0) ( hypothesis 3), there exists a neighborhood {(x, y) E X x Y l l x l "( a l\ IYI "( ,6} of (0, 0) E X x Y <
in which
<
<
<
ll g� (y) ll � II (F� (0, 0)) - 1 II · II F� (O, O) - F� (x, y) ll
Here we are using the relation
<
�.
(10.96)
( F� (o, o)) - 1 E C ( Z; Y ) ,
that is, II (F� (0, 0)) - 1 11 < oo . Throughout the following we shall assume that l x l < "( and IYI < "(, so that estimate (10.96) holds. Thus, at any x E Bx (O, "f) and for any Y 1 , Y2 E By(O, "f), by the mean value theorem, we indeed now find that sup ll g'(�) II IY 1 - Y2 l l gx ( YI ) - 9x ( Y2 ) 1 � � E]y, , y2 [
<
� IY1 - Y2 l ·
(10.97) 4°. In order to assert the existence of a fixed point Yx for the mapping gx , we need a complete metric space that maps into ( but not necessarily onto ) itself under this mapping. We shall verify that for any c satisfying 0 c "f there exists o = o (c) in the open interval ] O, "f[ such that for any X E Bx (0, o) the mapping 9x maps the closed ball B y (O, E: ) into itself, that is, 9x ( By(O, c)) By(O, E:). Indeed, we first choose a number 0 E ]O, "f[ depending on E: such that l 9x (O) I = I (F� (O, 0)) - 1 · F(x, 0) 11 � II (F� (O, 0)) - 1 II I F(x, O) l E: (10.98) for l x l 0. This can be done by hypotheses 1 and 2, which guarantee that F(O, 0) = 0 and F (x, y) is continuous at (0, 0). Now if l x l o (c) "( and IYI � E: "(, we find by (10.97) and (10.98) that l gx ( Y ) I � l9x ( Y ) - 9x (O) I + l gx (O) I 21 IYI + 21 c c , and hence for l x l o (c) (10.99) 9x ( By (O, c) ) By(O, c) . <
<
C
<
<
<
<
<
<
c
<
<
�
10.7 The General Implicit Function Theorem
99
Being a closed subset of the complete metric space Y, the closed ball By (O, c) is itself a complete metric space. 5° Comparing relations (10.97) and ( 10.99) , we can now assert by the fixed-point principle (Sect . 9.7) that for each x E Bx (O, J(c) ) =: U there exists a unique point y = Yx =: f(x) E By (O, c) =: V that is a fixed point of the mapping 9x : By (O, c) --+ By (O, E) . By the basic relation ( 10.94) , it follows from this that the function f : U --+ V so constructed has property 2' and hence also property 3 ', since F ( O, 0 ) = 0 by hypothesis 1 . Property 1 ' of the neighborhoods U and V follows from the fact that, by construction, U x V c Bx ( O, ) x By (O, (3) = W. Finally, the continuity of the function y = f(x) at x = 0, that is, property 4', follows from 2' and the fact that, as was shown in part 4° of the proof, for every E > 0 ( E < 1 ) there exists J ( E ) > 0 ( J ( E ) < 1 ) such that 9x ( By ( 0, E )) C By (O, E) for any x E Bx ( O, J ( c )) , that is, the unique fixed point Yx = f(x) of the mapping 9x : By (O, c) --+ By (O, c) satisfies the condition lf(x) l < E for l x l < J (c) . D o:
We have now proved the existence of the implicit function. We now prove a number of extensions of these properties of the function, generated by properties of the original function F.
If in addition to hypothe ses 2 and 3 of the theorem it is known that the mappings F : W --+ Z and F� are continuous not only at the point (x0, y0) but in some neighborhood of this point, then the function f : U --+ V will be continuous not only at x 0 E U but in some neighborhood of this point. Proof. By properties of the mapping .C(Y; Z) A A- 1 E .C(Z; Y) it follows from hypotheses 3 and 4 of the theorem (see Example 6 of Sect . 10. 3) that at each point (x, y) in some neighborhood of (x0, y0) the transformation f�(x, y) E .C(Y; Z) is invertible. Thus under the additional hypothesis that F is continuous all points (x, y) of the form (x, f(x)) in some neighborhood of (x0, y0) satisfy hypotheses 1 -4 , previously satisfied only by the point ( x0, y0). Repeating the construction of the implicit function in a neighborhood of these points (:i:, y) , we would obtain a function y ](x) that is continuous at x and by 2 ' would coincide with the function y = f(x) in some neighborhood of x. But that means that f itself is continuous at :i:. D Extension 2 . (Differentiability of the implicit function. ) If in addition to the hypotheses of the theorem it is known that a partial derivative F� (x, y) exists in some neighborhood W of (x0, y0) and is continuous at (x0, y0), then the function y f ( x) is differentiable at x0, and Extension 1 . (Continuity of the implicit function. )
3
H
=
=
( 10.100)
100
10 *Differential Calculus from a General Viewpoint
Proof. We verify immediately that the linear transformation L E .C(X; Y) on the right-hand side of formula (10.100) is indeed the differential of the function y = f(x) at xo. As before, to simplify the notation, we shall assume that x0 = 0 and oY = 0, so that f(O) = 0. We begin with a preliminary computation. l f(x) - f(O) - Lx l = l f(x) - Lx l = = l f(x) + ( F� (0, 0) ) - 1 · (F� (O, O))x l = = I ( F� ( 0 , 0) ) - 1 ( F� (O, O)x + F� (O, O)f(x)) l = = I ( F� (O, 0) ) - \ F (x, f(x)) - F (O, 0) - F� (O, O)x - F� (O, O)f(x)) I ::; ::; I I (F� (0, 0) ) - 1 II I ( F (x, f(x)) - F (O, O) - F� (O, O)x - F� (O, O)f(x) ) l ::; ::; II (F� (0, 0) ) - 1 11 · a (x, f(x) ) (lxl + l f(x) l) , where a(x, y) ---+ 0 as (x, y) ---+ (0, 0). These relations have been written taking account of the relation F( x, f(x)) 0 and the fact that the continuity of the partial derivatives F� and F� at (0, 0) guarantees the differentiability of the function F (x, y) at that point. For convenience in writing we set a := Il L II and b := II (F�(O, 0) ) - 1 11 · =
Taking account of the relations
l f(x) l = l f(x) - Lx + Lx l ::; l f(x) - Lx l + I Lx l ::; l f(x) - Lx l + a l x l , we can extend the preliminary computation just done and obtain the relation
l f(x) - Lx l ::; ba(x, f(x)) ( (a + 1) l x l + l f(x) - Lx l ) , or
+ 1)b a(x, f(x)) l x l . l f(x) - Lx l ::; 1 _ (aba(x, f(x)) Since f is continuous at x = 0 and f(O) = 0, we also have f(x) ---+ 0 as x ---+ 0, and therefore a(x, f(x)) ---+ 0 as x ---+ 0. It therefore follows from the last inequality that
l f(x) - f(O) - Lx l = l f(x) - Lx l = o ( l x l)
as
x ---+ 0. D
If in addition to the hypotheses of the theorem it is known that the mapping F has continuous partial derivatives F� and F� in some neighborhood W of (xo, Yo), then the function y = f(x) is continuously differentiable in some neighborhood of x0, and its derivative is given by the formula f'(x) = - {-F� (x, f(x))) - 1 · (F� (x, f(x))) . (10.101)
Extension 3 . (Continuous differentiability of the implicit function.)
10.7 The General Implicit Function Theorem
101
Proof. We already know from formula (10.100) that the derivative f' (x) exists and can be expressed in the form ( 10.101) at an individual point x at which the transformation F� ( x, f ( x)) is invertible. It remains to be verified that under the present hypotheses the function f'(x) is continuous in some neighborhood of x = x0. The bilinear mapping (A , B) 1--t A · B - the product of linear transforma tions A and B is a continuous function. The transformation B = - F� ( x, f ( x)) is a continuous function of x, being the composition of the continuous functions x 1--t (x, f(x)) 1--t - F� (x, f(x)). The same can be said about the linear transformation A - I = F�(x, f(x)). It remains only to recall ( see Example 6 of Sect . 10.3) that the mapping A - I 1--t A is also continuous in its domain of definition. Thus the function f'(x) defined by formula (10.101) is continuous in some neighborhood of x = x0, being the composition of continuous functions. 0 -
We can now summarize and state the following general proposition.
If in addition to the hypotheses of the implicit function theorem it is known that the function F belongs to the class C ( k ) (W, Z ) , then the function y f(x) defined by the equation F (x, y) 0 belongs to C ( k ) (U, Y ) in some neighborhood U of xo . Proof. The proposition has already been proved for k 0 and k 1. The general case can now be obtained by induction from formula ( 10.101) if we observe that the mapping .C ( Y; Z ) A A - I E .C ( Z; Y ) is ( infinitely ) differentiable and that when Eq. ( 10.101) is differentiated, the right-hand side always contains a derivative of f one order less than the left-hand side. Thus, successive differentiation of Eq. (10.101) can be carried out a number times equal to the order of smoothness of the function F . 0 Proposition. =
=
3
1--t
=
=
In particular, if
f'(x)h i - ( F� (x, f(x))) - I · ( F� (x, f(x)))hi , =
then
(x)(h i , h2 ) = - d ( F� (x, f(x )) r i h2 F� (x, f(x) )h i - ( F� (x, f(x))) - I d ( F� (x, f(x))h i) h2 = I I = ( F� (x, f(x))) - d F� (x, f(x))h 2 ( F� (x, f(x)) ) - F� (x, f(x))hi - ( F� (x, f(x)) r i (( F�x (x, f(x)) + F�� (x, f(x))J'(x))h i) h2 ( F� (x, f(x))) - I (( F�� (x, f(x)) + F�� (x, f(x))J'(x)))h2 ) x I I x ( F� (x, f(x)) ) - F� (x, f(x ))h i ) ( I;� (x, f(x))) - x x ( ( F�'x (x, f(x)) + F� (x, f(x) )f' (x) )h i) h 2 . Y
f"
=
=
102
10 *Differential Calculus from a General Viewpoint
j" ( ) ( ) ( � ) [( ( �� �� ' ) ) ( � ) � x h 1 , h2 F - 1 F + F( ( :f h2 ��FJ' )- 1 F) h]1 - F x + F h 1 h2 .
In less detailed, but more readable notation, this means that =
(10.102)
In this way one could theoretically obtain an expression for the derivative of an implicit function to any order; however, as can be seen even from formula ( 10.102), these expressions are generally too cumbersome to be conveniently used. Let us now see how these results can be made specific in the important special case when X = JR m , Y = R n , and Z = R n . In this case the mapping z = y) has the coordinate representation
F(x ,
( 10.103) The partial derivatives E .C(JR m ; R n ) and ping are defined by the matrices
F�
(x ,
F� E .C(JRn ; Rn ) of the map
computed at the corresponding point y). As we know, the condition that and be continuous is equivalent to the continuity of all the entries of these matrices. The invertibility of the linear transformation y E .C(JRn ; Rn ) is equivalent to the nonsingularity of the matrix that defines this transforma tion. Thus, in the present case the implicit function theorem asserts that if
F� F�
F� (x0, 0 )
(6 1) F 1 x ' . . . ' x(J' YB' . . . ' Yo ) 0 ' Fni ((x16, . . . ,xm0 , y16 , . . . , y0)n = 0 ; 2 ) (F x , . . . , x , y , . . . , y ),mi �:11,; m functions at the 1 , x0m , Yo1 , . . . , Yon ) E m x ( n. are continuous pom t x0, 3 ) all the partial derivatives x 1 , . . . ,xm( ,y61 , . . . , yn ), i 1, j 1, are defined in a neighborhood of x , . . . ,x0,y5, . . . , y0) and are continuous at this point; =
·
.
•
.
JN..
=
. . . , n,
JN..
,
=
=
. . . , n,
. . . , n,
10.7 The General Implicit Function Theorem
103
4) the determinant i) F
n
8iJ'
n i) Fn y i)
yJ, . . . , y0) ;
of the matrix F; is nonzero at the point ( x6, . . . , x()', then there exist a neighborhood U of x0 = ( x6, . . . , x()') E � m , a neighbor E � n , and a mapping f : U ---+ V having a hood V of y0 = coordinate representation
(yJ, . . . , yQ')
(10.104) such that 1') inside the neighborhood U x V of ( x6, . . . , x()', the system of equations
{
y) . � � ' . . '. � � ' : : : : : : � .
y F (x . · F (X , . . . , X , y
,. . . ,y )
YB , . . . , y0) E �m x �n
.. ..
0,
0
is equivalent to the functional relation f : U ---+ V expressed by
J l (xJ , . . . , x()') ,
2')
(10.104);
r (xJ , . . . , x()') ; 3') the mapping ( 10.104) is continuous at ( x6, , . . . , x()', YB, . . . , Yo ). If in addition it is known that the mapping (10.103) belongs to the class C ( k ) , then, as follows from the proposition above, the mapping (10.104) will also belong to C ( k ) , of course within its own domain of definition. In this case formula (10.101) can be made specific, becoming the matrix
Yo
equality
( : .. . : J ax
ox "'
( �- g�� l n iJ Fx l iJ
-··.... 0
0
0
n iJoxF"'
10 *Differential Calculus from a General Viewpoint
104
in which the left-hand side is computed at (x 1 , . . . , x m ) and the right hand side at the corresponding point ( x 1 , . . . , x m , y 1 , . . . , yn ) , where yi = f i ( x , . . . , xm ) , i = 1, . . . , n . If n = 1, that is, when the equation F( x , . . . , xm , y) = O
l
l
is being solved for y, the matrix F� consists of a single entry - the number �� (x 1 , . . . , x m , y) . In this case y = f(x 1 , . . . , x m ) , and
( ::1 , . . . , 8�) ( �:) ( -1
)
(10.105) :� , . . . , :; In this case formula (10.102) also simplifies slightly; more precisely, it can =
be written in the following more symmetric form:
2 - (F�� + F�� J')h 2 F�h 1 . (10 . 106) ! " ( X )(h 1 , h 2 ) = _ (F�'x + F�'y !')h 1 F�h(F�) 2 And if = 1 and m = 1, then y = f(x) is a real-valued function of one real argument , and formulas (10.105) and (10.106) simplify to the maximum n
extent , becoming the numerical equalities
J'(x) - F' F�y (x, y) , (F" + F"xy f')F'y - (F"YX + F"yy f')F' ( x, y ) J" (x) = (F�) 2 XX
X
for the first two derivatives of the implicit function defined by the equation F(x, y) = 0. 10. 7. 1 Problems and Exercises 1. a ) Assume that, along with the function f : U -+ Y given by the implicit function theorem, we have a function j : fJ -+ Y defined in some neighborhood fJ of xo and satisfying yo = j(xo ) and F(x, j(x)) = 0 in fJ. Prove that if j is continuous at x 0 , then the functions f and j are equal on some neighborhood of xa . b ) Show that the assertion in a) is generally not true without the assumption that j is continuous at xo . 2 . Analyze once again the proof of the implicit function theorem and the extensions to it, and show the following. a ) If z = F(x, y) is a continuously differentiable complex-valued function of the complex variables x and y, then the implicit function y = f(x) defined by the equation F(x, y) = 0 is differentiable with respect to the complex variable x. b ) Under the hypotheses of the theorem X is not required to be a normed space, and may be any topological space.
10.7 The General Implicit Function Theorem
105
3. a) Determine whether the form j" (x)(h 1 , h 2 ) defined by relation (10. 102) is symmetric. b) Write the forms (10.101) and (10.102) for the case of numerical functions F(x 1 , x 2 , y) and F(x, y 1 , y 2 ) in matrix form. c) Show that if JR. 3 t >-+ A(t) E £(1R. n ; IR. n ) is family of nonsingular matrices A ( t ) depending on the parameter t in an infinitely smooth manner, then
denotes the inverse of the matrix A = A(t) . 4. a) Show that Extension 1 to the theorem is an immediate corollary of the sta bility conditions for the fixed point of the family of contraction mappings studied in Sect. 9.7. b) Let {A t : X -+ X} be a family of contraction mappings of a complete normed space into itself depending on the parameter t, which ranges over a domain ft in a normed space T. Show that if At (x) = �.p(t, x) is a function of class c
a) Using the implicit function theorem, prove the following inverse function theorem. Let g : G -+ X be a mapping from a neighborhood G of a point yo in a complete normed space Y into a normed space X. If 1 ° the mapping x = g(y) is differentiable in G, 2 ° g' (y) is continuous at Yo , 3 ° g' (yo) is an invertible transformation, then there exists a neighborhood V C Y of Yo and a neighborhood U C X of Xo such that g : V -+ U is bijective, and its inverse mapping f : U -+ V is continuous in U and differentiable at x 0 ; moreover, 5.
b) Show that if it is known, in addition to the hypotheses given in a) , that the mapping g belongs to the class c
c
C > 0 with a constant C that is independent of x. Show that f is a bijective mapping. d) Using your experience in solving c) , try to give an estimate for the radius of a spherical neighborhood U = B(x 0 , r ) centered at x 0 in which the mapping f : U -+ V studied in the inverse function theorem is necessarily defined. 6. a) Show that if the linear mappings A E £(X; Y) and B E £ ( X ; JR.) are such that ker A C ker B (here ker, as usual, denotes the kernel of a transformation) , then there exists a linear mapping A E £(Y; JR.) , such that B = A A. ·
106
10 *Differential Calculus from a General Viewpoint
b ) Let X and Y be normed spaces and f : X -t ffi. and g : X -t Y smooth functions on X with values in ffi. and Y respectively. Let S be the smooth surface defined in X by the equation g(x) = y0 . Show that if x 0 E S is an extremum of the function ti s ' then any vector h tangent to S at Xo simultaneously satisfies two conditions: f'(x 0 )h = 0 and g' (xo ) h = 0. c ) Prove that if xo E S is an extremum of the function f I s then f ' ( xo) = g' (xo ) , where A E £(Y; ffi.) . A d ) Show how the classical Lagrange necessary condition for an extremum with constraint of a function on a smooth surface in ffi. n follows from the preceding result. ·
7 . As is known, the equation z n + c 1 z n - l + + Cn = 0 with complex coefficients has in general n distinct complex roots. Show that the roots of the equation are smooth functions of the coefficients, at least where all the roots are distinct. ·
·
·
1 1 Mult iple Integrals
1 1 . 1 The Riemann Integral over an n- Dimensional Interval 1 1 . 1 . 1 Definition of the Integral
JR.n and their Measure Definition 1 . The set I = {x E JR. n l a i � x i � bi , i = 1 , . . . } is called an interval or a coordinate paralellepiped in JR.n . If we wish to note that the interval is determined by the points a = (a!, . . . , an ) and b = (b 1 , . . . , bn ), we often denote it Ia, b , or, by analogy with the one-dimensional case, we write it as a � x � b. Definition 2. To the interval I = {x E JR. n l a i � x i � b i , i = 1, } we n i assign the number I I I : = Il W - a ), called the volume or measure of the i= l interval. a. Intervals in
, n
. . . , n
The volume ( measure ) of the interval
I is also denoted v(I) and 1-l(I). Lemma 1 . The measure of an interval in JR. n has the following properties. a) It is homogeneous, that is, if A. Ia, b := ha, >. b , where A. 2: 0, then k b) It is additive, that is, if the intervals I, I . . . , h are such that I = i=U Ii l and no two of the intervals It , . . . , Ik have common interior points, then k I I I = i=2:l I Ii l · c) If the interval I is covered by a finite system of intervals It , . . . , Ik , that is, I i=Uk Ii , then I I I � i=2:k I Ii l · l l 1,
C
All these assertions follow easily from Definitions 1 and 2.
108
11 Multiple Integrals
b. Partitions of an Interval and a Base in the Set of Partitions
Suppose we are given an interval I = {x E !R n l ai ::::; x i ::::; bi, i = l , . . . , n } . Partitions of the coordinate intervals [a i , bi ], i = 1 , . . . , n , induce a partition of the interval I into finer intervals obtained as the direct products of the intervals of the partitions of the coordinate intervals.
k
I (as the union I = j=U Ij l of finer intervals I1 ) just described will be called a partition of the interval I, Definition 3 . The representation of the interval
and will be denoted by
P.
>-.(P) := l:'O;maxj :'O;k d(I1 ) (the maximum among the diameters of the intervals of the partition P) is called the mesh of the parti tion P. Definition 5 . If in each interval Ij of the partition P we fix a point �j E I1 , we say that we have a partition with distinguished points. The set { 6 , . . . , �k } , before, will be denoted by the single letter �, and the partition with distinguished points by (P, �). In the set P = { ( P, �)} of partitions with distinguished points on an interval I we introduce the base >-.(P) ---+ 0 whose elements Bd (d > 0) , as in the one-dimensional case, are defined by Bd : = { (P, �) E P I >-.(P) d}. The fact that B = { Bd} really is a base follows from the existence of Definition 4. The quantity
as
<
partitions of mesh arbitrarily close to zero.
I ---+ be a real-valued 1 func tion on the interval I and = { Ir , . . . , h} a partition of this interval with distinguished points = { 6 , . . . k } .
c . Riemann Sums and the Integral Let
P
�
Definition 6 . The sum
,�
IR
k
f(�i ) I Ii l u (f, P, 0 : = L i= l
is called the Riemann sum of the function of the interval I with distinguished points Definition 7. The quantity
J f(x) dx := I
lim
.>.. ( P ) ---+ 0
f corresponding to the partition
(P, �).
u (f, P,�) ,
provided this limit exists, is called the Riemann integral of the function f over the interval I. 1 Please note that in the following definitions one could assume that the values of f lie in any normed vector space. For example, it might be the space C of complex numbers or the spaces JR:. n and e n .
1 1 . 1 The Riemann Integral over an n-Dimensional Interval
109
We see that this definition, and in general the whole process of construct ing the integral over the interval C !R n is a verbatim repetition of the procedure of defining the Riemann integral over a closed interval of the real line, which is already familiar to us. To highlight the resemblance we have even retained the previous notation f(x) dx for the differential form. Equiv alent, but more expanded notations for the integral are the following:
I
J
f(x 1 , . . . , xn ) dx 1 · . . . · dxn or
J J ···
f(x 1 ,
.
•
•
, x n ) dx 1 ·
.
•
.
· dxn .
I �
I
n To emphasize that we are discussing an integral over a multidimensional domain we say that this is a multiple integral (double, triple, and so forth, depending on the dimension of
I
I).
d. A necessary Condition for Integrability Definition 8. If the finite limit in Definition 7 exists for a function f :
then f is
Riemann integrable over the interval I.
I -+ IR,
R( I).
We shall denote the set of all such functions by We now verify the following elementary necessary condition for integra bility. Proposition 1. f E
R( I)
=}
f
is bounded on I.
I.
Proof. Let P be an arbitrary partition of the interval If the function f is unbounded on then it must be unbounded on some interval Ii o of the partition P. If (P, e) and (P, e') are partitions P with distinguished points such that t;,' and t;," differ only in the choice of the points !;,� 0 and t;,� � , then
I,
By changing one of the points !;,�0 and t;,�� , as a result of the unboundedness of f in Iio , we could make the right-hand side of this equality arbitrarily large. By the Cauchy criterion, it follows from this that the Riemann sums of f do not have a limit as .>.. ( P) -+ 0. D 1 1 . 1 . 2 The Lebesgue Criterion for Riemann Integrability
When studying the Riemann integral in the one-dimensional case, we ac quainted the reader (without proof) with the Lebesgue criterion for the ex istence of the Riemann integral. We shall now recall certain concepts and prove this criterion.
1 10
1 1 Multiple Integrals
JRn Definition 9. A set E JR n has (n-dimensional) measure zero or is a set of measure zero (in the Lebesgue sense) if for every c > 0 there exists a covering of E by an at most countable system { Ii } of n-dimensional intervals for which the sum of the volumes I: I Ii I does not exceed c. Lemma 2 . a ) A point and a finite set of points are sets of measure zero. b) The union of a finite or countable number of sets of measure zero is a set of measure zero. ) A subset of a set of measure zero is itself of measure zero. d) A nondegenemte2 interual Ia, b JRn is not a set of measure zero. a. Sets of Measure Zero in C
c
C
The proof of Lemma 2 does not differ from the proof of its one-dimensional version considered in Subsect. 6 . 1 .3, paragraph d. Hence we shall not give the details. The set of rational points in JR n (points all of whose coordinates are rational numbers) is countable and hence is a set of measure zero.
Example 1 .
Example 2. Let f : I ---+ lR be a continuous real-valued function defined on an (n - I )-dimensional interval I C JR n - l . We shall show that its graph in ]Rn is a set of n-dimensional measure zero.
Proof. Since the function f is uniformly continuous on I, for c > 0 we find (j > 0 such that l f(x i ) - f(x 2 ) I c for any two points x 1 , x 2 E I such li. If we now take a partition P of the interval I with that l x 1 - x 2 l mesh .\(P) li, then on each interval Ii of this partition the oscillation of the function is less than c. Hence, if X i is an arbitrary fixed point of the interval h the n-dimensional interval t = Ii x [f ( x i ) - c , f ( x ) + c] obviously contains the portion of the graph of the function lying over the interval h and the union U t covers the whole graph of the function over I. But I: I ii I = I: I Ii I · 2c 2c iii (here I Ii I is the volume of Ii in JR n - l and I t I i the volume of t in JR n ). Thus, by decreasing c, we can indeed make the total <
<
<
i
=
volume of the covering arbitrarily small.
0
Remark 1 . Comparing assertion b) in Lemma 2 with Example 2 , one can conclude that in general the graph of a continuous function f : JR n - l ---+ lR or a continuous function f : M ---+ JR, where M C JR n - l , is a set of n-dimensional measure zero in JR n . 2 That is, an interval I ,b = {x E IR n l a i :::; x i :::; bi , i a strict inequality a i < bi holds for each i E { 1 , . . . , n } .
=
1 , . . . , n } such that the
1 1 . 1 The Riemann Integral over an n-Dimensional Interval
111
a ) The class of sets of measure zero remains the same whether the intervals covering the set E in Definition 9, that is, E Ui Ji , are interpreted as an ordinary system of intervals { Ji }, or in a stricter sense, requiring that each point of the set be an interior point of at least one of the intervals in the covering. 3 b) A compact set K in !Rn is a set of measure zero if and only if for every c > 0 there exists a finite covering of K by intervals the sum of whose volumes is less than c. Proof. a ) I f {Ji } i s a covering o f E ( that i s , E Ui Ji and L:i I Ii I < c), then, replacing each Ii by a dilation of it from its center, which we denote ii , we obtain a system of intervals {ii } such that L: I ii I < A n c, where ,\ is a dilation coefficient that is the same for all intervals. If ,\ > 1 , it is obvious that the system {ii } will cover E in such a way that every point of E is interior to one of the intervals in the covering. b) This follows from a ) and the possibility of extracting a finite covering from any open covering of a compact set K. ( The system {ii \ oii } consisting of open intervals obtained from the system { ii } considered in a ) may serve as such a covering. )
Lemma 3 .
C
C
D
b. A Generalization of Cantor's Theorem We recall that the oscillation : --+ lR on the set has been defined as := z , and the oscillation at the point E as :=
f E E w(f; E) f(x ) x E w(f; x) l X ! ,X 2 E E lim w(f; U�(x)), where U�(x) is the 15-neighborhood of x in the set E. 6 --tO Lemma 4. If the relation w(f; x) � w0 holds at each point of a compact set K for the function f : K IR, then for every c > 0 there exists 15 > 0 such that w(f; Uf<(x)) < wo + c for each point x E K. When w0 = 0, this assertion becomes Cantor ' s theorem o n uniform conti nuity of a function that is continuous on a compact set . The proof of Lemma 4 is a verbatim repetition of the proof of Cantor ' s theorem ( Subsect. 6.2.2) of a function sup l f(x i )
-
--+
and therefore we do not take the time to give it here.
at almost all points of a set M or almost everywhere on M if the subset of M
c. Lebesgue's Criterion As before, we shall say that a property holds
where this property does not necessarily hold has measure zero.
( ) f E R(I) (! is continuous almost everywhere on I).
Theorem 1 . Lebesgue ' s criterion .
<=?
(! is bounded on I) 1\
3 In other words, it makes no difference whether we mean closed or open intervals in Definition 9.
1 12
1 1 Multiple Integrals
Proof. N e c e s s i t y. If f E R(J), then by Proposition 1 the function f is bounded on I. Suppose l f l :::; M on I. We shall now verify that f is continuous at almost all points of I. To do this, we shall show that if the set E of its points of discontinuity does not have measure zero, then f � R(J). (X) Indeed, representing E in the form E = U En , where En = {x E I I
n= lif E does not have measure w(f; x) ?: 1/n }, we conclude from Lemma 2 that zero, then there exists an index n 0 such that Eno is also not a set of measure zero. Let P be an arbitrary partition of the interval I into intervals {Ii }. We break the partition P into two groups of intervals A and B, where A = { Ii E P j Ii n Eno f:. 0 1\ w(f; Ii ) ?: -2no1- } , and B = P \ A. The system of intervals A forms a covering of the set Eno . In fact, each point of Eno lies either in the interior of some interval Ii E P, in which case obviously Ii E A, or on the boundary of several intervals of the partition 1 on P. In the latter case, the oscillation of the function must be at least 2 no at least one of these intervals (because of the triangle inequality) , and that interval belongs to the system A. We shall now show that by choosing the set � of distinguished points in the intervals of the partition P in different ways we can change the value of the Riemann sum significantly. To be specific, we choose the sets of points e and (' such that in the intervals of the system B the distinguished points are the same, while in the intervals Ii of the system A, we choose the points �: and �:' so that f(W - ! (��') > 3;,0 • We then have
I
l
I O' (f, P, e ) - O' (f, P, e' ) l = I: ( f(�D - J(�n) I Ii l > 3�0 I: I Ii I > > o . �EA
�EA
c
The existence of such a constant c follows from the fact that the intervals of the system A form a covering of the set En 0 , which by hypothesis is not a set of measure zero. Since P was an arbitrary partition of the interval I, we conclude from the Cauchy criterion that the Riemann sums O' (j, P, �) cannot have a limit as >.(P) -+ 0, that is, f � R(I) . S u f f i c i e n c y. Let E be an arbitrary positive number and EE = { x E I I w(f; x) ?: E }. By hypothesis, EE is a set of measure zero. Moreover, EE is obviously closed in I, so that EE is compact. By Lemma 3
k I1 , . . . , h of intervals in IR n such that EE i=U I; l k k and I: I Ii l < E. Let us set cl = u Ii and denote by c2 and c3 the unions i= l i= l
there exists a finite system
C
1 1 . 1 The Riemann Integral over an n-Dimensional Interval
1 13
of the intervals obtained from the intervals Ii by dilation with center at the center of Ii and scaling factors 2 and 3 respectively. It is clear that E6 lies strictly in the interior of c2 and that the distance d between the boundaries of the sets c2 and c3 is positive. We note that the sum of the volumes of any finite system of intervals lying in C3 , no two of which have any common interior points is at most 3 n E, where n is the dimension of the space JR. n . This follows from the definition of the set C3 and properties of the measure of an interval ( Lemma 1). We also note that any subset of the interval I whose diameter i s less than d is either contained in C3 or lies in the compact set K = I \ ( C2 \ 8C2 ), where 8C2 is the boundary of C2 ( and hence C2 \ 8C2 is the set of interior points of C2 ). By construction E6 c I \ K, so that at every point x E K we must have w(f; x) < E. By Lemma 4 there exists J > 0 such that l f(x 1 ) - f(x 2 ) 1 < 2E for every pair of points x 1 , x 2 E K whose distance from each other is at most J. These constructions make it possible now to carry out the proof of the sufficient condition for integrability as follows. We choose any two partitions P' and P" of the interval I with meshes >.. ( P') and >.. ( P") less than >.. = min { d, J}. Let P be the partition obtained by intersecting all the intervals of the partitions P' and P", that is, in a natural notation, P = {Iij = I: n I'/ } · Let us compare the Riemann sums a(!, P, �) and a(!, P', �'). Taking into account the equality I II ! = 2:: I Iij l , we can write
j
l a(f, P, ( ) - a (f, P, �) I =
1 2.:: (f(�D - f(�ij)) I Iij l l :::; t)
Here the first sum 2:: 1 contains the intervals of the partition P lying in the intervals I: of the partition P' contained in the set C3 , and the remaining intervals of P are included in the sum 2:: 2 , that is, they are all necessarily contained in K ( after all, >.. ( P) < d). Since l f l :::; M on I, replacing l f(W - f(�ij)l in the first sum by 2M, we conclude that the first sum does not exceed 2M · 3 n E. Now, noting that �� ' �ij E Ij c K in the second sum and >.. ( P') < J, we conclude that IJ (W - f(�ij) l < 2E, and consequently the second sum does not exceed 2E I I I · Thus l a(J, P' , �') - a(f, P, �) I < (2M · 3 n + 2 I I I )E, from which ( in view of the symmetry between P' and P"), using the triangle inequality, we find that
I a(!, P', ( ) - a(!, P" , (') l < 4(3n M + I I I )E for any two partitions P' and P" with sufficiently small mesh. By the Cauchy criterion we now conclude that f E R(I). 0
1 1 Multiple Integrals
1 14
Remark 2. Since the Cauchy criterion for existence of a limit is valid in any complete metric space, the sufficiency part of the Lebesgue criterion ( but not the necessity part ) , as the proof shows, holds for functions with values in any complete normed vector space. 1 1 . 1 . 3 The Darboux Criterion
Let us consider another useful criterion for Riemann integrability of a func tion, which is applicable only to real-valued functions.
f be a real-valued function on I and P {Ii } a partition of the interval I. We set mi inf f(x), Mi sup f(x) .
a. Lower and Upper Darboux Sums Let
the interval
=
=
xe �
=
xe�
Definition 10. The quantities
s(f, P) = L miiii l and S(f, P) L M I Ii l i i are called the lower and upper Darboux sums of the function f over the interval I corresponding to the partition P of the interval. Lemma 5 . The following relations hold between the Darboux sums of a func tion f : I IR: a ) s(f, P) = inf a(f, P, �) :::; a(f, P, �) :::; sup a (f , P, �) S(f, P) ; � � b) if the partition P' of the interval I is obtained by refining intervals of the partition P, then s(f, P) :::; s(f, P') :::; S(f, P') :::; S(f, P) ; ) the inequality s(f, P1) :::; S(f, P2 ) holds for any pair of partitions P1 and P2 of the interval I. Proof Relations a ) and b) follow immediately from Definitions 6 and 10, =
---t
=
c
taking account , of course, of the definition of the greatest lower bound and least upper bound of a set of numbers. To prove c ) it suffices to consider the auxiliary partition P obtained by intersecting the intervals of the partitions P1 and P2 . The partition P can be regarded as a refinement of each of the partitions P1 and P2 , so that b) now implies s(f, PI ) :::; s(f, P) :::; S(f, P) :::; S(f, P2 ) . D b. Lower and Upper Integrals
lower and upper Darboux integmls of the function f : I are respectively inf S(f, P) , :1 = sup s( ! , P) , :1 supremum and infimum are taken over all partitions P of the
Definition 1 1 . The
I
---t
lR over the interval
where the interval I.
p
=
p
1 1 . 1 The Riemann Integral over an n-Dimensional Interval
1 15
It follows from this definition and the properties of Darboux sums exhib ited in Lemma 3 that the inequalities
s(f, P) ::; J ::; J ::; S(f, P)
P of the interval. (Darboux) . For any bounded function f : I -+ JR., s(f, P) = -:r ) ; s(f, P) ) 1\ ( .>. (lim (3 .>. (lim P ) -+ 0 P ) -+ 0 S(f, P) = :r ) . S(f, P) ) 1\ ( .>. ( lim ( 3 .>. (lim P ) -+ 0 P ) -+ 0
hold for any partition Theorem 2 .
Proof. If we compare these assertions with Definition 1 1 , it becomes clear that in essence all we have to prove is that the limits exist . We shall verify this for the lower Darboux sums. Fix E: > 0 and a partition PE of the interval I for which s(f; PE) > J - E:. Let rE be the set of points of the interval I lying on the boundary of the intervals of the partition PE. As follows from Example 2, rE is a set of measure zero. Because of the simple structure of rE, it is even obvious that there exists a number >.E such that the sum of the volumes of those intervals that intersect rE is less than E: for every partition p such that >.(P) < Now taking any partition P with mesh >.(P) < >.E, we form an auxiliary partition P ' obtained by intersecting the intervals of the partitions P and PE. By the choice of the partition PE and the properties of Darboux sums (Lemma 5), we find
Ag.
s(f, PE) < s(f, P') ::; J . We now remark that the sums s(f, P ' ) and s(f, P) both contain all the terms that correspond to intervals of the partition P that do not meet rE. Therefore, if l f(x) l ::; M on I, then I s(!, P') - s(f, P) I < 2ME: J-
E: <
and taking account of the preceding inequalities, we thereby find that for >.(P) < >.E we have the relation J-
s(f, P) < (2M + l)c .
Comparing the relation just obtained with Definition 1 1 , we conclude that the limit lim s(f, P) does indeed exist and is equal to J. .>. ( P ) -+ 0 Similar reasoning can be carried out for the upper sums. D
1 16
1 1 Multiple Integrals
c. The Darboux Criterion for Integrability of a Real-valued Function Theorem 3. ( The Darboux criterion ) . A
real-valued function f I -+ JR. defined on an interval I JR.n is integrable over that interval if and only if it is bounded on I and its upper and lower Darboux integrals are equal. Thus, f E R(J) (f is bounded on I) 1\ (.J = .J) . Proof. N e c e s s i t y. If f E R(I), then by Proposition 1 the function f is bounded on I. It follows from Definition 7 of the integral, Definition 1 1 of C
:
-<====>
the quantities .J and .J, and part a ) of Lemma 5 that in this case .J = .J. S u f f i c i e n c y. Since when .J = .J, the extreme terms in these inequalities tend to the same limit by Theorem 2 as -+ 0. Therefore has the same limit as -+ 0. D
s( f, P) ::::; a( f, PJ ,) ::::; S( f, P) a( f, PJ,) >-.(P)
>-.(P)
Remark 3. It is clear from the proof of the Darboux criterion that if a function is integrable, its lower and upper Darboux integrals are equal to each other and to the integral of the function. 1 1 . 1 . 4 Problems and Exercises
a) Show that a set of measure zero has no interior points. b) Show that not having interior points by no means guarantees that a set is of measure zero. c) Construct a set having measure zero whose closure is the entire space JR n . d) A set E C I is said to have content zero if for every c > 0 it can be covered k by a finite system of intervals h , . . . , Ik such that I: I I; I < c. Is every bounded set i=l of measure zero a set of content zero? Show that if a set E C JR n is the direct product lR x e of the line lR and a set n- l o f (n - I)-dimensional measure zero, then E is a set of n-dimensional e C JR measure zero. 1.
a) Construct the analogue of the Dirichlet function in JR n and show that a bounded function f : I -+ lR equal to zero at almost every point of the interval I may still fail to belong to R(I) . b) Show that if f E R(I) and f(x) = 0, at almost all points of the interval I, then f f(x) dx = 0. 2.
I
3. There is a small difference between our earlier definition of the Riemann integral on a closed interval I C lR and Definition 7 for the integral over an interval of arbitrary dimension. This difference involves the definition of a partition and the measure of an interval of the partition. Clarify this nuance for yourself and verify that
1 1 .2 The Integral over a Set b
I f(x) dx and
I b
a
f ( x) dx =
=
I I
1 17
f(x) dx , if a < b
- I f ( x) dx , I
if a > b ,
where I is the interval on the real line R with endpoints a and b.
4. a) Prove that a real-valued function f : I R defined on an interval I C R n is integrable over that interval if and only if for every > 0 there exists a partition P of I such that S(f; P) - s(f; P) < b) Using the result of a) and assuming that we are dealing with a real-valued function f : I R, one can simplify slightly the proof of the sufficiency of the Lebesgue criterion. Try to carry out this simplification by yourself. -+
E.
E
-+
1 1 . 2 The Integral over a Set 11.2.1 Admissible Sets In what follows we shall be integrating functions not only over an interval, but also over other sets in R n that are not too complicated. Definition 1. A set E
C
Rn is admissible if it is bounded in Rn and its
boundary is a set of measure zero (in the sense of Lebesgue) .
Example 1 .
sets.
A cube, a tetrahedron, and a ball in
R3
(or
Rn ) are admissible
Example 2. Suppose the functions 'Pi : I -t R, i = 1, 2, defined on an ( n - I ) dimensional interval I c R n are such that cp 1 ( x ) < cp 2 ( x ) at every point x E I. If these functions are continuous, Example 2 of Sect. 1 1 . 1 makes it possible to assert that the domain in R n bounded by the graphs of these functions and the cylindrical lateral surface lying over the boundary 8I of I is an admissible set in R n .
We recall that the boundary 8E of a set E C R n consists of the points x such that every neighborhood of x contains both points of E and points of the complement of E in R n . Hence we have the following lemma.
Lemma 1. For any sets E, E1 , En 2 ) 8E is a closed subset of R ; b) 8(E1 E2 ) c 8E1 8E2 ; c) 8(E1 n E2 ) c 8E1 8E2 ; d) 8(E1 \ E2 ) c 8E1 8E2 . a
U
U
U
u
C
Rn , the following assertions hold:
1 18
1 1 Multiple Integrals
This lemma and Definition 1 together imply the following lemma.
The union or intersection of a finite number of admissible sets is an admissible set; the difference of admissible sets is also an admissible set. Remark 1. For an infinite collection of admissible sets Lemma 2 is generally
Lemma 2.
not true, and the same is the case with assertions b) and c) of Lemma 1 .
Remark 2. The boundary of an admissible set is not only closed, but also bounded in � n , that is, it is a compact subset of � n . Hence by Lemma 3 of Sect. 1 1 . 1 , it can even be covered by a finite set of intervals whose total content (volume) is arbitrarily close to zero.
{
We now consider the characteristic function X E (x) =
1 ' if x E E ,
0 , if X tj_ E ' of an admissible set E. Like the characteristic function of any set E, the function X E (x) has discontinuities at the boundary points of the set E and at no other points. Hence if E is an admissible set, the function X E (x) is continuous at almost all points of � n . 1 1 . 2. 2 The Integral over a Set Let be a function defined on a set E. We shall agree, as before, to denote the function equal to f (x) for x E E and to 0 outside E by fX E (x) (even though f may happen to be undefined outside of E) .
f
Definition 2. The
integral of f over E is given by
J
E where
f(x) dx : =
J
fX E (x) dx ,
I� E
I is any interval containing E.
If the integral on the right-hand side of this equality does not exist, we say that f is (Riemann) nonintegrable over E. Otherwise f is (Riemann) integrable over E. The set of all functions that are Riemann integrable over E will be denoted R(E). Definition 2 of course requires some explanation, which is provided by the following lemma.
119
1 1 . 2 The Integral over a Set Lemma 3.
the integrals
If h and I2 are two intervals, both containing the set E, then j fxe(x) dx and j fxe(x) dx h
I,
either both exist or both fail to exist, and in the first case their values are the same. Proof. Consider the interval I = h I2 . By hypothesis I ::) E. The points of discontinuity of fxe are either points of discontinuity of f on E, or the result of discontinuities of xe, in which case they lie on &E. In any case, all these points lie in I = h h . By Lebesgue ' s criterion (Theorem 1 of Sect . 11.1) it follows that the integrals of f over the intervals I, h , and h either all exist or all fail to exist. If they do exist, we may choose partitions of I, h , and I2 to suit ourselves. Therefore we shall choose only those partitions of h and I2 obtained as extensions of partitions of I = h I2 . Since the function is zero outside I, the Riemann sums corresponding to these partitions of h and I2 reduce to Riemann sums for the corresponding partition of I. It then results from passage to the limit that the integrals over h and h are equal to the integral of the function in question over I. D Lebesgue ' s criterion (Theorem 1 of Sect . 11.1) for the existence of the integral over an interval and Definition 2 now imply the following theorem. Theorem 1 . A function f E lR is integrable over an admissible set if and only if it is bounded and continuous at almost all points of E. Proof. Compared with f, the function fxe may have additional points of n
n
XE
n
---+
:
discontinuity only on the boundary &E of E, which by hypothesis is a set of measure zero. D 1 1 .2.3 The Measure (Volume) of an Admissible Set
Definition 3. The (Jordan) IS
measure or content of a bounded set E JRn f..l ( E) := 1 · dx
J
C
,
E
provided this Riemann integral exists. Since
J1
E
°
dx =
J xe(x) dx '
I�E
and the discontinuities of X E form the set &E, we find by Lebesgue ' s criterion that the measure just introduced is defined only for admissible sets.
120
11 Multiple Integrals
Thus admissible sets, and only admissible sets, are measurable in the sense of Definition 3. Let us now ascertain the geometric meaning of f..L ( E) . If E is an admissible set then f..L ( E) =
J XE x
( ) dx =
I�E
JJ�E XE (x)
dx =
J XE (x)
I�E
dx ,
where the last two integrals are the upper and lower Darboux integrals re spectively. By the Darboux criterion for existence of the integral (Theorem 3) the measure f..L ( E) of a set is defined if and only if these lower and upper integrals are equal. By the theorem of Darboux (Theorem 2 of Sect . 11.1 ) they are the limits of the upper and lower Darboux sums of the function X E corresponding to partitions P of I. But by definition of X E the lower Darboux sum is the sum of the volumes of the intervals of the partition P that are entirely contained in E (the volume of a polyhedron inscribed in E) , while the upper sum is the sum of the volumes of the intervals of P that intersect E (the volume of a circumscribed polyhedron) . Hence f..L ( E) is the common limit as >.(P) ---+ 0 of the volumes of polyhedra inscribed in and circumscribed about E, in agreement with the accepted idea of the volume of simple solids E c �n . For n = 1 content is usually called length, and for n = 2 it is called area.
Remark 3. Let us now explain why the measure f..L ( E) introduced in Definition 3 is sometimes called Jordan measure. Definition 4 . A set E �n is a set of measure zero in the sense of Jordan or a set of content zero if for every e > 0 it can be covered by a finite system k of intervals h , . . . , h such that I: I Ii I < e. i= l C
Compared with measure zero in the sense of Lebesgue, a requirement that the covering be finite appears here, shrinking the class of sets of Lebesgue measure zero. For example, the set of rational points is a set of measure zero in the sense of Lebesgue, but not in the sense of Jordan. In order for the least upper bound of the contents of polyhedra inscribed in a bounded set E to be the same as the greatest lower bound of the contents of polyhedra circumscribed about E (and to serve as the measure f..L ( E) or content of E) , it is obviously necessary and sufficient that the boundary 8E of E have measure 0 in the sense of Jordan. That is the motivation for the following definition. Definition 5 . A set E is Jordan-measurable if it is bounded and its bound ary has Jordan measure zero.
1 1 .2 The Integral over a Set
121
As Remark 2 shows, the class of Jordan-measurable subsets is precisely the class of admissible sets introduced in Definition 1. That is the reason the measure J.L(E) defined earlier can be called (and is called ) the Jordan measure of the (Jordan-measurable ) set E . 11.2.4 Problems and Exercises 1. a) Show that if a set E C JR:. n is such that p, (E) = 0, then the relation p,(E) = 0 also holds for the closure E of the set. b) Give an example of a bounded set E of Lebesgue measure zero whose closure E is not a set of Lebesgue measure zero. c) Determine whether assertion b) of Lemma 3 in Sect. 1 1 . 1 should be under stood as asserting that the concepts of Jordan measure zero and Lebesgue measure zero are the same for compact sets. d) Prove that if the projection of a bounded set E C JR:. n onto a hyperplane JR:. n - l has ( n - 1 )-dimensional measure zero, then the set E itself has n-dimensional measure zero. e) Show that a Jordan-measurable set whose interior is empty has measure 0. 2. a) Is it possible for the integral of a function f over a bounded set E, as in troduced in Definition 2, to exist if E is not an admissible (Jordan-measurable) set? b) Is a constant function f : E --+ JR:. integrable over a bounded but Jordan nonmeasurable set E? c) Is it true that if a function f is integrable over E, then the restriction f I A of this function to any subset A C E is integrable over A? d) Give necessary and sufficient conditions on a function f : E --+ JR:. defined on a bounded (but not necessarily Jordan-measurable) set E under which the Riemann integral of f over E exists. 3. a) Let E be a set of Lebesgue measure 0 and f : E --+ JR:. a bounded continuous function on E. Is f always integrable on E? b) Answer question a) assuming that E is a set of Jordan measure zero. c) What is the value of the integral of the function f in a) if it exists? 4. The Brunn-Minkowski inequality. Given two nonempty sets A, B C JR:. n , we form their (vector) sum in the sense of Minkowski A + B : = {a + bl a E A, b E B } . Let V(E) denote the content of a set E C JR:. n . a) Verify that if A and B are standard n-dimensional intervals (parallelepipeds) , then V l / n (A + B) 2 V l / n (A) + V l / n (B) . b) Now prove the preceding inequality (the Brunn-Minkowski inequality) for arbitrary measurable compact sets A and B. c) Show that equality holds in the Brunn-Minkowski inequality only in the following three cases: when V(A + B) = 0, when A and B are singleton (one-point) sets, and when A and B are similar convex sets.
11 Multiple Integrals
122
1 1 .3 General Properties of the Integral 1 1 .3 . 1 The Integral as a Linear Functional
a) The set R(E) of functions that are Riemann-integrable over a bounded set E �n is a vector space with respect to the standard operations of addition of functions and multiplication by constants. b) The integral is a linear functional
Proposition 1 .
c
J:
R(E) --+ �
on the set R(E) .
E
Proof.
Noting that the union of two sets of measure zero is also a set of measure zero, we see that assertion a) follows immediately from the definition of the integral and the Lebesgue criterion for existence of the integral of a function over an interval. Taking account of the linearity of Riemann sums, we obtain the linearity of the integral by passage to the limit. D
Remark 1 . If we recall that the limit of the Riemann sums as >-. (P) --+ 0 must be the same independently of the set of distinguished points �, we can conclude that ( ! E R(E) ) 1\ ( f(x) = 0 almost everywhere on E)
====?
( J f(x) dx = 0) . E
Therefore, if two integrable functions are equal at almost all points of E, then their integrals over E are also equal. Hence if we pass to the quotient space of R( E) obtained by identifying functions that are equal at almost all points of E, we obtain a vector space R(E) on which the integral is also a linear function. 1 1 .3.2 Additivity of the Integral Although we shall always be dealing with admissible sets E C �n , this as sumption was dispensable in Subsect. 1 1 .3. 1 (and we dispensed with it) . From now on we shall be talking only of admissible sets.
Let E1 and E2 be admissible sets in �n and f a function defined on E1 E2 . a) The following relations hold:
Proposition 2. U
(3
J
E 1 U E2
f(x) dx)
�
(3 J f(x) dx) 1\ (3 J f(x) dx) 3 J E,
E2
====?
E , n E2
f(x) dx .
1 1 .3 General Properties of the Integral
b) If in addition it is known that J..L ( El holds when the integmls exist:
n
123
= 0, the following equality J f(x) dx = J f(x) dx J f(x) dx . E2 )
+
Proof. Assertion a ) follows from Lebesgue ' s criterion for existence of the Rie mann integral over an admissible set ( Theorem 1 of Sect . 11.2). Here it is only necessary to recall that the union and intersection of admissible sets are also admissible sets ( Lemma 2 of Sect. 11.2). To prove b) we begin by remarking that
XE1UE2 = XE1 (x ) XE2 (x ) - XE1nE2 (x ) · Therefore, j f (x) dx = j f XE1 u E2 (x) dx = = J fXE1 (x) dx J fXE2 (x) dx - J XE1nE2 (x ) dx = = J f(x) dx J f(x) dx . +
+
I
I
I
+
The essential point is that the integral
J fXE1nE2 (x ) dx = J f(x) dx , E1nE2 as we know from part ) exists; and since J..L ( E l E2 ) = 0, it equals zero ( see Remark 1). 0 I
n
a ,
1 1 .3.3 Estimates for the Integral a . A General Estimate We begin with a general estimate of the integral that is also valid for functions with values in any complete normed space. Proposition 3. E R(E) , then E R(E) , and the inequality
Iff
holds. Proof.
lfl
lfl
I EJ f(x) dx l EJ l f l(x) dx ::;
The relation E R(E) follows from the definition of the integral over a set and the Lebesgue criterion for integrability of a function over an interval. The inequality now follows from the corresponding inequality for Riemann sums and passage to the limit . 0
1 24
1 1 Multiple Integrals
b. The Integral of a Nonnegative Function The following propositions
apply only to real-valued functions.
The following implication holds for a function f : E -+ lR : f(x) dx 2 0 . (! E R(E) ) (Vx E E (f(x) 2 0)) E Proof. Indeed, if f(x) 2 0 on E, then f X E (x) 2 0 in lRn . Then, by definition, f(x) dx = fX E (x) dx . E l"JE
Proposition 4.
A
====>
J
J
J
This last integral exists by hypothesis. But it is the limit of nonnegative Riemann sums and hence nonnegative. D From Proposition 4 just proved, we obtain successively the following corol laries. Corollary 1 . A
( f, g E R(E) ) (f ::=; g on E)
====>
( E/ f(x) dx :::; EJ g(x) dx) .
Corollary 2. If f E R(E) and the inequalities m :::; f(x) :::; M hold at every point of the admissible set E, then mJ.L (E) :::; f(x) dx :::; M J.L (E) . E Corollary 3. If f E R(E), m = inf f(x), and M = sup f(x), then there is x EE x EE a number () E [m, M] such that f(x) dx = O J.L (E) . E Corollary 4. If E is a connected admissible set and the function f : E -+ lR is continuous, then there exists a point � E E such that f(x) dx = f(�) J.L (E) . E Corollary 5 . If in addition to the hypotheses of Corollary 2 the function g E R( E ) is nonnegative on E, then m g(x) dx :::; fg(x) dx ::=; M g(x) dx . E E E
J
J
J
J
J
J
1 1 . 3 General Properties of the Integral
125
Corollary 4 is a generalization of the one-dimensional result and is usually called by the same name, that is, the mean-value theorem for the integral.
Proof. Corollary 5 follows from the inequalities mg ( x ) :::; f(x)g(x) :::; Mg ( x ) taking account of the linearity of the integral and Corollary 1 . It can also be proved directly by passing from integrals over E to the corresponding integrals over an interval, verifying the inequalities for the Riemann sums, and then passing to the limit. Since all these arguments were carried out in detail in the one-dimensional case, we shall not give the details. We note merely that the integrability of the product f · g of the functions f and g obviously follows from Lebesgue ' s criterion. D We shall now illustrate these relations in practice, using them to verify the following very useful lemma.
Linterval If the integral of a nonnegative function f : I -+ � over the emma.I a)equals zero, then f(x) = 0 at almost all points of the interval I.
b) Assertion a ) remains valid if the interval I in it is replaced by any admissible (Jordan-measurable) set E. Proof. B y Lebesgue ' s criterion the function f E R(E) i s continuous at almost all points of the interval I. For that reason the proof of a) will be achieved if we show that f(a) 0 at each point of continuity a E I of the function f. Assume that f(a) > 0. Then f(x) 2: c > 0 in some neighborhood U1 (a) =
of a (the neighborhood may be assumed to be an interval) . Then, by the properties of the integral just proved,
J f(x) dx = J I
Ur (a)
f(x) dx +
J
I\ Ur (a)
f(x) dx 2:
J
f(x) dx 2: ct-t ( UI (a) ) > 0 .
Ur (a)
This contradiction verifies assertion a) . If we apply this assertion to the function fX E and take account of the relation t-t( aE) = 0, we obtain assertion D
b). Remark 2.
It follows from the lemma just proved that if E is a Jordan measurable set in �n and R (E) is the vector space considered in Remark 1 , consisting of equivalence classes of functions that are integrable over E and differ only on sets of Lebesgue measure zero, then the quantity IIJII = J l f l (x) dx is a norm on R (E).
E Proof.
Indeed, the inequality J l f l (x) dx = 0 now implies that E same equivalence class as the identically zero function. D
f is in the
126
11 Multiple Integrals
1 1 .3.4 Problems and Exercises 1 . Let E be a Jordan-measurable set of nonzero measure, f : E -+ lR a continuous nonnegative integrable function on E, and M = sup f(x) . Prove that
}!_.�
2.
( I r (x) dx)
xEE
11n
E
=
M .
Prove that if j, g E R ( E ) , then the following are true. a ) Holder 's inequality
II
(f · g) (x) dx
E
l ( I I J I P (x) x) 11P ( I lgl q (x) dx) 1/q , d
S
where p 2: 1 , q 2: 1 , and � + � b ) Minkowski 's inequality
E
=
E
1;
( 1/p 1/ p ( I I f + giP dx) I IJ IP (x) dx) + ( I lgiP (x) dx) , 1/ p
E
S
E
E
if p :?: l . Show that c ) the preceding inequality reverses direction if 0 < p < 1 ; d ) equality holds i n Minkowski's inequality i f and only i f there exists >. 2: 0 such that one of the equalities f = >.g or g = >.j holds except on a set of measure zero in E; 11 e ) the quantity II J I I P = CJE) £ IJIP (x) dx ) P , where �-t ( E ) > 0, is a monotone function of p E lR and is a norm on the space R( E ) for p 2: 1 . Find the conditions under which equality holds i n Holder's inequality. 3. Let E be a Jordan-measurable set in IR n with �-t ( E ) > 0. Verify that if t.p E C ( E, IR) and f : lR -+ lR is a convex function, then
( � [ t.p (x) dx)
f J-t ( )
S
� [ (f
J-t ( )
o
t.p ) (x) dx .
4. a) Show that if E is a Jordan-measurable set in IR n and the function f : E -+ lR is integrable over E and continuous at an interior point a E E, then 1 lim f(x) dx = f ( a ) , 6--++ 0 J-t ua ( a ) ) E U� ( a )
(
I
where, as usual, U� ( a ) is the 8-neighborhood of the point in E. b) Verify that the preceding relation remains valid if the condition that a is an interior point of E is replaced by the condition �-t U� ( a ) ) > 0 for every 8 > 0.
(
1 1 .4 Reduction of a Multiple Integral to an Iterated Integral
127
1 1 .4 Reduction of a Multiple Integral to an Iterated Integral 1 1 .4 . 1 Fubini's Theorem
Up to now, we have discussed only the definition of the integral, the conditions under which it exists, and its general properties. In the present section we shall prove Fubini ' s theorem, 4 which, together with the formula for change of variable, is a tool for computing multiple integrals.
5 Let X X Y be an interval in JRm + n , which is the direct product of intervals X JRm and Y Rn . If the function f : X x Y -+ lR is integrable over X x Y, then all three of the integrals
Theorem.
C
c
J f ( x, y ) dx dy , J dx J f ( x, y ) dy , J dy J f (x, y ) dx
XxY
y
X
exist and are equal.
y
X
Before taking up the proof of this theorem, let us decode the meaning of the symbolic expressions that occur in the statement of it. The integral J f (x, y ) dx dy is the integral of the function f over the set X x Y, which
XxY
we are familiar with, written in terms of the variables x E X and y E Y. The iterated integral J dx J f (x, y ) dy should be understood as follows: For each X
y
E X the integral F(x) = J f (x, y ) dy is computed, and the resulting function F : X -+ lR is then to be integrated over X. If, in the process, the integral J f ( x, y ) dy does not exist for some x E X, then F ( x ) is set equal to any value between the lower and upper Darboux integrals .J (x ) = 1_ f (x, y ) dy and .J (x ) = J f (x , y ) dy, including the upper and lower integrals .J (x ) and .J (x ) themselves. It will be shown that in that case F E R(X ) . The iterated integral J limitsy dy J f (x, y ) dx has a similar meaning.
fixed x
y
y
y
y
X
It will become clear in the course of the proof that the set of values of x E X at which .J ( x ) =f. .J ( x ) is a set of m-dimensional measure zero in X . 4 G. Fubini ( 1870-1943) - Italian mathematician. His main work was in the area of the theory of functions and geometry. 5 This theorem was proved long before the theorem known in analysis as Fubini's theorem, of which it is a special case. However, it has become the custom to refer to theorems making it possible to reduce the computation of multiple integrals to iterated integrals in lower dimensions as theorems of Fubini type, or, for brevity, Fubini's theorem.
128
1 1 Multiple Integrals
Similarly, the set of y E Y at which the integral J f(x, y) dx may fail to X
exist will turn out to be a set of n-dimensional measure zero in Y. We remark finally that, in contrast to the integral over an ( m + n ) dimensional interval, which we previously agreed to call a multiple integral, the successively computed integrals of the function f(x, y) over Y and then over X or over X and then over Y are customarily called iterated integrals of the function. If X and Y are closed intervals on the line, the theorem stated here theoretically reduces the computation of a double integral over the interval X x Y to the successive computation of two one-dimensional integrals. It is clear that by applying this theorem several times, one can reduce the computation of an integral over a k-dimensional interval to the successive computation of k one-dimensional integrals. The essence of the theorem we have stated is very simple and consists of the following. Consider a Riemann sum l:: f(x i , Yj) I Xi l · l }j l corresponding
i ,j
to a partition of the interval X X into intervals xi X }j. Since the integral over the interval X x Y exists, the distinguished points f.ij can be chosen as we wish, and we choose them as the "direct product" of choices X i E Xi c X and Yj E }j E Y . We can then write
y
j
i ,j
j
and this is the prelimit form of theorem. We now give the formal proof.
Proof. Every partition P of the interval X x Y is induced by corresponding partitions Px and Py of the intervals X and Y. Here every interval of the partition p is the direct product Xi X }j of certain intervals Xi and }j of the partitions Px and Py respectively. By properties of the volume of an interval we have I Xi x }j l = I Xi i · I YJ I , where each of these volumes is computed in or in which the interval in question is situated. the space Using the properties of the greatest lower bound and least upper bound and the definition of the lower and upper Darboux sums and integrals, we now carry out the following estimates:
JR_m+n , JR_m , JR_n
(
}
s(f, P) L: }�L f(x, y) I Xi X }j l � L x�l L yi�$1 f(x, y) I YJ I Xi l � =
t,J y E Yj �
}
'
J
L }fl ( jf(x, y) dy Xi l � L }�l F(x) I Xi l � •
y
� l: :�I F(x) I Xi l � l: :�* i •
•
•
(] f(x, y) dy) I Xi l � y
1 1 .4 Reduction of a Multiple Integral to an Iterated Integral
(
)
129
L :�J, 2: :�� F ( x , y ) l YJ l l Xi l � ' J sup f(x, y ) l Xi x Yj l S(f, P) . � L �
=
i 1· x E X; ' y E Yj
Since f E R(X x Y), both of the extreme terms in these inequalities tend to the value of the integral of the function over the interval X x Y as .A(P) ---+ 0. This fact enables us to conclude that E R(X ) and that the following equality holds: f(x, y) dx dy = dx .
F
J
J F(x)
X
XxY
We have carried out the proof for the case when the iterated integration is carried out first over Y, then over X . It is clear that similar reasoning can be used in the case when the integration over X is done first . D 11.4.2 Some Corollaries
If f E R(X x Y), then for almost all x E X (in the sense of Lebesgue) the integral J f(x, y) dy exists, and for almost all y E Y the integral y J f(x, y) dx exists. Proof. By the theorem just proved, J f(x, y) dy - J f(x, y) dy dx = O .
Corollary 1.
X
(]
)
y
y
X
But the difference of the upper and lower integrals in parentheses is non negative. We can therefore conclude by the lemma of Sect . 1 1 .3 that this difference equals zero at almost all points x E X . Then by the Darboux criterion (Theorem 3 of Sect . 1 1 . 1 ) the integral J f(x, y) dy exists for almost all values of x E X.
y
The second half of the corollary is proved similarly. D Corollary 2. If the interval I C JR n is the direct product of the vals Ii = [ai , bi ], i 1 , . . . , n, then
closed inter
=
bn
bn -
J f(x) dx J dxn J =
I
an
an - 1
1
dx n l -
· · ·
bl
J f(x 1 , x2 , . . . , xn ) dx 1 .
al
Proof. This formula obviously results from repeated application of the the orem just proved. All the inner integrals on the right-hand side are to be understood as in the theorem. For example, one can insert the upper or lower integral sign throughout . D
1 1 Multiple Integrals
130
f(x,y, z) = zsin(x + y). I I Y I :::; 7r2/2 , z :::; 1 71" / 2 71" Ill f(x, y, z) dxdydz = I dz I dy I zsin(x + y) dx = I 71" / 2 1 71" / 2 1 71" / 2 = I dz I ( - zcos(x + y)l :=0 ) dy = I dz I 2zcosydy = - 71" / 2 - 71"/ 2 1 1 = I (2zsiny l �::!71"212 ) dz = I 4zdz = 2 .
Example 1 . Let We shall find the integral of the restriction of this function to the interval C JR 3 defined by the relations 1. 0 :::; X :::; 7f , 0 :::; B y Corollary 0
0
0
0
0
0
The theorem can also be used to compute integrals over very general sets.
be a bounded set in !Rn - 1 and E {(x, y) E !Rn l (x E (
Corollary 3.
D ) 1\
'P2 ( x )
Ex =
Ex =
The inner integral here may also fail to exist on a set of points in D of Lebesgue measure zero, and if so it is assigned the same meaning as in the theorem of Fubini proved above. D
Remark. If the set D in the hypotheses of Corollary 3 is Jordan-measurable and the functions : D -+ i 1 , are continuous, then the set E C is Jordan measurable.
IR, = 2 ,
!Rn
131
1 1 .4 Reduction of a Multiple Integral to an Iterated Integral
Proof. The boundary 8E of E consists of the two graphs of the continuous functions 'Pi : D ---+ JR, i = I, ( which by Example of Sect. Il.I) are sets of measure zero ) and the set Z, which is a portion of the product of the boundary aD of D c JR - l and a sufficiently large one-dimensional closed interval of length l. By hypothesis 8 D can be covered by a system of ( n - I) dimensional intervals of total ( n - I ) -dimensional volume less than Ejl . The direct product of these intervals and the given one-dimensional interval of length l gives a covering of Z by intervals whose total volume is less than c .
2
2,
n
0
Because of this remark one can say that the function f : E ---+ I E lR is integrable on a measurable set E having this structure ( as it is on any measurable set E) . Relying on Corollary 3 and the definition of the measure of a measurable set, one can now derive the following corollary.
If under the hypotheses of Corollary 3 the set D is Jordan measurable and the functions 'Pi : D ---+ JR, i I, 2 , are continuous, then the set E is measurable and its volume can be computed according to the formula (11.2) J-L(E) J ('P 2 (x) -
Corollary 4.
=
=
D
Example 2.
formula J-L(E)
=
=
For the disk E
=
{(x, y) E JR2 j x 2 + y 2 ::; r 2 } we obtain by this
r
r
J ( Jr2 - y2 - ( - Jr2 - y2 )) dy 2 J Jr2 - y2 dy
-r
=
r
-rr / 2
-r
-rr / 2
=
4 J Jr2 - y2 dy 4 J r cos
0
=
0
0
=
C
Let E be a measurable set contained in the interval I ]Rn . Represent I as the direct product I Ix x Iy of the ( n - I)-dimensional interval Ix and the closed interval Iy . Then for almost all values y0 E Iy the section Ey0 { (x, y) E E l y yo} of the set E by the ( n - I)-dimensional hyperplane y Yo is a measurable subset of it, and J-L(E) = J J-L(Ey ) dy , (I1.3) Corollary 5.
=
=
=
=
Iy
where J-L(Ey ) is the ( n - I)-dimensional measure of the set Ey if it is mea surable and equal to any number between the numbers J I · dx and J I · dx if Ey happens to be a nonmeasurable set. Ey
Ey
132
11 Multiple Integrals
Proof. Corollary 5 follows immediately from the theorem and Corollary 1, if we set(x).f =0 XE in both of them and take account of the relation XE (x, y) = X Ey A particular consequence of this result is the following. Corollary 6. ( Cavalieri ' s principle. ) 6 Let A and B be two solids in JR.3 having volume (that is, Jordan-measurable). Let Ae = {(x, y, z) E A l z = c} and Be = {(x, y, z) E B l z = c} be the sections of the solids A and B by the plane z = c. If for every c E JR. the sets Ae and Be are measurable and have the same area, then the solids A and B have the same volumes. It is clear that Cavalieri ' s principle can be stated for spaces JR.n of any dimension. Example Using formula (11.3), let us compute the volume V of the ball B = { x E JR.n l l x l ::::; r} of radius r in the Euclidean space JR.n . n It is obvious that V = 2 . In Example 2 we found that V2 = r2 . We 1 n shall show that Vn = Cn r , where Cn is a constant (which we shall compute below ) . Let us choose some diameter [ - r, r] of the ball and for each point x E [-r, r] consider the section Bx of the ball B by a hyperplane orthogonal toPythagorean the diameter.theorem, Since Bequals x is a ball of dimension n - 1, whose radim;, by the y'r 2 - x 2 , proceeding by induction and using (11.3), we can write /2 n l 2 2 Vn = J Cn � I (r - x ) ;- dx = (cn � l J cosn ipdip) rn . �-rr/ 2 ( In passing to the last equality, as one can see, we made the change of variThus able xwe= have r sin ip. ) shown that Vn = Cn rn , and -rr/ 2 ( 11.4) Cn = Cn �l J COSn ipdip . �-rr/ 2 We now find the constant Cn explicitly. We remark that for m ::::>: 2 -rr/ 2 -rr /2 Im = J cosrn ip dip = J cosm � 2 ip ( 1 - sin2 ip ) dip = � -rr/ 2 �-rr /2 -rr /2 1 1 �l = lm � 2 + -m - 1 J sin ip d cosm ip = Irn� 2 - m - 1 Im , �-rr/ 2 3.
n
1r
r
�r
--
6
B. Cavalieri { 1598� 1647) Italian mathematician, the creator of the so-called method of indivisibles for determining areas and volumes. �
1 1 .4 Reduction of a Multiple Integral to an Iterated Integral
that is, the following recurrence relation holds:
Im = mm- 1 Im - 2 · ---
133
(11.5)
In particular, h = /2. It is clear immediately from the definition of Im that h = 2. Taking account of these values of h and h we find by the recurrence formula (11.5) that - 1) !! 7r . (11.6) h k+l = (2k(2k+)!!1) !! · 2 , h k = (2k(2k)!! Returning to formula (11.4), we now obtain !! · 2 = C (2k) !! (2k - 1)!! 7r = . . = C1 . (2n) k C2k+l = C2k (2k(2k) 2k l ' (2k + 1) !! + 1)!! (2k + 1)!! ' (2k)!! - 1)!! 7r = C2k 2 (2k - 1)!! 7r· ( (2k - 2)!! 2 = · = C2 (2n) k - l · 2 C2k = C2k - l (2k(2k)!! - (2k) !! (2k 1)!! · ' ' (2k)!! · But, as we have seen above, c 1 = 2 and c2 = n, and hence the final 1r
_
formulas for the required volume Vn are as follows: k v2k+l - 2 (2k(2+7r ) 1)!! r 2k+l '
(11.7)
where k E N, and the first of these formulas is also valid for k = 0. 1 1 .4.3 Problems and Exercises 1. a) Construct a subset of the square I C JR 2 such that on the one hand its intersection with any vertical line and any horizontal line consists of at most one point, while on the other hand its closure equals I. b) Construct a function f : I -+ lR for which both of the iterated integrals that occur in Fubini's theorem exist and are equal, yet f tt R(I). c) Show by example that if the values of the function F(x) that occurs in Fubini's theorem, which in the theorem were subjected to the conditions :J(x) :S F(x) :S :J(x) at all points where :J(x) < :J(x) , are simply set equal to zero at those points, the resulting function may turn out to be nonintegrable. (Consider, for example, the function f(x, y) on JR2 equal to 1 if the point (x, y) is not rational and to 1 1 /q at the point (p/q, m/n) , both fractions being in lowest terms. ) -
2. a ) I n connection with formula ( 1 1 .3) , show that even i f all the sections o f a bounded set E by a family of parallel hyperplanes are measurable, the set E may yet be nonmeasurable. b) Suppose that in addition to the hypotheses of part a) it is known that the function Jt(Ey) in formula ( 1 1 .3) is integrable over the closed interval Iy. Can we assert that in this case the set E is measurable?
1 1 Multiple Integrals
134
3. Using Fubini's theorem and the positivity of the integral of a positive function, give a simple proof of the equality ::JY = ::Jx for the mixed partial derivatives, assuming that they are continuous functions.
4. Let f : Ia ,b --+ R be a continuous function defined on an interval Ia,b = {x E R n l a i :::; x i :::; bi , i 1 , . . . , n} , and let F : Ia ,b --+ R be defined by the equality =
F(x)
=
I
j(t) dt ,
Ia , x
where Ia , x C Ia,b · Find the partial derivatives of this function with respect to the variables x 1 , . . . , x n . 5 . A continuous function j(x, y) defined on the rectangle I = [a, b] x [c, d] C R2 has a continuous partial derivative U in /. a) Let F(y)
)
=
b
I j (x , y) dx. Starting from the equality F(y)
a
f (x , c ) dx , verify the Leibniz rule, according to which F' (y) b) Let G(x, y) c) Let H(y) 6.
=
X
=
=
a
y
c
l U Cx , y) dx.
I f(x, y) dx, where h E c <1 l [a, b] . Find H' (y) .
a
X
I f(y) dy ,
where f E C(R, R) . a ) Verify that F� (x) b) Show that
Fn (x)
Xl
=
=
X
I (x -n!y) n
Fn- 1 (x) , F�k \0)
d 2
..·
0
f(y) dy , n E N ,
0
Xn - 1
I dx 1 I x I f(xn ) dxn 0
b
h (y)
0
X
=
(
I I U Cx , t) dt+
I f(t, y) dt. Find �� and �; . a
Consider the sequence of integrals
Fo (x)
=
0
= =
0 i f k :::; n, and F�n+1) (x) X
�! l (x - Y tf(y) dy . 0
7. a) Let f : E --+ R be a function that is continuous on the set E R 2 1 0 :::; x :::; 1 t\ 0 :::; y :::; x}. Prove that X
I dx I 1
0
0
f(x, y) dy =
1
I dy I f(x, y) dx . 0
=
=
f(x) .
{ ( x, y) E
1
y
2tr s i n x b) Use the example of the iterated integral I dx I 1 dy to explain why not 0 0 every iterated integral comes from a double integral via Fubini's theorem. ·
1 1 .5 Change of Variable in a Multiple Integral
135
1 1 . 5 Change of Variable in a Multiple Integral 11.5.1 Statement of the Problem and Heuristic Derivation of the Change of Variable Formula
Inan our earlier formula study offorthechange integralof invariable the one-dimensional case, Our we obtained important in such an integral. problem now is to find a formula for change of variables in the general case. Let us makeLettheDxquestion more precise. be a set in JRn , f a function that is integrable over Dx , and Dt Dx a mapping t H r.p(t) of a set Dt JR n onto Dx . We seek a rule according to which, knowing f and we can find a function '1/J in Dt such that the equality j f(x) dx = j 'ljJ(t) dt holds, it possible reduceoverthe Dtcomputation of the integral over Dx to theWemaking computation of an tointegral . begin byofassuming that onto Dt is an interval I JRn and : I Dx a diffeomorphism this interval Dx . To every partition P of the interval , there corresponds a partition of Dxpairwise into theonly sets , I Ir.p(Iinto), iintervals I 1 2 = 1, . . . , k. If all these sets are measurable and intersect i of measure zero, then by the additivity of the integral we find in sets k (11.8) j f(x) dx = L j f(x) dx . t= l cp ( I; ) If f is continuous on Dx , then by the mean-value theorem j f(x) dx = f(�i )JL ( r.p(Ii ) ) , cp ( I; ) where �i E r.p(Ii ) · Since f(�i ) = f ( r.p(Ti ) ) , where Ti = r.p - 1 (�i ), we need only connect JL (r.p(Iai )linear ) with JL(Ii ) · If were transformation, thengeometry, r.p(Ji ) wouldwouldbe bea parallelepiped whose volume, as is known from analytic I detandr.p'I JLso,(Ii )if· But a diffeomorphism is locally a nearly linear transformation, dimensions Ii arerelative sufficiently small, webemay assume JLforthe( r.p(Isome )i ) choicel detr.pofof'the(Tthei )Ipoint Iintervals Ii l with small error (it can shown Ti E Ii actual equality will result). Thus that k k (11.9) L j f(x)dx L f ( r.p(Ti ) ) l det r.p' (Ti )I I Ii l · •=1 t=l cp ( � )
--7
C
D,
Dx
C
...,h
Dx
�
�
--7
1 1 Multiple Integrals
136
But,for thethe right-hand sidetheoffunction this approximate equality contains a Riemann integral of f ( -. (P)the partition 0 we obtain from (11.8) and (11.9) the relation I f(x) dx = I f (
-+
D.,
D,
:
c
:
-+
C
-+
bounded open set f E R(Dx ) , Rn of the'P sameof atype, Rn ontoIf a
Dt
C
I
C
o
f( x ) dx = I f
(11.10)
1 1 .5.2 Measurable Sets and Smooth Mappings -+
C
Let
C
C
C
7 Such functions are naturally called functions of compact support in the domain.
137
1 1 . 5 Change of Variable in a Multiple Integral
b) If a set Et contained in Dt along with its closure Et has Jordan measure zero, its image VJ(Et ) = Ex is contained in Dx along with its closure and also has measure zero. c) If a (Jordan) measurable set Et is contained in the domain Dt along with its closure Et , its image Ex = VJ(Et ) is Jordan measurable and Ex Dx . Proof. We begin by remarking that every open subset D in !Rn can be rep C
resentedhaveas any the interior union ofpoints a countable numberToofdoclosed intervals (noonetwocanof which in common). this, for example, partition the coordinate axesofinto closed intervals of length L1 and consider the corresponding partition !Rn into cubes with sides of length Fixing L1 =F 1,. Then take thetakingcubesL1 of= the partition contained in D.ofDenote theirpartition union bythat 1/2, adjoin to F the cubes the new 1 1 are contained in D \ F1 . In that way we obtain a new set F2 , and so forth. Continuing thisconsists process,of awefinite obtainor countable a sequencenumber F1 · · · Fn · · · of sets, each of which of intervals hav ing no interior points in common, and as one can see from the construction, U Fn = D. union zero, of an itatsuffices most countable collection of) for setsaofsetmeasure zero isa closed aSince set ofinthemeasure to verify assertion Et lying in terval I Dt . We shall now do this. Since E C ( l l (I) (that is, VJ1 E C (I)), there exists a constant M such that II VJ'(t) ll � M on I. By the finite-increment theorem the relation l x - x 1 l Mlt 2 - t 1 l must hold for every pair of points t 2 E I with images x 12 = VJ(t i ), X2 = VJ(t 2 ) · Now let { Ii } be a covering of Et by intervals such that 2::i I Ii I < E:. Without lossThe of generality we{ VJ(/may)}assume that Ii = Ii I I. collection i of sets VJ(/i ) obviously forms a covering of Ex = VJ(Et ) · If t i is the center of the interval h then by the estimate just given for the possible change in distances under the mapping VJ, the entire set VJ(/i ) can be covered by the interval L with center Xi = VJ(ti ) whose linear dimensions are M times those of the interval h Since I 'L l = Mn iii l , and VJ(Et ) ui L, weis lesshavethanobtained a covering of VJ(Et ) = Ex by intervals whose total volume Mn E:. Assertion ) is now established. Assertion b) follows from ) if we take into account the fact that E t hence by whatthathasEbeen(andproved, Ex = VJ(Et ) also) is a set of Lebesgue (and measure zero and hence also Ex ) is a compact set. Indeed, by t Lemma 3 of Sect. 11.1 every compact set that is of Lebesgue measure zero alsoFinally, has Jordan measurec) iszero. assertion ansetimmediate consequence of b),points if weofrecall the definition of a measurable and the fact that interior Et map to8Einterior points of its image Ex = VJ(Et ) under a diffeomorphism, so that VJ(8E ) D Ll.
C
C
a
C
£P
C
�
h,
n
C
c
a
a
x
=
t·
138
11 Multiple Integrals
Under the hypotheses of the theorem the integral on the right-hand side of formula (11.10) exists. Proof. Since I 1det cp' (t) I i:- 0 in D t , it follows that supp f o 'P · I det cp' l supp f o 'P ocp - (supp f) is a compactn subset in Dt . Hence the points at which the function f o 'P · I det cp' IX D, in JR is discontinuous have nothing to do with the function X D, , but are the pre-images of points of discontinuity of f in Dx . But isfaEsetR(Dof xLebesgue ) , and therefore the set Ex of points of discontinuity of f in by assertion a ) of the lemma zerozero.. ButBythen Dthe x set Et cp- 1 (Ex ) hasmeasure measure Lebesgue' s criterion, now conclude that f o 'P · I det cp' IXD, is integrable on any interval It weDcan t. D Corollary.
=
=
=>
1 1 . 5.3 The One-dimensional Case
a)
If 'P : It Ix 1is a diffeomorphism of a closed interval It JR 1 onto a closed interval Ix JR and f E R(J�,), then f o 'P · I 'P' I E R(It ) and (11.11) j f(x) dx j (f o 'P · I 'P' I )(t) dt .
Lemma 2.
--7
c
C
=
I,
b) Formula ( 11.10) holds in lR 1 . Proof. Although we essentially already know assertion a ) of this lemma, we
shall the Lebesgue criterion forproofthehere existence ofs independent an integral, which isproof now atgivenouruseindisposal, to give a short that i of the Part 1. Since f E R(Ix ) and 'P : It Ix is a diffeomorphism, the function fo'P I 'P' I is bounded on ft . Only thefunction preimages of points of discontinuity of f on Ix can discontinuities of the beform f o 'P I 'P' I · By Lebesgue ' s criterion, the latter a set of measure zero. The image of this set under the diffeomorphism cp - 1 : Ix --t It , as we saw in the proof of Lemma 1, has measure zero. Therefore f 'P I 'P' I E R(Jt ) · Now let Px be a partition of the closed interval IIx, . and Through the mapping cp - 1 it induces a partition Pt of the closed interval it follows from the t uniform continuity of the mappings 'P and cp - 1 that >.(Px ) 0 >. ( Pt ) 0. We now points write the Riemann sums for the partitions Px and Pt with distin guished �i cp(T; ): --7
0
--7
{:}
--7
=
L f(�; ) l x ; - X ; - 1 l L f 'P h ) I 'P(t; ) - cp(t;- 1 l L f o cp(T; ) I cp'( T;)I I t ; - t ; - 1 1 , =
0
=
=
andthethepoint pointsobtained assumed thechosen just so that T; �i tocp(theT; ),diwhere �i can bybe applying iscp(t mean-value theorem ff erence ) cp(t _ ). =
; - ; l
1 1 .5 Change of Variable in a Multiple Integral
139
both integrals in be( 11.11) exist, theourchoiceconvenience of the distinguished points inthetheSince Riemann sums can made to suit without affecting limit. Hence from the equalities just written for the Riemann sums, we findAssertion (11.11) for the integrals in the limit as >.(Px ) -+ 0 (>.(Pt ) -+ 0). b) of Lemma 2 follows from Eq. (11.11). We first note that in the one-dimensional case
o
D
1.
=
o
{ j f(x) dx,
•
if J f(x) dx - f f(x) dx, if a > b , hold, it becomes clear that when
I
a
b
a
2.
:
=
I,
o
o
1 1 Multiple Integrals
140
Given that(11.10) fact, weremains may take asforestablished thatfunction in the fone-dimensional case formula valid any bounded in it are understood as upper or lower Darboux integrals. if the integrals Proof. We shall assume temporarily that f is a nonnegative function bounded by aAgain, constant M. as in the proof of assertion ) of Lemma 2 , one may take partitions Px and Pt of the intervals Ix and It respectively that correspond to each other under theoscillation mapping of'P and write the following estimates,: in which E is the maximum 'P on intervals of the partition Pt L x supx f(x) l x i - X i - 1 1 L t supt f (tp(t)) t supt I 'P'(t) l l t i - t i - l l EL1 , i EL1 , i EL1 , L t supt ( f (tp(t)) · t supt I 'P'(t) l) IL1 t il EL1 , i EL1 , L sup ( f(tp(t)) ) (I'P'(t)l + s ) IL1t i l i t EL1 t , L t supt (! ( 'P( t)) I 'P' ( t) I) IL1 t i I + E L t supt f ( 'P( t) ) IL1 t i I i EL1 , i EL1 , sup ( f(tp(t)) I 'P'(t) I) IL1til + sM I It l . L t EL1 i t, Taking account of the uniform continuity of 'P we obtain from this the relation J f (x) dx J ( f 'P I 'P' I) ( t) dt asand>-.(Ptthe) function 0. Applying what has just been proved to the mapping 'P - l f 'P I 'P' I , we obtain the opposite inequality, and thereby establish the first equality in Remark 2 for a nonnegative function. But since any function can be written as fequality = max{can!, 0}be-considered max{- j, 0}to (abedifference of two nonnegative functions) the established in general. The second equality is verified similarly. D From ) theof Lemma equalities2 forjustreal-valued proved onefunctions can of course obtain once again as sertion f. a
:S:
:S:
:S:
:S:
:S:
:S:
:S:
:S:
:S:
:S:
--+
o
I,
o
a
1 1 .5.4 The Case of an Elementary Diffeomorphism in JR.n
Let 'P JR.�: DtwithDx(t 1 be, . . .a, tdiffeomorphism of a domain Dt lR.f onto a domain n ) and (x 1 , . . . , xn ) the coordinates of points t E lR.f and x E JR.� respectively. We recall the following definition. Definition 2. The diffeomorphism 'P Dt Dx is elementary if its coordi nate representation has the form Dx
C
--+
C
:
--+
1 1.5 Change of Variable in a Multiple Integral x 1 = IP 1 ( t l ' . . . ' t n ) = t 1 '
141
x k - 1 = ip k - 1 (t 1 , . . . , tn ) = t k - 1 , n . . . , tk , . . . , t ) ' x k +1 = ipk (t 1 , . . . , t n ) = t k +1
X k - ip k (t 1 , . . . , t n ) - ip k (t 1 , _
_
xn = ipn (t 1 ' . . . ' t n ) = tn .
only one coordinate is changed under an elementary diffeomorphism (theThus kth coordinate in this case). Lemma 3. Formula (11.10) holds for an elementary diffeomorphism. Proof. Up to a relabeling of coordinates we may assume that we are consider ing a diffeomorphism thatnotation: changes only the nth coordinate. For convenience we introduce the following (
1p
x 1 , . . . , xn - I , xn ) = : (x , x n ) ; (t 1 , . . . , t n - 1 , t n ) = : (t, t n ) ; Dxn (xo ) : = { (.X, x n ) E Dx l X = .Xo } ; Dt n (to ) : = {( l, t n ) E Dt l t = to } .
Thus Dxn (.X) and Dtn (t) are simply the one-dimensional sections of the sets Dx and Dt respectively by lines parallel to the nth coordinate axis. Let Ix be an interval in JR� containing Dx . We represent Ix as the direct product Ix = Ix X Ixn of an (n - I)-dimensional interval Ix and a closed interval Ixn ofa fixed the nthinterval coordinate axis. We give a similar representation It = ft x Itn for It in lRt' containing Dt . the definition of the integral over a set, Fubini 's theorem, and Re markUsing 2, we can write j f(x) dx = j f · xv,(x) dx = j dx j f · xvJ.X, xn ) dxn = Dx Ix Ix Ix n = j dx j f (x, x n ) dx n = I., D xn ( x) = J dt J J(l, ip n ( l, t n ) ) I ��: l(l, t n) dt n = It D,n = J dt J (! �P I det IP' I x v ,) ( l, t n ) dt n = I; I,n = j (/ o �Pi det !p' l xv, ) ( t ) dt = j ( / o �Pi de t !p 'l ) (t) dt . (t)
o
142
11 Multiple Integrals
In this computation we have used Dthe fact that det cp' q::L_ atn for the diffeomorphism under considertation. 1 1 .5.5 Composite Mappings and the Formula for Change of Variable Lemma 4. If Dr � D t � D x are two diffeomorphisms for each of which formula (11.10) for change of variable in the integral holds, then it holds also for the composition cp 7/J Dr -+ Dx of these mappings. Proof. It suffices to recall that ( cp 7/J)' = cp' 7/J' and that det ( cp 7/J)'(r) = det cp'(t) det 'ljJ'(r), where t = cp(r). We then have j f(x ) dx = j (f cp l det cp' l) dt = o
:
o
o
o
o
=
J ( f ( cp o
o
'I/J) I det ( cp 'I/J)' I) (r) dr . o
D
DT
1 1 .5.6 Additivity of the Integral and Completion of the Proof of the Formula for Change of Variable in an Integral
Lemmas 3 and 4assuggest that weofmight use thediffeomorphisms local decomposition of any diffeomorphism a composition elementary ( see Propo sition 2 from Subsect. 8.6.4 of Part 1) and thereby obtain the formula (11.10) in theThere general case. ways of reducing the integral over a set to integrals over are various neighborhoods of theits points. Forweexample, oneOnmaytheusebasistheofadditivity ofsmall the integral. That is procedure shall use. Lemmas 1, 3, and 4 we now carry out the proof of Theorem 1 on change of variable in a multiple integral. For eacha 8point t of the compact set Kt supp ( ( ! cp ) l det cp' l) Dt weProof.construct ( t ) -neighborhood U ( t ) of it in which the diffeomorphismc5 cp decomposes into a composition of elementary diffeomorphisms. From the �t) neighborhoods - U (t) U ( T ) of the points t E K1t we. choose a finite covering U ( hclosure ) , . . . , U(t k ) of the compact set Kt . Let 8 = mm { 8 ( h ) , . . . , 8(t k )}. Then the any set whose diameter is thesmaller than 8 andUwhich) , . intersects must letbeofIcontained in at least one of neighborhoods (t 1 . . , U (t k ) · Kt Now be an interval containing the set D and P a partition t found above andof dtheis interval I such that >-. (P) < min { 8, d}, where 8 was =
C
2
o
C
1 1 .5 Change of Variable in a Multiple Integral
143
the distanceP that from have Kt to the boundary of Dt . Let {Ji } be the intervals of the partition a nonempty intersection with Kt . It is clear that if Ii E {Ji } , then Ji Dt and J {f cpl det cp' l ) (t) dt = J ( (f cpl det cp' l )xD ) (t) dt = D, = L j (! 'PI det cp' l ) (t) dt . (11.13) = cp(Ji) of theandintervals set. ThenBytheLemma set E1=theUi Eiimage is alsoEi measurable supp f Ji Eis =a measurable E Dx . Using the additivity of the integral, we deduce from this that j f(x ) dx = j fXDJx ) dx = j fXDJ x ) dx + j fX D., (x ) dx = D., !.,CD., = j fX D., (x ) dx = j f(x) dx = L j f(x) dx . (11.14) By construction every interval Ji E { Ii } is contained in some neighbor hood U (xj ) inside which the diffeomorphism cp decomposes into a composition ofcanelementary di ff eomorphisms. Hence on the basis of Lemmas 3 and 4 we write J f(x ) dx = J {f o cpl det cp' l ) (t) dt . (11.15) Comparing relations (11.13), (11.14), and (11.15), we obtain formula (11.10). C
o
o
,
I
•
o
I;
C
E
I., \E
E
' E;
E
E;
C
I;
D
1 1 . 5 . 7 Corollaries and Generalizations of the Formula for Change of Variable in a Multiple Integral a. Change of Variable under Mappings of Measurable Sets
Let cp : Dt -+ Dx be a diffeomorphism of a bounded open set !Rn onto a set Dx !Rn of the same type; let Et and Ex be subsets of and Dx respectively and such that Et Dt , Ex Dx , and Ex = cp(Et ) · If f E R(Ex ) , then f cpl det cp' l E R(Et ) , and the following equality holds: (11.16) j f(x) dx = j (! cpl det cp' l ) (t) dt .
Proposition 1.
Dt Dt
C
o
C
C
o
C
144
11 Multiple Integrals
Indeed, j f(x) dx = j (fxex)(x) dx = j (((fxeJ o
Proof.
D,
o
o
E,
D,
X Ex
X E,
:
o
D
:::>
The value of the integral of a function f over a set E JR:n is independent of the choice of Cartesian coordinate system in JR:n . Proof. In fact the transition from one Cartesian coordinate system in JRn to anotherByCartesian has a Jacobian constantly equal to 1 in absolute value. Propositionsystem 1 this implies the equality c
Proposition 2 .
J f ( x) dx = J (f
E,
p
o
E,
D
C
1 1 .5 Change of Variable in a Multiple Integral
145
The changes of variable or formulas for transforming co ordi n ates used in practice sometimes have various singularities (for example, one-to-oneness may faiAsl ina some places,singularities or the Jacobian mayon avanish, ormeasure differ entiability may fail). rule, these occur set of zero useful.and so, to meet the demands of practice, the following theorem is very c. Negligible Sets
Let 'P : Dt -+ Dx be a mapping of a (Jordan) measurable set JR� onto a set Dx JR� of the same type. Suppose that there are subsets St and Bx of Dt and Dx respectively having (Lebesgue) measure zero and such that Dt \ St and Dx \ Bx are open sets and 'P maps the former diffeomorphically onto the latter and with a bounded Jacobian. Then for any function f E R( D x) the function (f 'P) I det cp' I also belongs to R( Dt \ St) and J f (x) dx = J ( (f 'P) I det cp' I) ( t) dt . (11.17) Theorem 2.
Dt
C
C
o
o
Dx
D , \ St
If, in addition, the quantity I det cp' l is defined and bounded in Dt , then (11.18) J f(x) dx = J (( f o cp)l det cp' l ) (t) dt . Dx
Dt
Proof. By Lebesgue ' s criterion the function f can have discontinuities in Dx and hence alsseto inofDxdiscontinuities \ Bx only on a set of measure zero.1 By Lemma 1, the image of this the mapping cp - : Dx \ x Dt \ St is a set of measure zero in Dt \ St . under Thus the relation (f 'P) I det cp' I E R( Dt \ St ) will follow immediately from Lebesgue ' s criterion for integrability if we establish that the setset Dwillt \ beSt ais by-product measurable.ofThethe fact that this is indeed a Jordan measurable reasoning below. By 8Sx hypothesis8DxDx \ xBxandis anconsequently open set, so8Dxthat (Dx \ x ) n 8Sx = 0 . Hence x = 8Dx Bx , where Bx = x 8Sx is the closure of Bx in lR� . As a result, 8Dx x is a closed bounded set, that is, it is compact in JRn , and, being the union of two sets of11.1measure zero,thatisthen itselftheof setLebesgue measure zero. From Lemma 3 of Sect. we know 8Dx x (and along with it, x ) has measure zero, that is, for every E > 0k there exists a finite covering . . . , h of this set by intervals such that iL= l l li l < E. Hence it follows, in particular, that the set Dx \ (and similarly the set Dt \ St ) is Jordan measurable: indeed, 8(Dx \ x ) 8Dx 8Sx 8Dx x . The covering . . . , h can obviously also be chosen so that every point x E 8Dx \ x is an interior point of at least one of the intervals of the covering. Let theUx =setU7Vx= 1ishsuchThethat set UxVxis measurable, as is Vx = Dx \ Ux . By construction Dx \ x and for every measurable set Ex Dx containing the compact set Vx we have the estimate B
o
U
B
C
U
U
B
UB
B
Bx
C
B
C
U
h,
C
U
B
B
U
B h,
B
C
B
U
B
-+
146
11 Multiple Integrals
I j f (x) dx - j f( x)dx l = I j f (x) dx l � M p (D x \ Ex ) < M · E , (11.19) where M = xsup f(x). The pre-image V t = ip - 1 (V x ) of the compact set V x is a compact subset ofWtDtsubject \ St . Reasoning as above, we can construct a measurable compact set to the conditions Vt Wt Dt \ St and having the property that the estimate I J ( (f o 'P)I det ip' l ) (t) dt - J ( (f o 'P)I det ip' l ) (t) dt l < E (11. 20) holdsNowforleteveryE measurable set Et such thatholds Wt Et Dt \ St . Formula (11.16) for the sets Ex Dx \ Sx = ip(E ) x t· and Et Dt \ St by Lemma 1. Comparing relations (11.16), (11.19), and (11.20) (11.17) and taking account of the arbitrariness of the quantity E > 0, we obtain . the last assertion of Theorem 2. If the function (fo'P) l det IP'I We now prove issetdefined on the entireof this set Dtfunction , then, since Dt \ St is open in lRf , the entire of discontinuities in Dt consists of the set A of points of of the original function discontinuity of (! o 'P) I det ip1 I I (the restriction to DtAs\weSt)have and seen, perhapsthe asetsubset B of St U 8Dt . A is a set of Lebesgue measure zero (since the integral onzero,thetheright-hand sidebe ofsaid(11.17) exists), and since St U 8Dt has measure same can of B. Hence it suffices to know that the functioncriterion ( ! o 'P) I det ip1 I is bounded on Dt ; it will then follow from the Lebesgue it is integrable over Dt . But I f o 'PI(t) M on Dt , so that'I theis bounded functionthat (! o ip) I det ip1 I is bounded on St , given that the function hypothesis. As forbounded. the set DtThus, on St by over the function function I deto IP'P) I det ip1 I is integrable \ St , the it and hence (! (! o ip) I det ip1 I is integrable over Dt . But the sets Dt and Dt \ St differ only by the measurable setofStthe, whose measure, asfacthasthat beentheshown, is zero. Therefore, bywe thecan additivity integral and the integral over St is zero, conclude that the right-hand sides of (11 17) and (11 18) are . . indeed equal in this case. The mapping of the rectangle I = { (r, 'P) E JR2 1 0 r R 1\ 0 'PExample. � 21r} onto the disk K = { (x, y) E JR 2 1 x 2 + y 2 R2 } given by the formulas (11. 2 1) x = r cos ip , y = r sin ip , ismapsnot toa diffeomorphism: the rectangle r=0 the single point (0,the0)entire undersidethisofmapping; the imIageson ofwhich the points �
Ex
Dx
D x \ Ex
E Dx
C
�
�\�
C
C
C
C
C
D , \ S,
�
D
�
�
�
�
1 1 .5 Change of Variable in a Multiple Integral
147
(r, 0) andand(r,K21r)\ E,arewhere the same. However, if we consider, for example, the sets E is the union of the boundary 8K of the disk K thedomain radius Iending at (0, R) , then the restriction of the mapping (11.21) toand the \ 8I turns out to be a diffeomorphism of it onto the domain K \ E. Hence by Theorem 2, for any function f E R(K) we can write I \ 8I
JJ f(x, y) dx dy JJ f(r cos tp , r sin tp)r dr dtp =
I
K
and, applying Fubini 's theorem 2 1r R JJK f(x,y) dxdy J0 dtp J0 f(rcos tp, rsin tp)r dr . Relationsto(11.21) are coordinates the well-known coordinates Cartesian in theformulas plane. for transition from polar What has beencoordinates said can innaturally be developed and extended to the polar (spherical) JRn that we studied in Part 1, where we also exhibited from polar coordinates to Cartesian coordinatestheinJacobian a space JRofn theof anytransition dimension. =
1 1 .5.8 Problems and Exercises 1. a) Show that Lemma 1 is valid for any smooth mapping '{! : D t -+ D x (also see Problem 8 below in this connection) . b) Prove that if D is an open set in ffi:.m and '{! E cC l l (D, ffi:. n ) , then i.f!(D) is a set of measure zero in ffi:. n when m < n . 2. a) Verify that the measure of a measurable set E and the measure of its image '{!(E) under a diffeomorphism '{! are connected by the relation �t ( i.f!(E) ) = 8�t(E) ,
[
]
where () = tinf l det '{! ' (t) l , sup l det '{!' (t) l · EE tEE b) In particular, if E is a connected set, there is a point T E E such that �t ( i.f!(E) ) = l det i.f!' (T) I�t(E) .
3. a) Show that if formula ( 1 1 . 10) holds for the function f = 1 , then it holds in general. b) Carry out the proof of Theorem 1 again, but for the special case f = 1 , simplifying it for this special situation.
4. Without using Remark 2, carry out the proof of Lemma 3, assuming Lemma 2 is known and that two integrable functions that differ only on a set of measure zero have the same integral.
148
11 Multiple Integrals
5. Instead of the additivity of the integral and the accompanying analysis of the measurability of sets, one can use another device for localization when reducing formula ( 1 1 . 10) to its local version (that is to the verification of the formula for a small neighborhood of the points of the domain being mapped) . This device is based on the linearity of the integral. a) If the smooth functions e 1 , . . . , e k are such that 0 ::; ei ::; 1 , i = 1 , . . . , k, and t e i (x) = 1 on Dx , then I t e ; f (x) dx = I f(x) dx for every function t= l t Dx Dx f E R(Dx ) · b) If supp ei is contained in the set U C Dx , then I ( e i f ) ( x ) dx = I (ed) (x) dx .
(
)
Dx
U
c) Taking account of Lemmas 3 and 4 and the linearity of the integral, one can derive formula ( 1 1 . 10) from a) and b) , if for every open covering {Ua } of the compact set K = supp f C Dx we construct a set of smooth functions e 1 , . . , e k k in Dx such that 0 ::; ei ::; 1 , i = 1 , . . . , k, 2::: ei = 1 on K, and for every function i= l ei E { ei } there is a set Ua i E { Ua } such that supp ei c Ua i · In that case the set of functions { ei } is said to be a partition of unity on the compact set K subordinate to the covering {Ua } · 6 . This problem contains a scheme for constructing the partition of unity discussed in Problem 5. = 1 and supp f c a) Construct a function f E c<=l (JR, JR) such that II [- 1 , 1] [ - 1 - 8, 1 + 8] , where 8 > 0. b) Construct a function f E c<=l (JR n , lR) with the properties indicated in a) for the unit cube in JR n and its 8-dilation. c) Show that for every open covering of the compact set K c JR n there exists a smooth partition of unity on K subordinate to this covering. d) Extending c) , construct a c<=l-partition of unity in JR n subordinate to a locally finite open covering of the entire space. (A covering is locally finite if every point of the set that is covered, in this case JR n , has a neighborhood that intersects only a finite number of the sets in the covering. For a partition of unity containing an infinite number of functions { ei } we impose the requirement that every point of the set on which this partition is constructed belongs to the support of at most finitely many of the functions {ei } . Under this hypothesis no questions arise as to the meaning of the equality 2::: ei = 1 ; more precisely, there are no questions as to i the meaning of the sum on the left-hand side.) 7. One can obtain a proof of Theorem 1 that is slightly different from the one given above and relies on the possibility of decomposing only a linear mapping into a composition of elementary mappings. Such a proof is closer to the heuristic considerations in Subsect. 1 1 .5.1 and is obtained by proving the following assertions. a) Verify that under elementary linear mappings L : lR n -+ lR n of the form ( x 1 , . . . , x k , . . . , x n ) I-t (x 1 , . . . , x k - \ .X. x k , x k +1 , . . . , x n ) , A i- 0, and .
1 1 .5 Change of Variable in a Multiple Integral
149
the relation 11 ( L(E) ) = I det L' I 11(E) holds for every measurable set E C IR.n ; then show that this relation holds for every linear transformation L : IR.n --+ IR.n . (Use Fubini's theorem and the possibility of decomposing a linear mapping into a composition of the elementary mappings just exhibited.) b) Show that if cp : D t --+ Dx is a diffeomorphism, then f1 ( cp( K) ) :=::; I l det cp' ( t ) l dt for every measurable compact set K C D t and its image cp(K) . (If K a E D t , then :3 ( cp' (a)) - 1 and in the representation cp( t) = ( cp 1 (a) o ( cp 1 (a) ) - 1 o cp ) (t) the mapping cp' (a) is linear while the transformation (cp' (a) ) - 1 o cp is nearly an isometry on a neighborhood of a.) c) Show that if the function f in Theorem 1 is nonnegative, then I f(x) dx :=::;
I ( ( f o cp) l det cp' l ) ( t ) d t .
Dx
Dt
d) Applying the preceding inequality to the function (f o cp) l det cp' l and the mapping cp - 1 : Dx --+ D, , show that formula ( 1 1 . 10) holds for a nonnegative func tion. e) By representing the function f in Theorem 1 as the difference of integrable nonnegative functions, prove that formula ( 1 1 . 10) holds.
S a r d' s l e m m a. Let D be an open set in IR.n , let cp E c
150
1 1 Multiple Integrals
=
x E ({'(D) , the complete pre-image ({J - 1 (x) Mx in D is a surface (manifold) of codimension n in JR m (that is, m - dim Mx = n for almost all x E D). 9 . Suppose we consider an arbitrary mapping ({! E C ( l ) (Dt , Dx ) such that det ((! 1 ( t) =/= 0 in Dt instead of the diffeomorphism ({! of Theorem 1. Let n( x) = card { t E supp o ({!) I ({'( t) x}, that is, n(x) is the number of points of the sup port of the function f o ({! that map to the point x E Dx under ({! : Dt -+ Dx . The following formula holds:
(!
I (! ·
=
n) (x) dx
= I (U Dt
Dx
o
({') I det ((J ' I ) (t) dt .
a) What is the geometric meaning of this formula for f = 1? b) Prove this formula for the special mapping of the annulus Dt = { t E JR; 1 1 < ltl < 2} onto the annulus Dx = {x E IR� I 1 < lxl < 2} given in polar coordinates ( r, ({!) and ( p, B) in the planes IR� and JR; respectively by the formulas r = p, ({! = 20. c) Now try to prove the formula in general.
1 1 .6 Improper Multiple Integrals 1 1 .6 . 1 Basic Definitions
An exhaustion of a set E JR.rn is a sequence of measurable sets {En } such that En En+l E for any n E N and nU= l En = E. Definition 1.
C
C
C
00
If {En } is an exhaustion of a measurable set E, then: a ) lim J.l. (En ) = J.l. (E); n-+oo b) for every function f E R(E) the function ! l En also belongs to R(En ), and lim f(x ) dx f(x ) dx . n-+oo En E
Lemma.
J
=
J
Proof. Since En En+l E, it follows that f.l. (En ) :::; f.l. (En +I ) :::; f.l. (E) and li-+oom J.l. (En ) :::; J.l. (E). To prove a) we shall show that the inequality n lim J.l. (En ) J.l. (E) also holds. n-+oo The number boundaryof &E ofintervals E has content zero, and hence can be covered by anumber finite open of total content less than any preassigned c ::;:: 0. Let L1 be the union of all these oeen intervals. Then the set E L1 =: E is open in JR.m and by construction E contains the closure of E andForf.l. (E)every:::; J.l. set(E)E+ f.l.of( L1the) <exhaustion f.l. (E) + c. n } the construction just described n can be repeated with the value en = c/2{En . We then obtain a sequence of open C
:2:
U
C
1 1 .6 Improper Multiple Integrals
151
sets En = En U L1n such that En En , J.L (En ) � J.L (En ) + J.L (Lln ) < J.L (En ) +cn , and nU= l En nU= l En E. The system of open sets L1 , E1 , E2 , . . . , forms an open covering of the compact set E. Let Ll, E1 , E2 ,. . . ,Ek be a finite covering of E extracted from this covering. Since E1 hence E2 Ek , the sets Ll, L1 1 , . . . , Ll k , Ek also form a covering of E and J.L ( E ) � J.L ( E ) � J.L (Ek ) + J.L (Ll) + J.L (Ll l ) + + J.L (Ll k ) < J.L (Ek ) + 2c; . It follows from this that J.L (E) � nli--+moo J.L(En ) · b) The relation f I E R( En ) is well known to us and follows from 's criterion for the existence of the integral over a measurable set. By Lebesgue hypothesis f E R(E), and so there exists a constant M such that lf(x)l � M onintegral E. From the additivity of the integral and the general estimate for the we obtain I J f(x) dx - J f(x) dx l = I J f(x) dx l � MJ.L (E \ En ) . From this, with what was proved in a) , we conclude that b) does indeed hold.together D Definition 2. Let {En } be an exhaustion of the set E and suppose the function f E lR is integrable on the sets En E {En }. If the limit J f(x) dx nlim--+ oo J f(x) dx and limit has aisvalue choiceofoff theoversetsE. in the exhaustion ofexists E, this calledindependent the improperof theintegral The integral sign onbuttheweleftsayin this lastthe equality isexists usuallyor converges written forif any function defined on E, that integral the limit in Definition 2 exists. If there is no common limit for all exhaustions of E,diverges. we say that the integral of f over E does not exist, or that the integral Theof anpurpose of Definition 2 is to extend the concept of integral to the caseThe unbounded integrand or ananunbounded domain isof theintegration. symbol introduced to denote improper integral sameremark as the symbol for an ordinary integral, and that fact makes the following necessary. Remark If E is a measurable set and f E R(E), then the integral off over Eintegral in theofsense of Definition 2 exists and has the same value as the proper f over E. 00
:J
�
C
00
C
:J
C · · · C
· · ·
E
E
En
:
--+
:=
E
1.
E \ En
En
152
11 Multiple Integrals
This is precisely the content of assertion b) in the lemma above. D The set exhaustions. of all exhaustions of any reasonably ricimproper h set is immense, and we do not use all The verification that an integral converges is often simplified by the following proposition. Proof
If a function f : E ---+ R is nonnegative and the limit in Def inition 2 exists for even one exhaustion {En } of the set E, then the improper integral of f over E converges. Proof Let { Ek} be a second exhaustion of E into elements on which f is integrable. E� := E� En , n = 1, 2, . . . form an exhaustion of the set E�, andThe so itsetsfollows from part b) of the lemma that Proposition 1 .
n
J f(x) dx nlim-+ oo J f(x) dx :5 nlim-+oo J f(x) dx =
k
=
A
.
E'
Since f 0 and E� �
c
E� +
l c
E, it follows that
3 k-too lim j f(x) dx = B :5 A . k
E'
But there is symmetry between the exhaustions {En } and { Ek}, so that A :5 B also, and hence A = B. D Example 1 . Let us find the improper integral JJ e -( x 2 + Y 2 ) dx dy. ]R2 We shall exhaust the plane R 2 by the sequence of disks En = {(x, y) E R2 1 x 2 + y 2 < n2 } . After passing to polar coordinates we find easily that 2 71" n 2 2 x +y ) -( dx dy = dcp e r\ dr 11'(1 - e - n2 ) ---+ 11' e
JJ En
J J 0
-
=
0
as nBy---+ Proposition oo . 1 we can now conclude that this integral converges and equals 11'. One can derive a useful corollary from this result ifRwe2 now nconsider the of the plane by the squares E� = {(x, y) E l l x l 1\ IYI n } . exhaustion By Fubini' s theorem :5
:5
1 1 .6 Improper Multiple Integrals
153
By Proposition this lastwequantity following Euler and 1Poisson, find thatmust tend to as n --+ oo . Thus, +oo J e - x2 dx = .jii . -oo additionalobvious propertiesat first of Definition improper which3 . are Some not completely glance, will2 ofbeangiven belowintegral, in Remark 1r
1 1 .6.2 The Comparison Test for Convergence of an Improper Integral
Let f and g be functions defined on the set E and integrable over exactly the same measurable subsets of it, and suppose l f(x) l � g(x) on E. If the improper integral J g(x) dx converges, then the integrals J l f l (x) dx E E and J f(x) dx also converge.
Proposition 2.
E
Proof. Let {En } be an exhaustion of E on whose elements both g and f are integrable. the Lebesgue criterion that the function l / 1 is integrable on theIt follows sets Enfrom , n E N, and so we can write J l f l (x) dx - J l f l (x) dx J l f l (x) dx < J g(x) dx = J g(x) dx - J g(x) dx , where k and n are any natural numbers. When we take account of Proposition 1 and the Cauchy criterion for the existence of a limit of a sequence, we conclude that the integral J l f l (x) dx converges. the functions f+ : = HI / I + f ) and f- : = HI / I - f ) . Now consider Obviously 0 � f+ � 1 ! 1 and 0 � f- l f l . By what has just been proved, the improperthe integrals ofintegral f+ and off fover E both converge. But f = f+ - f and hence improper over as well (and is equal to the difference of the integrals theof f+sameandsetf converges ). D In order tointegrals, make effective use oftoProposition 2 in studying the convergence ofcomparison. improper it is useful have a store of standard In this connection we consider the following example.functions for Example In the deleted n-dimensional ball ofa radius 1 , B !Rn with its center 0 removed, consider the function 1 / r , where r = d(O, x) is the distanceatfrom the point x E B \ 0 to the point 0. Let us determine the values =
�
En + k \ En
En + k
En
E
_
�
_,
_
2.
C
154
11 Multiple Integrals
ofTo doEthisIR forwe construct which theanintegral of r� a over the domain B \ 0 converges. exhaustion of the domain by the annular regions B(c:)Passing = {x EtoB polar l c: < d(O, x) < 1}. coordinates with center at 0, by Fubini ' s theorem, we obtain dx = f(cp) dcp !1 rn � 1 dr = c /1 �drn+l ' r<> r<> J r<>(x) J where dcp, .=. . ,dcp'P 1�. . .that d'f!n � 1 and f(cp) is a certain product of sines of the angles in the Jacobian of the transition to polar cp nn 2 1 coordinates in !R , while appears c is the magnitude of the integral over which depends only on n, not on r and c:. As c: ---+ +0 the value just obtained for the integral over B(c:) will have a finite limit if < n. In all other cases this last integral tends to infinity as c ---+ +0. Thus we have shown that the function da (�,x) , where d is the distance to infinity, n. this same function is integrable in the improper sense only for Example Let I = {x E IR n l 0 :::; xi :::; 1, i = 1, . . . , n} be the n-dimensional cube= and the k-dimensional face of it defined by the conditions x k +1 = n x = 0. On the set I \ we consider the function da(x) , where d(x) is EtheIR distance E I \ to the face Let usI \ determine the values of for whichfrom the xintegral of1 this function over converges. We remark that if x = (x , . . . , xk , x k +1 , . . . , xn ) then a
B(c)
E
E
S
s,
a
a a
· · ·
a
3. h
h
h
h.
h
I(c:) be Bythe Fubini' cube Isfrom which the c:-neighborhood of the face has beenLetremoved. theorem du J l u i <> ' where u = (x k +l , . . . , xn ) and In � k (c:) is the face In � k !Rn � k from which the But c:-neighborhood it is clear onofthe0 hasbasisbeenof theremoved. experience acquired in Example 1 that the lastconsideration integral converges onlyonlyfor for< n<-nk.-Hence thekimproper integral under converges k, where is the dimension of the face near which the function may increase without bound. h
C
a
a
1 1 .6 Improper Multiple Integrals
155
Remark In the proof of Proposition 2 we verified that the convergence of the integral I f ! implies the convergence of the integral of f. It turns out that the converse is also true for an improper integral in the sense of Definition 2, which was not the case previously when we studied improper integrals on the line.) Inconvergence the latter ofcase,an weimproper distinguished absolute and nonabsolute ( con ditional integral. To understand right away the essence of the new phenomenon that has arisen in connection with Definition 2, consider the following example. Example 4. Let the function f : JR + lR be defined on�the set JR + of nonneg ative numbers by the following conditions: f(x) if n - 1 ::; x < n, n E N. Since the series 2::: � converges, the integral J f(x) dx has a limit as AHowever, ---+ oo equal to the sum of this series. this series does not converge absolutely, and one can make it divergent to +oo, for example, by rearranging its terms. The partial sums of the newE series can be interpreted as the integrals of the function f over the union n of the closed intervals on the real line corresponding to the terms of thedomain series.JRTheonsetswhich En , taken all together, however, form an exhaustion of the f is defined. + Thus the improper integral J f(x) dx of the function f : JR+ lR exists in its earlier sense, but not in the sense of Definition 2. Wechoice see thatof thethe exhaustion condition inis Definition 2 that the limit be independent ofof the equi v alent tolatter, the independence ofis theexactly sum a series on the order of summation. The as we know, equiInvalent to absolute convergence. practice one nearly always has fto: DconsiderlR defined only special exhaustions ofbe theunbounded followingintype. Let a function in the domain D a neighborhood of some set E oD. We then remove from D the points lying in the E-neighborhood of E and obtain a domain D ( E ) D. As E ---+ 0 these domains generate an exhaustion of D. If the domain is unbounded, we can obtain an exhaustion of itspecial by taking the D complements of neighborhoods of infinity. These are the exhaustions wespecial mentioned earlierthat and lead studieddirectly in thetoone-dimensional case,ofand itnotion is theseof exhaustions the generalization the Cauchy principal value of an improper integralstudying to the case of a space of anyon dimension, which we discussed earlier when improper integrals the line. 2.
---+
=
n= l
=
( l)n-1
(-l n-1 A
0
=
---+
0
---+
C
,
C
1 56
1 1 Multiple Integrals
1 1 .6.3 Change of Variable in an Improper Integral
In conclusion we making obtain thea valuable, formula foralthough changevery of variable insupplement improper into tegrals, thereby simple, Theorems 1 and 2 of Sect. 11.5. �
C
Let r.p Dt Dx be a diffeomorphism of the open set Dt lRf onto the set Dx JR� of the same type, and let f Dx lR be integrable on all measurable compact subsets of Dx . If the improper integral J f(x) dx D., converges, then the integral J ( (f r.p) I det r.p' I) ( t) dt also converges and has D, the same value. Proof. The open set Dt lRf can be exhausted by a sequence of compact sets Ef, k E N, contained in N, each of which is the union of a finite number of intervals in lRf (in this connection, see the beginning of the proof of Lemma 1 in Sect. 11.5). Since r.p Dt Dx is a diffeomorphism, the exhaustion E; of Dx , where E; r.p(Ef), corresponds to the exhaustion {Ef} of Dt . Here sets Lemma E; r.p(Ef) are measurable compact sets in Dx (measurability followsthefrom 1 of Sect. 11.5). By Proposition 1 of Sect. 11.5 we can Theorem 1 .
C
:
�
:
o
c
=
=
write
�
:
J f (x) dx J ( (f r.p) I det r.p' I) (t) dt . Ef: Ef side side of thisalsoequality limit byDhypothesis as k HenceThetheleft-hand right-hand has the has samea limit. Remark By the reasoning just given we have verified that the integral on the right-hand side of the last equality has the same limit for any exhaustion Dt of the given special type. It is this proven part of the theorem that we shall bewithusing. But formally, to complete the proof of the theorem in accordance Definition 2 it is necessary to verify that this limit exists for every of astheandomain Dt . We leave this (not entirely elementary) proof toexhaustion the reader excellent exercise. Weintegral remark ofonly that one can already deduce the convergence of the improper I f 'P I I det r.p' l over the set Dt (see Problem 7). =
o
� oo .
3.
o
�
Let r.p Dt Dx be a mapping of the open sets Dt and Dx . As sume that there are subsets St and Bx of measure zero contained in Dt and Dx respectively such that Dt \ St and Dx \ Bx are open sets and r.p is a diffeomor phism of the former onto the latter. Under these hypotheses, if the improper integral J f(x) dx converges, then the integral J ( (for.p) I det r.p'l) (t) dt also D., D , \ St converges to the same value. If in addition I det r.p ' I is defined and bounded on Theorem 2.
:
1 1 .6 Improper Multiple Integrals
157
compact subsets of Dt , then (J rp) I det rp' I is improperly integrable over the set Dt , and the following equality holds: f ( x) dx = ( (J rp) I det rp' I ) ( t) dt . o
J
J
Dx
o
D, \ St
Proof. The assertion is a direct corollary of Theorem 1 and Theorem 2 of Sect. 11.5, provided we take account of the fact that when finding an improper integral an open setcompact one maysetsrestrict consideration to exhaustions that consist ofovermeasurable (see Remark 3). . Example 5. Let us compute the . t egra1 JJ wh"1ch . an lmintegral when a > 0, since the integrand is unbounded in that case in 2 + y 2 1. aproper neighborhood of the disk x Passing to polar coordinates, we obtain from Theorem 2 !! r dr drp2 "' r m
x 2 + y2 < l
dx d y ( 1 - x 2 - y2 ) " ,
IS
=
0O<<
For a v>e,0itthiscanlastbe computed integral is asalsotheimproper, but,the since theexhaustion integrand ofis nonnegati limit over special the rectangle I { ( r, rp) E JR2 1 0 < rp < 27T 1\ 0 < r < 1} by the rectangles In {(r, rp) E JR2 1 0 < rp < 27T 1\ 0 < r < 1 - � } , n E N. Using Fubini ' s theorem, we find that !! (�
=
�
O<
0
1T
0
By theforsame considerations, one can deduce that the original integral diverges a 2 1. Example Let us show that the integral JJ f �; converges only under the condition � + � < 1. Proof. In view of the obvious symmetry it suffices to consider the integral only over the that domain D in which x 2 0, y 2 0 and x + y 1. It is clear the simultaneous conditions > 0 and q > 0 are necessary forfollowing the integral to converge. Indeed, if p 0 for example, we would obtain the estimate for the integral over the rectangle I { ( x, y) E lR2 1 1 ::::; x A 1\ 0 y 1} alone, which is contained in D : !! dx dy Jdxj dy -> Jdxj dy - (A- 1) J dy ' lxiP + IYiq - 0 lxiP + IYi q 0 1 + IYiq 0 1 + IY i q 6.
lxl+lvl� l
p
::::;
::::;
::::;
::::;
A
lA
1
I
A
1
x lx v+
1q
2
A =
1
1
11 Multiple Integrals
158
whichnowshowson wethatmayas assume A this integral increases without bound. Thus from that > 0 and q > 0. The integrand has no singularities in the bounded portion of the domain D, so that studying the convergence of this integral is equivalent to studying the convergence of the integral of the same function over, for example, the portion G of the domain D where xP + a > 0. The number a can be assumed sufficiently large that the curve xP + a lies in D for x 0 and 0. Passing to generalized curvilinear coordinates r.p using the formulas x (r cos2 r.p) 1 1P , (r sin2 r.p? fq , by Theorem 2 we obtain /r r dx d � q JJ (r i + � -2 cos % - 1 r.psin % - 1 r.p) dr dr.p . J l x i P + IYI 9 O<
p
yq 2:
y ;:::
=
y
yq =
;:::
y =
p
G
=
c
:=:;
c
:=:;
A
:=:;
:=:; :=:;
=
O<
Tr / 2 - e
A 1 1 lim0 J cos % r.p sin % r.pdr.p A-+lim= J r i + � -2 dr . e---+ a e Sinceis finite > 0 and > 0, the first of these limits is necessarily finite and the second only when � + % < 1 . D =
p
q
1 1 .6.4 Problems and Exercises
Give conditions on verges. 1.
2.
p
and A
q
under which the integral
a) Does the limit Alim I cos x 2 dx exist? -> oo 0
II
O< l x i + I Y I
, �\�� l • con :'S l 1
b) Does the integral I cos x 2 dx converge in the sense of Definition 2?
c) By verifying that
Jl!. l
r r sin(x 2 + y 2 ) dx dy = 7r lim }}
n � oo
l x i :S n
1 1 .6 Improper Multiple Integrals and
n -+ oo
!!
lim
x 2 + y 2 S: 21Tn
159
sin(x 2 + y 2 ) dx dy = 0
verify that the integral of sin(x 2 + y 2 ) over the plane JR 2 diverges. a) Compute the integral J J J :���:: . 1 1 1
3.
0 0 0
b) One must be careful when applying Fubini's theorem to improper integrals (but of course one must also be careful when applying it to proper integrals) . Show 2 2 that the integral JJ ( :2;y1 ) 2 dx dy diverges, while both of the iterated integrals f dx f 00
00
1
1
x 2: 1 , y2: 1
2 2 dy J c :2;Y2 ( :2;Y1 y ) 2 dy and J y ) 2 dx converge. 1 1 2
2
00
00
c) Prove that if f E C(JR 2 , JR) and f :2: 0 in JR 2 , then the existence of either of 00 00 00 00 the iterated integrals J dx J f(x, y) dy and J dy J f(x, y) dx implies that the oo
oo
oo
integral JJ f(x, y) dx dy converges to the value of the iterated integral in question. -
JR2
4.
-
-
Show that if f E C(JR, JR) , then lim .!. h -+ 0 7r
1
J h2 +h X2 f(x) dx
-1
-
=
f(O) .
5. Let D be a bounded domain in JR n with a smooth boundary and S a smooth k-dimensional surface contained in the boundary of D. Show that if the function f E C(D, JR) admits the estimate IJI < d n _\ _ , , where d = d(S, x) is the distance from x E D to S and E > 0, then the integral of f over D converges.
As a supplement to Remark 1 show that it remains valid even if the set E is not assumed to be measurable. 7. Let D be an open set in JR n and let the function f : D -+ lR be integrable over any measurable compact set contained in D. a) Show that if the improper integral of the function If I over D diverges, then there exists an exhaustion {En } of D such that each set En is an elementary compact set, consisting of a finite number of n-dimensional intervals and JJ I f I ( x ) dx -+ +oo En as n -+ oo. b) Verify that if the integral of f over a set converges while the integral of l f l diverges, then the integrals of f+ = � ( I f I + f) and f- = � ( I f I - f) over the set both diverge. c) Show that the exhaustion { En } obtained in a) can be distributed in such a way that J f+ (x) dx > J IJI (x) dx for all n E N. 6.
E n + l \ En
En
d) Using lower Darboux sums, show that if J f+ (x) dx > A, then there exists E an elementary compact set F C E consisting of a finite number of intervals such that J f(x) dx > A. F
160
11 Multiple Integrals
e) Deduce from c) and d) that there exists an elementary compact set Fn En + 1 \ En for which J f(x) dx > J l f l (x) dx + n . Fn
En
C
f) Show using e) that the sets G n = Fn n En are elementary compact sets (that is, they consist of a finite number of intervals) contained in D that , taken together, constitute an exhaustion of D, and for which the relation J f(x) dx -+ +oo as Gn n -+ oo holds. Thus, if the integral of 1 / 1 diverges, then the integral of f (in the sense of Definition 2) also diverges. Carry out the proof of Theorem 2 in detail. 9 . We recall that if X = (x 1 , . . . , x n ) and e = (e , . . . , c ) , then (x, e) = x 1 e + · · · + x n C is the standard inner product in JR n . Let A = (a;i ) be a symmetric n x n matrix of complex numbers. We denote by Re A the matrix with elements Re aij · Writing Re A � 0 (resp. Re A > 0) means that ((Re A)x, x) � 0 (resp. ((Re A)x, x) > 0) for every x E JR n , x =f. 0. a) Show that if Re A � 0, then for A > 0 and e E JR n we have 8.
Here the branch of v'det A is chosen as follows: (det A) - 1 1 2 = l det AI - 1 1 2 exp - i lnd A , n � Ind A = 2"1 arg J,tj (A) , l arg J,tj (A) I :S 2 j= 1
L::
)
(
where J,tj (A) are the eigenvalues of A. b) Let A be a real-valued symmetric nondegenerate e E JR n and A > 0 we have
(n
,
x n)
matrix. Then for
j exp (i � (Ax, x) - i(x, e) ) dx = n/ 2 i (A - 1 , ) exp i� sgn A . = ( 2� ) l det AI _ 1 ; 2 exp ( - 2A ee) (4 )
JR n
T
Here sgn A is the signature of the matrix, that is,
sgn A = v+ (A) - v_ (A) , where v+ (A) is the number of positive eigenvalutes of A and v_ (A) the number of negative eigenvalues.
1 2 Surfaces and Different ial Forms in JRn
Inconsistent this chapter we discuss thesurface concepts ofits surface, boundary of aasurface, andfor orientation of a and boundary; we derive formula computing the area of a surface lying in lR n ; and we give some elementary information on diwith fferenti of these concepts is verychapter imporis tant in working linealandforms. surfaceMastery integrals, to which the next devoted. 1 2 . 1 Surfaces in
IRn
The standard model for a k-dimensional surface is JRk . Definition 1 . A surface of dimension k (or k-dimensional surface or k in JRn is a subset 1 in S homeomorphic 2 to JRSk . JRn each point of which has adimensional neighborhoodmanifold) S provided by the homeomor Definition 2. The mapping
C
:
=
=
1 As before, a neighborhood of a point x E S C JE. n in S is a set Us (x) = S n U(x) , where U(x) is a neighborhood of x in JE. n . Since we shall be discussing only neighborhoods of a point on a surface in what follows, we shall simplify the notation where no confusion can arise by writing U or U(x) instead of Us (x) . 2 On S C JE.n and hence also on U C S there is a unique metric induced from JE. n , so that one can speak of a topological mapping of U into JE. n .
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12 Surfaces and Differential Forms in Rn
To carry out certain analogies and in order to make a number of the following constructions easier to visualize, we shall as a rule take a cube I k as the canonical parameter domain for local charts on a surface. Thus a chart
I k -+ U S (12.1) gives a local parametric equation x =
c
c
c
The parametric definition of a surface is especially important for com putational purposes, as will become clear below. Sometimes one can define the entire surface by a single chart . Such a surface is usually called elemen tary. For example, the graph of a continuous function f : I k -+ JR. in JR.k +1 is an elementary surface. However, elementary surfaces are more the exception than the rule. For example, our ordinary two-dimensional terrestrial sphere cannot be defined by only one chart. An atlas of the surface of the Earth must contain at least two charts (see Problem 3 at the end of this section) . I n accordance with this analogy we adopt the following definition.
A(S) := {
surface
The union of two atlases of the same surface is obviously also an atlas of the surface. If no restrictions are imposed on the mappings ( 12. 1 ) , the local parametrizations of the surface, except that they must be homeomorphisms, the surface may be situated very strangely in JR. n . For example, it can happen that a surface homeomorphic to a two-dimensional sphere, that is, a topolog ical sphere, is contained in JR. 3 , but the region it bounds is not homeomorphic to a ball (the so-called Alexander horned sphere). 3 To eliminate such complications, which have nothing to do with the ques tions considered in analysis, we defined a smooth k-dimensional surface in JR.n in Sect. 8. 7 to be a set S C JR.n such that for each x0 E S there exists a neighborhood U ( x o ) in JR. n and a diffeomorphism '¢ : U(x 0 ) -+ I n = {t E JR.n l l t l < 1 , i = 1n, . . . , n} under which the set Us (xo) := S n U(x 0 ) maps into the cube I k = I n {t E JR. n l t k +1 = · · · = t n = 0}. It is clear that a surface that is smooth in this sense is a surface in the sense of Definition 1 , since the mappings x = 'lj; - 1 (t 1 , . . . , t k , 0, . . . , 0) =
12.1 Surfaces in !R n
163
if the mappings ( 1 2 . 1 ) are sufficiently regular, the concept of a surface is actually the same in both the old and new definitions. In essence this has already been shown by Example 8 in Sect . 8.7, but considering the importance of the question, we give a precise statement of the assertion and recall how the answer is obtained.
If the mapping (12. 1 ) belongs to class C ( l ) (I k , JRn ) and has maximal rank at each point of the cube J k , there exists a number > 0 and a diffeomorphism I;" n lRn of the cube I� {t E JRn l l t i l :<:::: E i , i = 1 , . . . , n } of dimension n in JR such that
:=
-t
Jk n i;' =
c
In other words, it is asserted that under these hypotheses the mappings
( 1 2 . 1 ) are locally the restrictions of diffeomorphisms of the full-dimensional cubes to the k-dimensional cubes 1: = n
J k J�.
I�
Proof. Suppose for definiteness that the first k of the n coordinate functions x k
=
=
=
X
k -
=
xn
In this case the mapping
X
n fn ( 1 , -
_
X
.
•
.
,X
k)
•
164
12 Surfaces and Differential Forms in ffi. n
t n = xn - f n (x i , . . . ' x k )
is a diffeomorphism of a full-dimensional neighborhood of the point x0 E JR n . As tpc; we can now take the restriction to some cube I';' of the diffeomorphism inverse to it . D
By a change of scale, of course, one can arrange to have E = 1 and a unit cube l';' in the last diffeomorphism. Thus we have shown that for a smooth surface in R n one can adopt the following definition, which is equivalent to the previous one.
Rn
introduced by Definition 1 Definition 4. The k-dimensional surface in is m 2': if it has an atlas whose local charts are smooth mappings ( of class m 2': and have rank k at each point of
smooth (of class C ( m) ,
their domains of definition.
C (m ) 1),
1)
We remark that the condition on the rank of the mappings (12.1) is es sential. For example, the analytic mapping lR 3 t H ( x 1 , x 2 ) E JR 2 defined by x 1 = t 2 , x 2 = t 3 defines a curve in the plane JR2 having a cusp at 2(0, 0) . It is clear that this curve is not a smooth one-dimensional surface in JR , since the latter must have a tangent ( a one-dimensional tangent plane ) at each point .4 Thus, in particular one should not conflate the concept of a smooth path of class and the concept of a smooth curve of class In analysis, as a rule, we deal with rather smooth parametrizations (12.1) of rank k. We have verified that in this case Definition 4 adopted here for a smooth surface agrees with the one considered earlier in Sect . 8.7. However, while the previous definition was intuitive and eliminated certain unnecessary complications immediately, the well-known advantage of Definition 4 of a surface, in accordance with Definition 1, is that it can easily be extended to the definition of an abstract manifold, not necessarily embedded in R n . For the time being, however, we shall be interested only in surfaces in R n . Let us consider some examples of such surfaces.
C (m) .
C (m)
Example 1 . We recall that if p i E (Rn , JR), i = smooth functions such that the system of equations
C(m)
4 For the tangent plane see Sect. 8.7.
1, . . . , n - k, is a set of
{
1201 Surfaces in Rn F 1 ( X 1 1 o o o , X k , X k + 1 1 o o o 1 Xn ) _- 0 ,
165 (1202)
Fn - k ( X 1 , o o o , X k , X k + 1 1 o o o 1 X n ) - 0 _
has rank n - k at each point in the set S of its solutions, then either this system has no solutions at all or the set of its solutions forms a k-dimensional C ( m ) _smooth surface S in !R n 0
Proof. We shall verify that if S =/= 0, then S does indeed satisfy Definition 40 This follows from the implicit function theorem, which says that in some neighborhood of each point x 0 E S the system (1202) is equivalent, up to a
relabeling of the variables, to a system
where
{
Xn - Jn ( X 1 1 o o o 1 X k ) _
J k + 1 , 0 0 0 , f n E C ( m ) o By writing this last system as
we arrive at a parametric equation for the neighborhood of the point x 0 E S on So By an additional transformation one can obviously turn the domain into a canonical domain, for example, into J k and obtain a standard local chart (1201)0 D
Example 20
In particular, the sphere defined in
!Rn by the equation (1203) ( r > 0)
is an ( n - I )-dimensional smooth surface in !R n since the set S of solutions of Eqo (1203) is obviously nonempty and the gradient of the left-hand side of (1203) is nonzero at each point of So
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12 Surfaces and Differential Forms in lR n
When n =
2, we obtain the circle in IR2 given by
{
which can easily be parametrized locally by the polar angle () using the polar coordinates x 1 = r cos () , x2
=
r sin O . For fixed r > 0 the mapping () f-t (x i , x 2 ) (0) is a diffeomorphism on every interval of the form ()0 < () < ()0 + 2 7!' , and two charts (for example, those corresponding to values ()0 = 0 and ()0 = - 7!' ) suffice to produce an atlas of the circle. We could not get by with one canonical chart (12.1) here because a circle is compact , in contrast to IR 1 or J 1 = B 1 , and compactness is invariant under topological mappings. Polar (spherical) coordinates can also be used to parametrize the two dimensional sphere (x i ) 2 + (x 2 ) 2 + (x 3 ) 2 = r 2 in IR 3 . Denoting by 7/J the angle between the direction of the vector ( x 1 , x 2 , x 3 ) and the positive x 3 -axis (that is, 0 ::; 7/J ::; 7!' ) and by t.p the polar angle of the projection of the radius-vector (x 1 , x 2 , x 3 ) onto the (x 1 , x 2 )-plane, we obtain
{
x3 x2 x1
r cos 7/J , r sin 7/J sin t.p , r sin 7/J cos t.p .
In general polar coordinates ( r, 0 1 , . . . , On- I ) in !R n are introduced via the relations r cos 0 1 , x1 r sin 0 1 cos 02 , x2 (12.4) x n- I xn We recall the Jacobian
r sin 0 1 sin 0 2 r sin 0 1 sin 02
· . . . ·
· . . . ·
J = r n- l sin n- 2 () 1 sin n- 3 ()2
sin On- 2 cos On- 1 , sin On- 1 sin On- 1 .
°
0
0
°
0
sin ()n- 2
( 1 2.5)
for the transition ( 1 2.4) from polar coordinates ( r, 0 1 , . . . , On- 1 ) to Cartesian coordinates ( x 1 , . . . , x n ) in !R n . It is clear from the expression for the J aco bian that it is nonzero if, for example, 0 < ()i < 7!', i = 1 , . . . , n - 2, and r > 0. Hence, even without invoking the simple geometric meaning of the
12.1 Surfaces in Rn
167
parameters fh ' . . . ' en - 1 ' one can guarantee that for a fixed r > 0 the map ping ( e 1 , . . . , en - 1 ) H ( x 1 , . . . , Xn), being the restriction of a local diffeomor phism (r, e 1 , . . . , en - d H ( x l , . . . , x n ) is itself a local diffeomorphism. But the sphere is homogeneous under the group of orthogonal transformations of JRn, so that the possibility of constructing a local chart for a neighborhood of any point of the sphere now follows.
Example 3.
The cylinder (
x 1 ) 2 + . . . + (x k ) 2 = r 2 (r > 0) ' for k < n is an ( n - I)-dimensional surface in JRn that is the direct product of the (k - I)-dimensional sphere in the plane of the variables ( x 1 , . . . , x k ) and the ( n - k)-dimensional plane of the variables ( x k + 1 , . . . , x n ) . A local parametrization of this surface can obviously be obtained if we take the first k - I of the n - I parameters (t 1 , . . . , tn - 1 ) to be the polar coordinates e 1 ' . . . ' ek - 1 of a point of the ( k - I )-dimensional sphere in JR k and set t k , . . . , tn- 1 equal to x k + 1 , . . . , x n respectively. Example 4. If we take a curve (a one-dimensional surface) in the plane x = 0 of JR 3 endowed with Cartesian coordinates ( x, z) , and the curve does not y,
intersect the z-axis, we can rotate the curve about the z-axis and obtain a 2-dimensional surface. The local coordinates can be taken as the local coor dinates of the original curve (the meridian) and, for example, the angle of revolution (a local coordinate on a parallel of latitude) . In particular, if the original curve is a circle of radius a with center at (b, 0, 0) , for a < b we obtain the two-dimensional torus (Fig. I2. I ) . Its para metric equation can be represented in the form x = ( b + a cos 'lj;) cos <.p ,
{
y
= ( b + a cos 'lj;) sin <.p ,
z = a sin 'lj; , where 'lj; is the angular parameter on the original circle - the meridian - and <.p is the angle parameter on a parallel of latitude.
It is customary to refer to any surface homeomorphic to the torus of revolution just constructed as a torus (more precisely, a two-dimensional torus). As one can see, a two-dimensional torus is the direct product of two circles. Since a circle can be obtained from a closed interval by gluing together (identifying) its endpoints, a torus can be obtained from the direct product of two closed intervals (that is, a rectangle) by gluing the opposite sides together at corresponding points (Fig. I2.2) . In essence, we have already made use of this device earlier when we estab lished that the configuration space of a double pendulum is a two-dimensional torus, and that a path on the torus corresponds to a motion of the pendulum.
168
12 Surfaces and Differential Forms in ffi.n
- -
I I
I 1-
1
- - - - - - - - - -
Fig. 1 2 . 2 .
Fig. 1 2 . 1 .
Example 5 . If a flexible ribbon (rectangle) is glued along the arrows shown in Fig. 1 2.3,a, one can obtain an annulus (Fig. 12.3,c) or a cylindrical surface (Fig. 12.3,b) , which are the same from a topological point of view. (These two surfaces are homeomorphic. ) But if the ribbon is glued together along the arrows shown in Fig. 12.4a, we obtain a surface in JR. 3 (Fig. 12.4,b) called a Mobius band.5 Local coordinates on this surface can be naturally introduced using the coordinates on the plane in which the original rectangle lies.
[---------]
/
/
J[
- - - - - - - - a.
'
'
b.
Fig. 1 2 . 3 .
I
I
I
\
/
'
, ,.. - ,
'
-
', ...
-
-..
c.
-
....... '
'
\
'------1 ,--, -
/
/
I
[1] 1�1 [1] II ] [I ] [I} a.
Fig. 1 2 . 4 .
Example 6. Comparing the results of Examples 4 and 5 in accordance with the natural analogy, one can now prescribe how to glue a rectangle (Fig. 12.5,a) that combines elements of the torus and elements of the Mobius band. But, just as it was necessary to go outside JR. 2 in order to glue the 5 A.F. Mobius ( 1790-1868) - German mathematician and astronomer.
1 2 . 1 Surfaces in Rn
a.
169
b.
Fig. 1 2 . 5 .
Mobius band without tearing or self-intersections, the gluing prescribed here cannot be carried out in R 3 . However, this can be done in R 4 , resulting in a surface in R 4 usually called the Klein bottle. 6 An attempt to depict this surface has been undertaken in Fig. 12.5,b. This last example gives some idea of how a surface can be intrinsically described more easily than the same surface lying in a particular space Rn. Moreover, many important surfaces ( of different dimensions ) originally arise not as subsets of Rn, but, for example, as the phase spaces of mechanical systems or the geometric image of continuous transformation groups of auto morphisms, as the quotient spaces with respect to groups of automorphisms of the original space, and so on, and so forth. We confine ourselves for the time being to these introductory remarks, waiting to make them more pre cise until Chapter 15, where we shall give a general definition of a surface not necessarily lying in Rn. But already at this point, before the definition has even been given, we note that by a well-known theorem of Whitney7 any k-dimensional surface can be mapped homeomorphically onto a surface lying in R 2 k + 1 . Hence in considering surfaces in Rn we really lose nothing from the point of view of topological variety and classification. These questions, however, are somewhat off the topic of our modest requirements in geometry.
6
F.Ch. Klein (1849-1925) - outstanding German mathematician, the first to make a rigorous investigation of non-Euclidean geometry. An expert in the history of mathematics and one of the organizers of the "Encyclopiidie der mathematischen Wissenschaften." 7 H. Whitney (1907-1989) - American topologist, one of the founders of the theory of fiber bundles.
170
12 Surfaces and Differential Forms in lRn
1 2. 1 . 1 Problems and Exercises 1.
For each of the sets
En En En En
En given by the conditions
{ (x, y) E 1R 2 I x 2 - y 2 = a} , { (x, y, z) E 1R 3 I x 2 - y 2 = a} , { (x, y, z) E 1R 3 I x 2 + l - z 2 = a} , { z E C I I z 2 - 1 l = a} ,
depending on the value of the parameter a E JR, determine a) whether En is a surface; b) if so, what the dimension of En is; c) whether En is connected. Let f : lR n --t lR n be a smooth mapping satisfying the condition f o f = f. a) Show that the set j(JR n ) is a smooth surface in lR n . b) By what property of the mapping f is the dimension of this surface deter mined? 3. Let e 0 , e 1 , . . . , e n be an orthonormal basis in the Euclidean space JR n + I , let x = x 0 eo + x 1 e 1 + · · · + x n e n , let { x} be the point ( x 0 , x 1 , , x n ) , and let e 1 , , e n be a basis in lR n C JR n + I . The formulas 0 x - x0 eo for x ..J. "
•
•
•
• . •
from the points {eo } and { -eo } respectively. a) Determine the geometric meaning of these mappings. b) Verify that if t E lR n and t # 0, then (1/; 2 o 1/;� 1 ) (t) W• where 1/;� 1 = ( 1/J I I Sn \ {eo } ) - I · c) Show that the two charts 1/;� 1 = 'P I : lR n --t s n \ {eo } and 1/;2 1 = 'P 2 : lRn --t s n \ { - eo } form an atlas of the sphere s n C JR n + I . d) Prove that every atlas of the sphere must have at least two charts.
1 2 . 2 Orientation of a Surface We recall first of all that the transition from one frame e1 , . . . , en in JRn to a second frame el , . . . , e n is effected by means of the square matrix obtained from the expansions ej = aj ei . The determinant of this matrix is always nonzero, and the set of all frames divides into two equivalence classes, each
12.2 Orientation of a Surface
171
class containing all possible frames such that for any two of them the de terminant of the transition matrix is positive. Such equivalence classes are called orientation classes of frames in JR n . To define an orientation means to fix one of these orientation classes. Thus, the oriented space JR n is the space JR n itself together with a fixed orientation class of frames. To specify the orientation class it suffices to exhibit any of the frames in it, so that one can also say that the oriented space JR n is JR n together with a fixed frame in it. A frame in JR n generates a coordinate system in JR n , and the transition from one such coordinate system to another is effected by the matrix ( ai ) that is the transpose of the matrix ( aj ) that connects the two frames. Since the determinants of these two matrices are the same, everything that was said above about orientation can be repeated on the level of orientation classes of coordinate systems in JRn , placing in one class all the coordinate systems such that the transition matrix between any two systems in the same class has a positive Jacobian. Both of these essentially identical approaches to describing the concept of an orientation in JR n will also manifest themselves in describing the orien tation of a surface, to which we now turn. We recall, however, another connection between coordinates and frames in the case of curvilinear coordinate systems, a connection that will be useful in what is to follow. Let G and D be diffeomorphic domains lying in two copies of the space JRn endowed with Cartesian coordinates ( x 1 , . . . , xn ) and ( t 1 , . . . , t n ) respec tively. A diffeomorphism r.p : D --+ G can be regarded as the introduction of curvilinear coordinates (t 1 , . . . , t n ) into the domain G via the rule x = r.p(t), that is, the point x E G is endowed with the Cartesian coordinates (t 1 , . . . , t n ) of the point t = r.p - 1 (x) E D. If we consider a frame e 1 , . . . , e n of the tan gent space TlRf at each point t E D composed of the unit vectors along the coordinate directions, a field of frames arises in D, which can be regarded as the translations of the orthogonal frame of the original space JR n containing D, parallel to itself, to the points of D. Since r.p : D --+ G is a diffeomor phism, the mapping r.p'(t) : TDt --+ TGx =
172
12 Surfaces and Differential Forms in ffi. n
Fig. 1 2 . 6 .
mutual transitions between these coordinate systems. The Jacobians of these mappings at corresponding points of D 1 and D 2 are mutually inverse to each other and consequently have the same sign. If the domain G ( and together with it D 1 and D 2 ) is connected, then by the continuity and nonvanishing of the Jacobians under consideration, they have the same sign at all points of the domains D 1 and Dz respectively. Hence the set of all curvilinear coordinate systems introduced in a con nected domain G by this method divide into exactly two equivalence classes when each class is assigned systems whose mutual transitions are effected with a positive Jacobian. Such equivalence classes are called the orientation classes of curvilinear coordinate systems in G. To define an orientation in G means by definition to fix an orientation class of its curvilinear coordinate systems. It is not difficult to verify that curvilinear coordinate systems belonging to the same orientation class generate continuous fields of frames in G ( as described above ) that are in the same orientation class of the tangent space TGx at each point x E G. It can be shown in general that, if G is connected, the continuous fields of frames on G divide into exactly two equivalence classes if each class is assigned the fields whose frames belong to the same orientation class of frames of the space TGx at each point x E G ( in this connection, see Problems 3 and 4 at the end of this section ) . Thus the same orientation of a domain G can be defined in two com pletely equivalent ways: by exhibiting a curvilinear coordinate system in G, or by defining any continuous field of frames in G, all belonging to the same orientation class as the field of frames generated by this coordinate system. It is now clear that the orientation of a connected domain G is completely determined if a frame that orients TGx is prescribed at even one point x E G. This circumstance is widely used in practice. If such an orienting frame is defined at some point x o E G, and a curvilinear coordinate system
173
12.2 Orientation of a Surface
If G is an open set , not necessarily connected, since what has just been said is applicable to any connected component of G, it is necessary to define an orienting frame in each component of G in order to orient G. Hence, if there are m components, the set G admits 2 m different orientations. What has just been said about the orientation of a domain G C JR n can be repeated verbatim if instead of the domain G we consider a smooth k-dimensional surface S in JR n defined by a single chart (see Fig. 12.7) . In this case the curvilinear coordinate systems on S also divide naturally into two orientation classes in accordance with the sign of the Jacobian of their mutual transition transformations; fields of frames also arise on S; and the orientation can also be defined by an orienting frame in some tangent plane TSx0 to S.
Fig. 1 2 . 7 .
The only new element that arises here and requires verification is the implicitly occurring proposition that follows.
The mutual transitions from one curvilinear coordinate sys tem to another on a smooth surface S JRn are diffeomorphisms of the same degree of smoothness as the charts of the surface. Proof In fact , by the proposition in Sect . 12. 1 , we can regard any chart Jk U S locally as the restriction to J k n O(t) of a diffeomorphism :F : O(t) O(x) from some n-dimensional neighborhood O(t) of the point t E J k JRn to an n-dimensional neighborhood O(x) of x E S JRn , :F being of the same degree of smoothness as r.p. If now 'P I If UI and 'P2 : J� U2 are two such charts, then the action of the mapping r.p:; I 'P I (the transition from the first coordinate system to the second) which arises in the common domain of action can be represented locally as r.p2 I 'P I (t l , . . . , t k ) :F:; I :F(t l , . . . , t k , 0, . . . , 0) , where :FI and :F2 are Proposition 1.
--+
C
c
--+
C
o
o
:
--+
=
o
C
--+
the corresponding diffeomorphisms of the n-dimensional neighborhoods.
D
174
12 Surfaces and Differential Forms in JRn
We have studied all the essential components of the concept of an orien tation of a surface using the example of an elementary surface defined by a single chart . We now finish up this business with the final definitions relating to the case of an arbitrary smooth surface in R n . Let S be a smooth k-dimensional surface in R n , and let
consistent if their domains of action either do not intersect , or have a nonempty intersection for which the mutual transitions are effected by diffeomorphisms with positive Jacobian in their common domain of action.
Definition 1 . Two local charts of a surface are
Definition 2 . An atlas of a surface is an
consists of pairwise consistent charts. Definition 3 . A surface is
it is
nonorientable.
orienting atlas of the surface if it
orientable if it has an orienting atlas. Otherwise
In contrast to domains of R n or elementary surfaces defined by a single chart , an arbitrary surface may turn out to be nonorientable.
Example 1 . The Mobius band, as one can verify (see Problems 2 and the end of this section) , is a nonorientable surface.
3 at
Example 2. The Klein bottle is also a nonorientable surface, since it contains a Mobius band. This last fact can be seen immediately from the construction of the Klein bottle shown in Fig. 12.5. Example 3 . A circle and in general a k-dimensional sphere are orientable, as can be proved by exhibiting directly an atlas of the sphere consisting of consistent charts (see Example 2 of Sect . 12 . 1 ) . Example 4. The two-dimensional torus studied in Example 4 of Sect . 12.1 is also an orientable surface. Indeed, using the parametric equations of the torus exhibited in Example 4 of Sect . 12. 1 , one can easily exhibit an orienting atlas for it. We shall not go into detail, since a more visualizable method of controlling the orientability of sufficiently simple surfaces will be exhibited below, making it easy to verify the assertions in Examples 1-4. The formal description of the concept of orientation of a surface will be finished if we add Definitions 4 and 5 below to Definitions 1 , 2, and 3.
175
12.2 Orientation of a Surface
Two orienting atlases of a surface are equivalent if their union is also an orienting atlas of the surface. This relation is indeed an equivalence relation between orienting atlases of an orientable surface. Definition 4. An equivalence class of orienting atlases of a surface under this relation is called an orientation class of atlases or simply an orientation
of the surface.
Definition 5. An oriented surface is a surface with a fixed orientation class of atlases ( that is, a fixed orientation of the surface ) .
Thus orienting a surface means exhibiting a particular orientation class of orienting atlases of the surface by some means or other. Some special manifestations of the following proposition are already fa miliar to us. Proposition 2.
entable surface.
There exist precisely two orientations on a connected ori
They are usually called opposite orientations. The proof of Proposition 2 will be given in Subsect . 15.2.3. If an orientable surface is connected, an orientation of it can be defined by specifying any local chart of the surface or an orienting frame in any of its tangent planes. This fact is widely used in practice. When a surface has more than one connected component, such a local chart or frame is naturally to be exhibited in each component . The following way of defining an orientation of a surface embedded in a space that already carries an orientation is widely used in practice. Let S be an orientable ( n 1 ) -dimensional surface embedded in the Euclidean space JRn with a fixed orienting frame e 1 , . . . , en in JRn . Let TSx be the ( n 1 ) dimensional plane tangent to S at x E S, and n the vector orthogonal to TSx , that is, the vector normal to the surface S at x. If we agree that for the given vector n the frame e 1 , . . . , e n _ 1 is to be chosen in T Sx so that the frames (e 1 , . . . , en ) and (n, e 1 , . . . , e n _ 1 ) = (e 1 , . . . , en ) belong to the same orientation class on JR n ' then, as one can easily see, such frames ( e 1 ' . . . ' e n ) of the plane T Sx will themselves all turn out to belong to the same orientation class for this plane. Hence in this case defining an orientation class for TSx and along with it an orientation on a connected orientable surface can be done by defining the normal vector n ( Fig. 12 .8) . It is not difficult to verify ( see Problem 4) that the orientability of an (n 1 ) -dimensional surface embedded in the Euclidean space JRn is equiva lent to the existence of a continuous field of nonzero normal vectors on the surface. Hence, in particular, the orientability of the sphere and the torus follow obviously, as does the nonorientability of the Mobius band, as was stated in Examples 1-4. -
-
-
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12 Surfaces and Differential Forms in JR n
Fig. 1 2 . 8 .
In geometry the connected ( n 1 ) -dimensional surfaces in the Euclidean space JR n on which there exists a ( single-valued ) continuous field of unit normal vectors are called two-sided. Thus, for example, a sphere, torus, or plane in JR 3 is a two-sided surface, in contrast to the Mobius band, which is a one-sided surface in this sense. To finish our discussion of the concept of orientation of a surface, we make several remarks on the practical use of this concept in analysis. In the computations that are connected in analysis with oriented surfaces in JR n one usually finds first some local parametrization of the surface S with out bothering about orientation. In some tangent plant T Bx to the surface one then constructs a frame e 1 ' . . . ' e n- 1 consisting of ( velocity ) vectors tan gent to the coordinate lines of a chosen curvilinear coordinate system, that is, the orienting frame induced by this coordinate system. If the space JR n has been oriented and an orientation of S has been defined by a field of normal vectors, one chooses the vector n of the given field at the point X and compares the frame n, e 1 , . . . , e n- 1 with the frame e 1 , . . . , en that orients the space. If these are in the same orientation class, the local chart defines the required orientation of the surface in accordance with our convention. If these two frames are inconsistent , the chosen chart defines an orientation of the surface opposite to the one prescribed by the normal -
n.
It is clear that when there is a local chart of an ( n - 1 ) -dimensional surface, one can obtain a local chart of the required orientation ( the one prescribed by the fixed normal vector n to the two-sided hypersurface embedded in the oriented space JR n ) by a simple change in the order of the coordinates. In the one-dimensional case, in which a surface is a curve, the orientation is more often defined by the tangent vector to the curve at some point; in that case we often say the direction of motion along the curve rather than "the orientation of the curve." If an orienting frame has been chosen in JR 2 and a closed curve is given, the positive direction of circuitaround the domain D bounded by the curve is taken to be the direction such that the frame n, v, where n is the exterior
12.2 Orientation of a Surface
177
normal to the curve with respect to D and v is the velocity of the motion, is consistent with the orienting frame in � 2 . This means, for example, that for the traditional frame drawn in the plane a positive circuit is "counterclockwise" , in which the domain is always "on the left" . In this connection the orientation of the plane itself or of a portion of the plane is often defined by giving the positive direction along some closed curve, usually a circle, rather than a frame in JR. 2 . Defining such a direction amounts to exhibiting the direction of shortest rotation from the first vector in the frame until it coincides with the second, which is equivalent to defining an orientation class of frames on the plane. 12.2. 1 Problems and Exercises 1. Is the atlas of the sphere exhibited in Problem 3 c) of Sect. 12. 1 an orienting atlas of the sphere? 2. a) Using Example 4 of Sect. 12. 1 , exhibit an orienting atlas of the two dimensional torus. b) Prove that there does not exist an orienting atlas for the Mobius band. c) Show that under a diffeomorphism f : D -+ D an orientable surface S C D maps to an orientable surface S C D.
a) Verify that the curvilinear coordinate systems on a domain G C JR n belong ing to the same orientation class generate continuous fields of frames in G that determine frames of the same orientation class on the space TGx at each point 3.
X E G. b) Show that in a connected domain G C JR n the continuous fields of frames divide into exactly two orientation classes. c) Use the example of the sphere to show that a smooth surface S C JR n may be orientable even those there is no continuous field of frames in the tangent spaces to S. d) Prove that on a connected orientable surface one can define exactly two different orientations. 4. a) A subspace lR n - 1 has been fixed, a vector E lR n \ lR n - 1 has been chosen, along with two frames (e 1 , . . . , e n _ 1 ) and (l 1 , . . . , ln _ 1 ) of the subspace lR n - 1 . Verify that these frames belong to the same orientation class of frames of JR n - 1 if and only if the frames (v, e 1 , . . . , e n _ 1 ) and (v, l 1 , . . . , ln _ 1 ) define the same orientation on lR n . b) Show that a smooth hypersurface S C JR n is orientable if and only if there exists a continuous field of unit normal vectors to S. Hence, in particular, it follows that a two-sided surface is orientable. c) Show that if grad F =/= 0, then the surface defined by F( x 1 , . . . , x m ) = 0 is orientable (assuming that the equation has solutions) . v
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12 Surfaces and Differential Forms in JR. n
d ) Generalize the preceding result to the case of a surface defined by a system of equations. e ) Explain why not every smooth two-dimensional surface in JR3 can be defined by an equation F(x, y, z ) = 0, where F is a smooth surface having no critical points ( a surface for which grad F i= 0 at all points ) .
1 2 . 3 The Boundary of a Surface and its Orientation 1 2 . 3 . 1 Surfaces with Boundary
Let !R n be a Euclidean space of dimension k endowed with Cartesian coordi nates t 1 , . . . , t k . Consider the half-space H k : = { t E JR k I t 1 :<::; 0 of the space JR k . The hyperplane 8H k := { t E JR k I t 1 = 0} will be called the boundary of the half-space H k . We remark that the set H k : = H k \ 8H k , that is, the open part of H k , is an elementary k-dimensional surface. The half-space H k itself does not formally satisfy the definition of a surface because of the presence of the boundary points from 8H k . The set H k is the standard model for surfaces with boundary, which we shall now describe.
!Rn is a ( k-dimensional ) surface with boundary if every point x E S has a neighborhood U in S homemorphic either to JR k or to H k . Definition 2 . If a point x E U corresponds to a point of the boundary 8H k under the homeomorphism of Definition 1 , then x is called a boundary point of the surface ( with boundary ) S and of its neighborhood U. The set of all such boundary points is called the boundary of the surface S. Definition 1. A set S
c
As a rule, the boundary of a surface S will be denoted aS. We note that for k = 1 the space 8H k consists of a single point. Hence, preserving the relation 8H k = JR k -l , we shall from now on take IR 0 to consist of a single point and regard 8IR 0 as the empty set . We recall that under a homeomorphism 'Pij : Gi --+ Gj of the domain Gi c JR k onto the domain Gj c JR k the interior points of Gi map to interior points of the image 'Pij ( G i ) ( this a theorem of Brouwer ) . Consequently, the concept of a boundary point of the surface is independent of the choice of the local chart, that is, the concept is well defined. Formally Definition 1 includes the case of the surface described in Defi nition 1 of Sect. 1 2 . 1 . Comparing these definitions, we see that if S has no boundary points, we return to our previous definition of a surface, which can now be regarded as the definition of a surface without boundary. In this connection we note that the term "surface with boundary" is normally used when the set of boundary points is nonempty.
12.3 The Boundary of a Surface and its Orientation
179
The concept of a smooth surface S ( of class C ( m ) ) with boundary can be introduced, as for surfaces without boundary, by requiring that S have an atlas of charts of the given smoothness class. When doing this we assume that for charts of the form
The boundary of a k-dimensional surface of class C ( m ) is itself a surface of the same smoothness class, and is a surface without bound ary having dimension one less than the dimension of the original surface with boundary. Proof. Indeed, if A(S) = {(H k ,
U
ously an atlas of the same smoothness class for as. 0
We now give some simple examples of surfaces with boundary.
Example 1 . A closed n-dimensional ball 1f' in Rn is an n-dimensional surface with boundary. Its boundary aB" is the ( n - 1 ) -dimensional sphere ( see Figs. 12.8 and 12.9,a ) . The ball 1f' , which is often called in analogy with the two-dimensional case an n-dimensional disk, can be homeomorphically mapped to half of an n-dimensional sphere whose boundary is the equatorial ( n - I ) -dimensional sphere ( Fig. 12.9,b) .
a.
b.
Fig. 1 2 . 9 .
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12 Surfaces and Differential Forms in JRn
The closed cube Jn in !R n can be homeomorphically mapped to the closed ball aF along rays emanating from its center. Consequently Jn , like B n is an n-dimensional surface with boundary, which in this case is formed by the faces of the cube (Fig. 12. 10) . We note that on the edges, which are the intersections of the faces, it is obvious that no mapping of the cube onto the ball can be regular (that is, smooth and of rank n) . Example 2.
L L 1
1
Fig. 1 2 . 1 0 .
Example 3 . If the Mobius band is obtained by gluing together two opposite sides of a closed rectangle, as described in Example 5 of Sect. 12. 1 , the result is obviously a surface with boundary in IR 3 , and the boundary is homeomorphic to a circle (to be sure, the circle is knotted in JR3 ) . Under the other possible gluing of these sides the result i s a cylindrical surface whose boundary consists of two circles. This surface is homeomorphic to the usual planar annulus (see Fig. 12.3 and Example 5 of Sect. 12. 1 ) . Figures 1 2 . l la, 12. l lb, 12. 1 2a, 1 2 . 1 2b, 12. 13a, and 12. 13b, which we will use below, show pairwise homeomorphic surfaces with boundary embedded in IR 2 and IR 3 . As one can see, the boundary of a surface may be disconnected, even when the surface itself is connected.
a.
Fig. 1 2 . 1 1 .
b.
Fig. 1 2 . 1 2 .
€@ 1
1
12.3 The Boundary of a Surface and its Orientation
a.
181
Fig. 1 2 . 13.
12.3.2 Making the Orientations of a Surface and its Boundary Consistent
If an orienting orthoframe e 1 , . . . , e k that induces Cartesian coordinates , x k is fixed in � k , the vectors e2 , . . . , e k define an orientation on the boundary aH k = � k - 1 of aH k = {x E � k l x 1 ::::; 0 } which is regarded as the orientation of the half-space H k consistent with the orientation of the half-space H k given by the frame e 1 , . . . , e k . In the case k = 1 where aH k = � k - 1 = � 0 is a point, a special convention needs to be made as to how to orient the point . By definition, the point is oriented by assigning a sign + or - to it. In the case aH 1 = � 0 , we take (�0 , +), or more briefly +�0 . We now wish to determine what is meant in general by consistency of the orientation of a surface and its boundary. This is very important in carrying out computations connected with surface integrals, which will be discussed below. We begin by verifying the following general proposition.
x1 ,
.
.
•
The boundary as of a smooth orientable surface S is itself a smooth orientable surface (although possibly not connected). Proof. After we take account o f Proposition 1 all that remainsk i s t o ver ify that as is orientable. We shall show that if A(S) {H ,
U
182
1 2 Surfaces and Differential Forms in �n
0
aw 2
7fi2
J(t o ) =
0
aw 2
aw 2
aw 2
7fi2
7ft�<
7ft�<
aw k
aw k
7ft�<
7ft�<
since for t 1 0 we must also have f:1 = 'lj; 1 (0 , t 2 , . . . , t k ) 0 (boundary points map to boundary points under a diffeomorphism) . It now remains only to remark that when t 1 < 0 we must also have F 'lj; 1 (t 1 , t 2 , . . . , t k ) < 0 (since t 'lj;(t) E H k ), so that the value of �(0 , t 2 , . . . , t k ) cannot be negative. By hypothesis J(t0) > 0 , and since � (O, t 2 , , t k ) > 0 it follows from the equality given above connecting the determinants that the Jacobian of the mapping '1/J i a k '1/J(O, t 2 , . . . , t k ) is positive. D H We note that the case of a one-dimensional surface (k = 1 ) in Proposi =
=
=
•
u
.
=
.
=
tion 2 and Definition 3 below must be handled by a special convention in accordance with the convention adopted at the beginning of this subsection.
= {(Hk , cpi , Ui )} n {(JR.k , cpj , Uj)} is an orienting atlas of standard local charts of the surface S with boundary aS, then A( aS) = {(JR. k - 1 , cp l a H k =JR k - u aUi )} is an orienting atlas for the boundary. The orientation of aS that it defines is said to be the orientation consistent with the orientation of the surface. Definition 3. If A(S)
To finish our discussion of orientation of the boundary of an orientable surface, we make two useful remarks.
Remark 1 . In practice, as already noted above, an orientation of a surface embedded in JR. n is often defined by a frame of tangent vectors to the surface. For that reason, the verification of the consistency of the orientation of the surface and its boundary in this case can be carried out as follows. Take a k-dimensional plane T Sx0 tangent to the smooth surface S at the point x 0 E aS. Since the local structure of S near xo is the same as the structure of the half-space H k near 0 E aH k , directing the first vector of the orthoframe e 1 , e 2 , . . . , e k E T Sxo along the normal to aS and in the direction exterior to the local projection of S on T Sxo , we obtain a frame e 2 , . . . , e k in the (k - 1 ) dimensional plane Ta Sx o tangent to aS at X o , which defines an orientation of TaSxo , and hence also of aS, consistent with orientation of the surface S defined by the given frame e l , e 2 , . . . , e k . Figures 12.9-1 2 . 1 2 show the process and the result of making the orien tations of a surface and its boundary consistent using a simple example. We note that this scheme presumes that it is possible to translate a frame that defines the orientation of S to different points of the surface and its boundary, which, as examples show, may be disconnected.
12.3 The Boundary of a Surface and its Orientation
183
Remark 2. 1In the oriented space JR k we 1consider the half-space H� H k {x E JR k l x :::; 0} and H! {x E JR k l x ;:::1 0} with the orientation induced {x E JR k j x 0} is the common boundary from JR k . The hyperplane =
F
=
=
=
=
of H� and H! . It is easy to see that the orientations of the hyperplane r consistent with the orientations of H� and H! are opposite to each other. This also applies to the case k = I , by convention. Similarly, if an oriented k-dimensional surface is cut by some ( k - I) dimensional surface ( for example, a sphere intersected by its equator ) , two opposite orientations arise on the intersection, induced by the parts of the original surface adjacent to it. This observation is often used in the theory of surface integrals. In addition, it can be used to determine the orientability of a piecewise smooth surface. We begin by giving the definition of such a surface. Definition 4. ( Inductive definition of a piecewise-smooth surface ) . We agree to call a point a zero-dimensional surface of any smoothness class. A piecewise smooth one-dimensional surface ( piecewise smooth curve ) is a curve in JRn which breaks into smooth one-dimensional surfaces ( curves ) when a finite or countable number of zero-dimensional surfaces are removed from it . A surface S C JRn of dimension k is piecewise smooth if a finite or count able number of piecewise smooth surfaces of dimension at most k - I can be removed from it in such a way that the remainder decomposes into smooth k-dimensional surfaces si ( with boundary or without ) .
Example 4. The boundary of a plane angle and the boundary of a square are piecewise-smooth curves. The boundary of a cube or the boundary of a right circular cone in JR3 are two-dimensional piecewise-smooth surfaces.
Let us now return to the orientation of a piecewise-smooth surface. A point ( zero-dimensional surface ) , as already pointed out, is by conven tion oriented by ascribing the sign + or - to it. In particular, the boundary of a closed interval [ a, b] C JR, which consists of the two points a and b is by convention consistent with the orientation of the closed interval from a to b if the orientation is ( a, - ) , ( b, +), or, in another notation, -a, +b. Now let us consider a k-dimensional piecewise smooth surface S C JRn ( k > 0) . We assume that the two smooth surfaces Sit and Si 2 in Definition 4 are oriented and abut each other along a smooth portion r of a ( k - I) dimensional surface ( edge ) . Orientations then arise on r, which is a boundary, consistent with the orientations of sit and si 2 . If these two orientations are opposite on every edge r c sit n si 2 , the original orientations of sit and si 2 are considered consistent. If sit n si 2 is empty or has dimension less than ( k - I ) , all orientations of sit and si 2 are consistent.
184
12 Surfaces and Differential Forms in ne
Definition 5 . A piecewise-smooth k-dimensional surface ( k > 0) will be con sidered orientable if up to a finite or countable number of piecewise-smooth surfaces of dimension at most ( k 1) it is the union of smooth orientable surfaces Si any two of which have a mutually consistent orientation. Example 5. The surface of a three-dimensional cube, as one can easily verify, is an orientable piecewise-smooth surface. In general, all the piecewise-smooth surfaces exhibited in Example 4 are orientable. -
Example 6. The Mobius band can easily be represented as the union of two orientable smooth surfaces that abut along a piece of the boundary. But these surfaces cannot be oriented consistently. One can verify that the Mobius band is not an orientable surface, even from the point of view of Definition 5.
1 2 . 3 . 3 Problems and Exercises 1 . a) Is it true that the boundary of a surface S C Rn is the set S \ S, where S is the closure of S in Rn? b) Do the surfaces sl = { (x, y) E 1R 2 I l < x 2 + y 2 < 2 } and s2 = { (x, y) I O < 2 x + y 2 have a boundary? c) Give the boundary of the surfaces S1 = { (x, y) E 1R 2 I l :::; x 2 + y 2 < 2 } and S2 = { (x, y) E 1R 2 I l :::; x 2 + y 2 } . 2 . Give an example of a nonorientable surface with an orientable boundary. 3. a) Each face I k = {x E R k l lx i l < 1 , i = 1 , . . . , k} is parallel to the corresponding (k - I)-dimensional coordinate hyperplane in R k , so that one may consider the same frame and the same coordinate system in the face as in the hyperplane. On which faces is the resulting orientation consistent with the orientation of the cube I k induced by the orientation of R k , and on which is it not consistent? Consider successively the cases k = 2, k = 3, and k = n . b) The local chart (t l , t 2 ) H (sin e cos t 2 , sin e sin t 2 , cos t 1 ) acts in a certain domain of the hemisphere S = { (x, y, z) E JR 3 I x 2 + y 2 + z 2 = 1 1\ z > 0} , and the local chart t H (cos t, sin t, 0) acts in a certain domain of the boundary aS of this hemisphere. Determine whether these charts give a consistent orientation of the surface s and its boundary as. c) Construct the field of frames on the hemisphere S and its boundary as induced by the local charts shown in b) . d) On the boundary as of the hemisphere S exhibit a frame that defines the orientation of the boundary consistent with the orientation of the hemisphere given in c) . e) Define the orientation of the hemisphere S obtained in c) using a normal vector to S C JR 3 . 4. a) Verify that the Mobius band is not an orientable surface even from the point of view of Definition 5. b) Show that if S is a smooth surface in Rn , determining its orientability as a smooth surface and as a piecewise-smooth surface are equivalent processes.
12.4 The Area of a Surface in Euclidean Space
185
5. a) We shall say that a set S C JRn is a k-dimensional surface with boundary if for each point x E S there exists a neighborhood U(x) E JRn and a diffeomorphism 'ljJ : U(x) --t rn of this neighborhood onto the standard cube In C JRn under which '1/J(S n U(x)) coincides either with the cube I k = {t E In I t k +1 = · · · = tn = 0} or with a portion of it I k n {t E JRn l t k :::; 0} that is a k-dimensional open interval with one of its faces attached. Based on what was said in Sect. 12. 1 in the discussion of the concept of a surface, show that this definition of a surface with boundary is equivalent to Definition 1 . b ) Is it true that if f E cCZ l (H\ JR) , where H k = {x E 1R k l x 1 :::; 0}, then for every point x E 8H k one can find a neighborhood of it U(x) in JR k and a function :F E cCZ l (U(x) , JR) such that :F I k ? = II k
H nU(x)
H nU(x)
c) If the definition given in part a) is used to describe a smooth surface with boundary, that is, we regard 'ljJ as a smooth mapping of maximal rank, will this definition of a smooth surface with boundary be the same as the one adopted in Sect. 12.3?
12.4 The Area of a Surface in Euclidean Space We now turn to the problem of defining the area of a k-dimensional piecewise smooth surface embedded in the Euclidean space JR n , n :2 k. We begin by recalling that if e 1 , . . . , e are k vectors in Euclidean space ]Rk ' then the volume v (e l ' . . . ' e k ) of thek parallelepiped spanned by these vectors as edges can be computed as the determinant ( 12.6) of the matrix J = ( �{) whose rows are formed by the coordinates of these vectors in some orthonormal basis e 1 , . . . , e k of JR k . We note, however, that in actual fact formula ( 12.6) gives the so-called oriented volume of the paral lelepiped, rather than simply the volume. If V "I 0, the value of V given by ( 12.6) is positive or negative according as the frames e l ' . . . ' e k and e l ' . . . ' e k belong to the same or opposite orientation classes of JR k . We now remark that the product J J* of the matrix J and its transpose J* has elements that are none other than the matrix G = (g;j ) of pairwise inner products 9ij = (e ; , ej ) of these vectors, that is, the Gram matrix 8 of the system of vectors e 1 , . . . , e k . Thus det G = det ( J J* )
=
det J det J*
and hence the nonnegative value of the volume as 8
See the footnote on p. 503.
=
( det J ) 2 ,
( 1 2 . 7)
V ( e , . . . , e k ) can be obtained 1
( 12.8)
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12 Surfaces and Differential Forms in JRn
This last formula is convenient in that it is essentially coordinate-free, con taining only a set of geometric quantities that characterize the parallelepiped under consideration. In particular, if these same vectors e 1 , . . . , e k are re garded as embedded in n-dimensional Euclidean space !R n (n 2: k) , formula ( 12.8) for the k-dimensional volume (or k-dimensional surface area) of the parallelepiped they span remains unchanged. Now let r : D ---+ S C IR n be a k-dimensional smooth surface S in the Euclidean space !Rn defined in parametric form r = r(t 1 , . . . , t k ), that is, as a smooth vector-valued function r(t) = (x 1 , . . . , x n )(t) defined in the domain D c JR k . Let e 1 , . . . , e k be the orthonormal basis in JR k that generates the coordinate system W , . . . , t k ). After fixing a point t 0 = ( t6, . . . , t�) E D, we take the positive numbers h 1 , . . . , h k to be so small that the parallelepiped I spanned by the vectors h i e i E T Dt0 , i = 1 , . . . , k, attached at the point t 0 is contained in D. Under the mapping D ---+ S a figure Is on the surface S, which we may pro visionally call a curvilinear parallelepiped, corresponds to the parallelepiped I (see Fig. 12. 14, which corresponds to the case k = 2, n = 3) . Since
r (t0l , . . . , t 0i - l , t0i + hi , t 0i + l , . . . , t 0k ) - r(to1 , ' t�. - 1 ' t�,. t�+1 ' t ok ) = otori (to) h'. + o(h' ) ' a displacement in !R n from r(t0) that can be replaced, up to o( h i ), by the partial differential 8�; (t 0 ) h i =: r i h i as h i ---+ 0 corresponds to displacement from t 0 by h i e i . Thus, for small values of h i , i 1 , . . . , k, the curvilinear parallelepiped Is differs only slightly from the parallelepiped spanned by the vectors h 1 r 1 , . . . , h i r k tangent to the surface S at r(t0). Assuming on that basis that the volume L1 V of the curvilinear parallelepiped Is must also be .
0
0
0
'
0
0
0
=
close to the volume of the standard parallelepiped just exhibited, we find the
Fig. 1 2 . 14.
12.4 The Area of a Surface in Euclidean Space
187
approximate formula ( 12.9) where we have set 9ij (t0) = (r i , rj )(t 0 ) and Llt i = h i , i, j = 1 , . . . ,k. If we now tile the entire space JR k containing the parameter domain D with k-dimensional parallelepipeds of small diameter d, take the ones that are contained in D, compute an approximate value of the k-dimensional volume of their images using formula (12.9), and then sum the resulting values, we arrive at the quantity a
which can be regarded as an approximation to the k-dimensional volume or area of the surface S under consideration, and this approximate value should become more precise as d -+ 0. Thus we adopt the following definition.
area ( or k-dimensional volume) of a smooth k dimensional surface S given parametrically by D 3 t H r(t) E S and embed ded in the Euclidean space JRn is the quantity
Definition 1. The
Vk (S) := j Jdet ((ri , rj)) dt 1 · . · dt k .
(12. 10)
D
Let us see how formula (12. 10) looks in the cases that we already know about. For k = 1 the domain D c lR 1 is an interval with certain endpoints a and b ( a < b) on the line lR 1 , and S is a curve in JRn in this case. Thus for k = 1 formula (12. 10) becoms the formula b
b
V1 (S) = j l r(t) l dt = j )(x 1 ) 2 + · · · + (xn) 2 (t) dt a
a
for computing the length of a curve. If k = n, then S is an n-dimensional domain in JRn diffeomorphic to D. In this case the Jacobian matrix J = x'(t) of the mapping D 3 (t 1 , . . . , tn) = t H r(t) = (x 1 , . . . , xn)(t) E S is a square matrix. Now using relation ( 1 2 . 7) and the formula for change of variable in a multiple integral, one can write
j
Vn (S) = J )det G(t) dt = l det x'(t) l dt = J dx = V(S) , D
D
S
That is, as one should have expected, we have arrived at the volume of the domain S in JRn.
188
12 Surfaces and Differential Forms in JR n
We note that for k = 2 , n = 3, that is, when S is a two-dimensional surface in IR3 , one often replaces the standard notation 9ij = (r i , rj ) by the following: a : = V2 ( S ) , E : = 91 1 = (r 1 , r 1 ) , F : = 91 2 = 92 1 = (r 1 , r 2 ) , G : = 922 = (r 2 , r 2 ) ; and one writes u, v respectively instead of t l , t 2 . In this notation formula ( 12 . 10) assumes the form a=
JJ J
EG
-
F 2 du dv .
D
In particular, if u = x, v = y, and the surface S is the graph of a smooth real-valued function z = f ( x, y ) defined in a domain D C IR 2 , then, as one can easily compute, a=
jj )
1 + ( !�) 2 + (!� ) 2 dx dy .
D
We now return once again to Definition 1 and make a number of remarks that will be useful later.
Remark 1 . Definition 1 makes sense only when the integral on the right-hand side of ( 1 2 . 10) exists. It demonstrably exists, for example, if D is a Jordan measurable domain and r E c ( l l (D, !Rn ). Remark 2. If the surface S in Definition 1 is partitioned into a finite number of surfaces S1 , . . . , Sm with piecewise smooth boundaries, the same kind of partition of the domain D into domains D 1 , . . . , Dm corresponding to these surfaces will correspond to this partition. If the surface S had area in the sense of Eq. ( 1 2 . 10) , then the quantities Vk ( Sa ) =
JJ
Da
det (r i , r1 ) (t) dt
are defined for each value of a = 1 , . . . , m. By the additivity of the integral, it follows that
We have thus established that the area of a k-dimensional surface is ad ditive in the same sense as the ordinary multiple integral.
Remark 3. This last remark allows us to exhaust the domain D when neces sary, and thereby to extend the meaning of the formula ( 12. 10) , in which the integral may now be interpreted as an improper integral. Remark 4. More importantly, the additivity of area can be used to define the area of an arbitrary smooth or even piecewise smooth surface ( not necessarily given by a single chart ) .
12.4 The Area of a Surface in Euclidean Space
189
Definition 2 . Let S be an arbitrary piecewise smooth k-dimensional surface in JR.n . If, after a finite or countable number of piecewise smooth surfaces of
dimension at most k - 1 are removed, it breaks up into a finite or countable number of smooth parametrized surfaces S1 , . . . , Sm , . . . , we set
The additivity of the multiple integral makes it possible to verify that the quantity Vk (S) so defined is independent of the way in which the surface S is partitioned into smooth pieces S1 , . . . , Sm , . . . , each of which is contained in the range of some local chart of the surface S. We further remark that it follows easily from the definitions of smooth and piecewise smooth surfaces that the partition of S into parametrized pieces, as described in Definition 2, is always possible, and can even be done while observing the natural additional requirement that the partition be locally finite. The latter means that any compact set K C S can intersect only a finite number of the surfaces S 1 , . . . , Sm , . . . . This can be expressed more vividly in another way: every point of S must have a neighborhood that intersects at most a finite number of the sets S 1 , . . . , Sm , . . . .
Remark 5. The basic formula (12. 10) contains a system of curvilinear coor dinates t 1 , . . . , t k . For that reason, it is natural to verify that the quantity Vk (S) defined by (12. 10) (and thereby also the quantity Vk (S) from Defi nition 2) is invariant under a diffeomorphic transition D 3 (? , . . . , tk ) f--t t = ( t i , . . . , t k ) E D to new curvilinear coordinates ? , . . . , tk varying in the domain D C JR. k . Proof.
For the verification it suffices to remark that the matrices and
0
( / or or ) ) g - ,1 - \ ofi ' ov .
_
(
-·
·
)
_
at corresponding points of the domains D and D are connected by the re lation G = J*GJ, where J = ( �!� ) is the Jacobian matrix of the map ping D 3 t f--t t E D and J* is the transpose of the matrix J. Thus, det G (t) = det G(t) (det J) 2 (t) , from which it follows that
J y'
det G(t) dt =
D
JJ
i5
det G ( t(t) ) I J(t) l dt =
JV
i5
det G(t) dt . D
Thus, we have given a definition of the k-dimensional volume or area of a k-dimensional piecewise-smooth surface that is independent of the choice of coordinate system.
Remark 6.
We precede the remark with a definition.
190
12 Surfaces and Differential Forms in !Rn
Definition 3. A set E embedded in a k-dimensional piecewise-smooth sur face S is a set of k-dimensional measure zero or has area zero in the Lebesgue sense if for every c: > 0 it can be covered by a finite or countable sys tem 81 , . . . , Sm , . . . of (possibly intersecting) surfaces 801 c S such that I: Vk (Sa ) < c: . 01
As one can see, this is a verbatim repetition of the definition of a set of measure zero in JR k . It is easy to see that in the parameter domain D of any local chart cp : D """""* S of a piecewise-smooth surface S the set cp - 1 (E) C D C JR k of k dimensional measure zero corresponds to such a set E. One can even verify that this is the characteristic property of sets E C S of measure zero. In the practical computation of areas and the surface integrals introduced below, it is useful to keep in mind that if a piecewise-smooth surface has been obtained from a piecewise-smooth_ surface S by removing a set E of measure zero from S, then the areas of S and S are the same.
S
The usefulness of this remark lies in the fact that it is often easy to remove such a set of measure zero fro� a piecewise-smooth surface in such a way that the result is a smooth surface S defined by a single chart. But then the area of S and hence the area of S also can be computed directly by formula (12. 10) . Let us consider some examples. t-t
Example 1 . The mapping ]0, 2n(3 t (R cos t, R sin t) E JR2 is a chart for the arc S of the circle x 2 + y 2 = R 2 obtained by removing the single point E = (R, 0) from that circle. Since E is a set of measure zero on S, we can
write
2 71'
S j VR2 sin2 t + R2 cos2 t dt
V1 (S) = V1 ( ) =
0
=
2n R .
Example 2. In Example 4 of Sect. 12. 1 we exhibited the following parametric representation of the two-dimensional torus S in IR 3 : r (cp, '!j; )
=
( ( b + a cos '!j; ) cos cp, ( b + a cos '!j; ) sin cp, a sin '!j; ) .
{
In the domain D = (cp, 'l/J) I O < cp < 2n, 0 < 'lj; < 2n} the mapping ( cp, 'lj;) t-t r ( cp, 'lj;) is a diffeomorphism. The image of the domain D un der this diffeomorphism differs from the torus by the set E consisting of the coordinate line cp = 2n and the line 'lj; = 2n. the set E thus consists of one parallel of latitude and one meridian of longitude of the torus, and, as one can easily see, has measure zero. Hence the area of the torus can be found by formula (12. 10) starting from this parametric representation, considered within the domain D.
S
12.4 The Area of a Surface in Euclidean Space
191
Let us carry out the necessary computations: r "' = ( - ( b + a cos ¢) sin cp, ( b + a cos ¢) cos cp, 0) , r 'lj; (-asin¢) coscp, -asin¢sincp, acos¢) , 911 = (r"' , r"' ) = (b + acos¢) 2 , 91 2 = 92 1 = (r"' , r'lj; ) 2 o , 922 = (r1f; , r'lj; ) = a , detG = 1 992111 99221 2 1 = a2 (b + acos¢) 2 . Consequently, 27r 27r V2 (S) = V2 (S) J dcp J a(b + a cos¢) d¢ 4n2 ab . 0 0 In conclusion that ofthepiecewise-smooth method indicatedcurves in Definition 2 can now be used to computewe thenoteareas and surfaces. =
=
=
=
12.4. 1 Problems and Exercises
a) Let P and P be two hyperplanes in the Euclidean space JR n , D a subdomain of P, and i5 the orthogonal projection of D on the hyperplane P. Show that the (n- 1 ) dimensional areas of D and i5 are connected by the :_elation O"(D) = O"(D) cos a , where a is the angle between the hyperplanes P and P. b) Taking account of the result of a) , give the geometric meaning of the formula dO" = y'1 + (f� ) 2 + (f� ) 2 dx dy for the element of area of the graph of a smooth function z = f(x, y) in three-dimensional Euclidean space. c) Show that if the surface S in Euclidean space JR 3 is defined as a smooth vector-valued function = defined in a domain D C JR 2 , then the area of the surface S can be found by the formula 1.
r r (u, v) O" ( S )
[r�, r�]
=
// l [r�,r�J i dudv , D
where i:s the vector product of g: and g: . d) Verify that if the surface S C JR 3 is defined by the equation F(x, y, z ) = 0 and the domain U of the surface S projects orthogonally in a one-to-one manner onto the domain D of the xy-plane, we have the formula grad F O"(U) = � dx dy .
!! D
2. Find the area of the spherical rectangle formed by two parallels of latitude and two meridians of longitude of the sphere S C JR 3 .
12 Surfaces and Differential Forms in !R n
192
3. a) Let (r, r.p, h) be cylindrical coordinates in IR 3 • A smooth curve lying in the plane r.p = r.po and defined there by the equation r = r ( s ) , where s is the arc length parameter, is revolved about the h-axis. Show that the area of the surface obtained by revolving the piece of this curve corresponding to the closed interval [ s 1 , s 2 ] of variation of the parameter s can be found by the formula •2
u
= 27r
J r ( s) ds .
81
b) The graph of a smooth nonnegative function y = f(x) defined on a closed interval [a, b] C IR + is revolved about the x-axis, then about the y-axis. In each of these cases, write the formula for the area of the corresponding surface of revolution as an integral over the closed interval [a, b] . 4. a) The center of a ball of radius 1 slides along a smooth closed plane curve of length L. Show that the area of the surface of the tubular body thereby formed is
21r · 1 · L. b) Based on the result of part a) , find the area of the two-dimensional torus obtained by revolving a circle of radius a about an axis lying in the plane of the circle and lying at distance b > a from its center. 5 . Describe the helical surface defined in Cartesian coordinates (x, y, z) in IR 3 by the equation y = x tan hz = O , l z l ::; 27r h , and find the area of the portion of it for which r 2 < x 2 + y 2 ::; R2 . 6. a) Show that the area !tn - 1 of the unit sphere in !R n is where r(a) =
rG)) '
f e - x - dx. (In particular, if n is even, then r( � ) = ( � ) ! , while if n is odd, r( �) = <;�" fo.) b) By verifying that the volume Vn (r) of the ball of radius r in !Rn is � rn , r \. � ) "'
"'
1
n
2
0
show that � 1 r= 1 = !tn- 1 · c) Find the limit as n -+ oo of the ratio of the area of the hemisphere { x E !R n l lxl = 1 A x n > 0} to the area of its orthogonal projection on the plane x n = 0. d) Show that as n -+ oo , the majority of the volume of the n-dimensional ball is concentrated in an arbitrarily small neighborhood of the boundary sphere, and the majority of the area of the sphere is concentrated in an arbitrarily small neighborhood of its equator. e) Show that the following beautiful corollary on concentration phenomena fol lows from the observation made in d). A regular function that is continuous on a sphere of large dimension is nearly constant on it (recall pressure in thermodynamics) .
193
12.4 The Area of a Surface in Euclidean Space
Specifically, Let us consider, for example, functions satisfying a Lipschitz condition with a fixed constant. Then for any e > 0 and 6 > 0 there exists N such that for n > N and any function f : s n -+ lR there exists a value c with the following properties: the area of the set on which the value of f differs from c by more than e is at most 6 times the area of the whole sphere. 7. a) Let X 1 , . . . , X k be a system of vectors in Euclidean space JR n , that the Gram determinant of this sytem can be represented as
det ( ( x; , Xj ) )
where
. =
P,1 . . , k
= 1 :5 il < .. · < ik :5 n
det
(
x ;•
x' ;
n
2::
k. Show
)
x �l . . . x �k b) Explain the geometric meaning of the quantities P; 1 .. . ; k from a) and state the result of a) as the Pythagorean theorem for measures of arbitrary dimension k, 1 ::; k ::; n. c) Now explain the formula
a=I D
det 2 L 1 :5 i l < . . · < ik :5 n
(
8tr
axil
.)
axil . . {itT< . . ' . . . . . .. . . . . ' '
8x" k 8x" k 8tr · · · a t k
dt 1 . . . dt n
for the area of a k-dimensional surface given in the parametric form x x ( t \ . . . , t k ) , t E D C JR k .
8. a) Verify that the quantity Vk (S) in Definition 2 really is independent of the method of partitioning the surface S into smooth pieces S1 , . . . , Sm , . . . . b) Show that a piecewise-smooth surface S admits the locally finite partition into pieces S1 , . . . , Sm , . . . described in Definition 2. c) Show that a set of measure 0 can always be removed from a piecewise-smooth surface S so as to leave a smooth surface S S \ E that can be described by a single standard local chart r.p : I -+ S.
=
9. The length of a curve, like the high-school definition of the circumference of a circle, is often defined as the limit of the lengths of suitably inscribed broken lines. The limit is taken as the length of the links in the inscribed broken lines tend to zero. The following simple example, due to H. Schwarz, shows that the analogous procedure in an attempt to define the area of even a very simple smooth surface in terms of the areas of polyhedral surfaces "inscribed" in it, may lead to an absurdity. In a cylinder of radius R and height H we inscribe a polyhedron as follows. Cut the cylinder into m equal cylinders each of height H/m by means of horizontal planes. Break each of the m + 1 circles of intersection (including the upper and lower bases of the original cylinder) into n equal parts so that the points of division on each
194
1 2 Surfaces and Differential Forms in ffi:. n
circle lie beneath the midpoints of the points of division of the circle immediately above. We now take a pair of division points of each circle and the point lying directly above or below the midpoint of the arc whose endpoints they are. These three points form a triangle, and the set of all such triangles forms a polyhedral surface inscribed in the original cylindrical surface (the lateral surface of a right circular cylinder) . In shape this polyhedron resembles the calf of a boot that has been crumpled like an accordion. For that reason it is often called the Schwarz boot. a) Show that if m and n are made to tend to infinity in such a way that the ratio n 2 /m tends to zero, then the area of the polyhedral surface just constructed will increase without bound, even though the dimensions of each of its faces (each triangle) tend to zero. b) If n and m tend to infinity in such a way that the ratio m/n 2 tends to some finite limit p, the area of the polyhedral surfaces will tend to a finite limit, which may be larger than, smaller than, or (when p = 0) equal to the area of the original cylindrical surface. c) Compare the method of introducing the area of a smooth surface described here with what was just done above, and explain why the results are the same in the one-dimensional case, but in general not in the two-dimensional case. What are the conditions on the sequence of inscribed polyhedral surfaces that guaranteee that the two results will be the same?
The isoperimetric inequality Let V(E) denote the volume of a set E C ffi:. n , and A + B the (vector) sum of the sets A, B C ffi:. n . (The sum in the sense of Minkowski is meant. See Problem 4 in Sect. 1 1 .2.) Let B be a ball of radius h. Then A + B = : A h is the h-neighborhood of the set A. The quantity . V(Ah ) - V(A) _ .. f.J+ (aA) lIm h h -+ 0 is called the Minkowski outer area of the boundary 8A of A. a) Show that if aA is a smooth or sufficiently regular surface, then Ji+ (aA) equals the usual area of the surface aA. b) Using the Brunn-Minkowski inequality (Problem 4 of Sect. 1 1 .2), obtain now the classical isoperimetric inequality in ffi:.n : 1 0.
-
here V is the volume of the unit ball in ffi:. n , and Ji(SA ) the area of the ( (n - I) dimensional) surface of the ball having the same volume as A. The isoperimetric inequality means that a solid A C ffi:. n has boundary area Ji+ (aA) not less than that of a ball of the same volume.
12.5 Elementary Facts about Differential Forms
195
1 2 . 5 Elementary Facts about Differential Forms
We now known give anaselementary description of theparticular convenientattention mathematical maits chinery differential forms, paying here to algorithmic than 15.the details of the theoretical constructions, which will bepotential discussedrather in Chap. 1 2 . 5 . 1 Differential Forms: Definition and Examples
Having studiedandalgebra, thealready readermade is wellextensive acquainted with theconcept conceptin con of a listructing near form, we have use of that the differential calculus. In thatweprocess wediscussing encountered mostly sym metric forms. In the present subsection will be skew-symmetric (anti-symmetric) forms. : X k -+ Y of degree or order k defined on ordered We recall that a form L sets e 1 , . . . , e k of vectors of a vector space X and assuming values in a vector space Y is skew-symmetric or anti-symmetric if the value of the form changes sign when any pair of its arguments are interchanged, that is, of theIn particular, other vectors.if ei = ej then the value of the form will be zero, regardless Example The vector (cross) product [e 1 , e 2 J of two vectors in JR3 is a skew symmetric bilinear form with values in JR3 . Example The oriented volume V(e 1 , . . . , e k ) of the parallelepiped spanned bysymmetric the vectorsreal-valued e 1 , . . . , e k of JR k , defined by Eq. ( 1 2.6) of Sect. 12.4, is a skew k-form on JRk . ForcasetheY time being we shall be interested only in real-valued k-forms (the = !R) , even though everything that will be discussed below is tonumbers. the more general situation, for example, when Y is the field C ofapplicable complex A linear combination of skew-symmetric forms of the same degree is in turn a constitute skew-symmetric form,space.that is, the skew-symmetric forms of a given degree a vector In addition, inwhich algebraassigns one introduces thepairexterior product 1\ of skew symmetric forms, to an ordered AP, Bq of such forms (of degrees p and respectively) a skew-symmetric form AP 1\ Bq of degree p + This operation is associative: (AP 1\ Bq) 1\ cr AP 1\ (Bq 1\ cr ) , distributive: (AP + BP ) 1\ cq = AP 1\ cq + BP 1\ cq, skew-commutative: AP 1\ Bq ( - l ) Pq Bq 1\ AP . 1.
2.
q
q.
=
=
196
12 Surfaces and Differential Forms in IR.n
In particular, in the case of 1-forms A and B, we have anticommutativity A 1\ B = - B 1\ A, for the operations, like the anticommutativity of the vector
product shown in Example 1. The exterior product of forms is in fact a generalization of theintovector product.of the definition of the exterior product, we Without going the details take as knownthatforinthethetimecasebeing theexterior properties of thisof 1-forms operation£ just listed and observe of the product ,1 . . . , L k E .C(JRn , JR) the result 1\ 1\ is a k-form that assumes the value L1
· · ·
Lk
on the set of vectors �1 , . . . , �k If relation (12.11) is taken as thethatdefinition of theof left-hand side,A,it B,follows from properties of determinants in the case linear forms and C, we do indeed have A 1\ B = -B 1\ A and (A + B) 1\ C = A 1\ C + B 1\ C. Let us now consider some examples that will be useful below. Example Let 1r i E .C(JRn , JR),i i = 1, . . . , n , be the projections. More precisely, the nlinear function 1r JRn -+ lR isi such that on each vector n � = (e , . . . , � ) E JR it assumes the value 1r (� ) = �i of the projection ofwiththatformula vector(12.11) on thewecorresponding obtain coordinate axis. Then, in accordance ·
3.
:
(12.12) Example 4. The Cartesian coordinates of the vector product [� 1 , �2 ] of the vectors (�}, �?,by�?)theandequality � 2 = (�� ' �i, ��) in the Euclidean space JR3 , as is known, �are1 =defined
Thus, in accordance with the result of Example 3 we can write 7r 1 ( [� 1 ' � 2 ] ) 7r 2 1\ 7r 3 (� 1 ' �2 ) ' 7r 2 ( [� 1 , � 2 l ) = 7r3 1\ 7r 1 ( � 1 , � 2 ) ' 7r3 ( [� 1 , � 2 l ) = 7r 1 /\ 7r2 (� 1 , �2 ) . =
197
12.5 Elementary Facts about Differential Forms
Example Let f D lR be a function that is defined in a domain D !Rn and differentiable atisx0a linear E D. Asfunction is known, the ondifferential df(xvectors 0 ) of thee function at a point defined displacement from Moreunder precisely, on vectorsWeofrecall the tangent space 1 , . . . T, xDn xare0 totheD (orcoordinates !Rnthat ) at point. thein point consideration. that if x !Rn and e (e ' ' �n ) ' then df(xo)(e) ooxfl (xo)� 1 + · · · + ooxfn (xo) �n Dd ( xo ) . i (e) �i , or, more formally, dxi (x0 )(e) �i . If fi , . . . , fk In particular dx are real-valued functions defined wein Gobtain and differentiable at the point x0 E G, then in accordance with (12.11) --+
:
5.
C
=
0
0
0
=
=
=
=
(12.13) at the point Xo for the set e l , ,ek of vectors in the space TGxo ; and, in particular, 0
0
0
(12.14)
��1 ��k In this way skew-symmetric forms of degree k defined on the space TDxo TIR�0 !Rn have been obtained from the linear forms dfi , . . . , dfk defined on this space. Example If f E C( l ) ( D, lR), where D is a domain in !Rn , then the differen tial df(x) ofbeen the functions fa islinear definedfunction at anydf(x) point x E D, andT!Rfthis differen tial, as has stated, is TDx ( x ) lR on the tangent space TDx to D at x. In general the form df(x) f'(x) varies in passage one point to another in D. Thus a smooth scalar-valued func tigenerates on f from D lR generates a linear form df(x) at each point, or, as we say, spaces TDax .field of linear forms in D, defined on the corresponding tangent Definition 1 . We shall say that a real-valued differential p-form w is defined n P indefined the domain at each Dpoint !Rx E ifD.a skew-symmetric form w(x) (TDx ) lR is The number p is usually called the degree or order of w. In this connection the form w is often denoted wP. �
�
6.
:
:
�
:
--+
=
--+
C
p-
--+
198
1 2 Surfaces and Differential Forms in .!R n
Thus, thein Example field of the6 is differential df1-form of a smooth function f D � considered a differential in D, and w dxi 1 /\· ·1\dx i v is the simplest example of a differential form of degree p. Example Suppose a vector field D C �n is defined, that is, a vector F(x) isthisattached to eachgenerates point xtheE D.following When differential there is a Euclidean structure in �n vector field 1-form w} in D. If e is a vector attached to X E D, that is, e E TDx , we set w�(x)(e) ( F(x),e ) . It follows fromformproperties of thex inner product that w}(x) (F(x), · ) is indeed a linear at each point E D. Suchforce differential forms arise very frequently. For example, if F is a con tinuous field in D and e an infinitesimal displacement vector from the point xfrom E D, the element of work corresponding to this displacement, as is known physics, isFdefined preciselyD ofbythetheEuclidean quantity (space F(X) , �e)n. naturally Thus a force fiel d in a domain generates a differential 1-form w} in D, which it is natural to call the work form of the field F in this case. We remark that in Euclidean space then differential df of a smooth func fgenerated D � in the domain D � can also be regarded as the 1tion form by a isvector in this case the field F forgradevery f. Invector fact, by definition grad/ such field, that df(x)(e) gradf(x),e ) ( e E TDx . Example A vector field V defined in a domain1 D of the Euclidean space �n - of degree n - 1 . If at a point can al s o be regarded as a differential form Wy X E D we take the vector field V(x) and n - 1 additional vectors e 1 , . . . , en E TDx attached to the point x, then the oriented volume of the parallelepiped spannedwhose by therowsvectors V (X)coordinates , e 1 , . . . , e n - 1 , which is the determinant of the matrix are the of these vectors, will obviously be a skew-symmetric ( n - 1)-form with respect to the variables e 1 , . . . ,e n - 1 · For n one3 theof which form w� is isthegiven, usualresulting scalar triple product (V(x),e2-form 1 ,e2 ) ofw�vectors, V(x) in a skew-symmetric (V,example, ·, ·). if a steady flow of a fluid is taking place in the domain D and For V(x) the velocity vectorofatthethefluid poinpassing t x E D,through the quantity (V(x),e 1 ,e2 ) is the iselement of volume the (parallelogram) area spanned by thevectors smallevectors e, 1weE shall TDx and e 2 E TDx in unit time. By choosing different and e 1 different configuration, differently 2situated inobtain space,areas all havi(parallelograms) ng one vertex atof x.of For each such areaAstherehaswillbeenbe, stated, in general, avalue differentshowsvaluhow e (V(x),e ,e2 ) 1fluid the form w�(x). this much has throughelement the surface time,reason that is,weitoften characterizes the flux acrossflowed the chosen of areain. unit For that call the form w� :
=
7.
=
=
--+
:
C
=
=
8.
=
=
•
--+
12.5 Elementary Facts about Differential Forms
199
-l the flux form of the vector (and indeed its multidimensional analogue w� vield V in D. 12.5.2 Coordinate Expression of a Differential Form
Let usdifferential now investigate the coordinate expressionthatof skew-symmetric algebraic and forms and show, in particular, every differential is in a certain sense a linear combination of standard differential formsk-form of the formTo(12.14). abbreviate thefor notation, weoccur shall asassume summation over the range of(asallowable values indices that both superscripts and subscripts we did earlier in similar situations). Letvector L be aEk-linear form in JRn . If arepresentation basis e1 , . . . , en is�fixed inthatJRn ,basis, then i n each lR gets a coordinate in ei e e and the form L acquires the coordinate expression L (e 1 , . . . , e k ) L (��' ei, , . . . , ��k �ik ) = L( ei, , . . . , eik )��1 ��k . (12.15) numbers a i, , ... ,ik L( e i, , . . . , eik ) characterize the form L completely ifobviously theThebasis in which they have been obtained is known.to These numbersif and are symmetric or skew-symmetric with respect their indices onlyInifthethe case form ofL possesses the corresponding typecoordinate of symmetry. a skew-symmetric form L the representation can be transformed slightly. To make the direction of that transformation clear natural, let us 2-form considerintheJR3 .special case of (12.15) that occurs when L is aandskew-symmetric Then for the vectors i ' ei, and e � 1 e2 ��2 ei2 ' where i l ' i2 1, 2, 3, we obtain L (e l , e2 ) L (��' ei, , ��2 ei2 ) L (ei, , ei2 ) ��, ��2 L (e1 , e1 ) �}�� + L (e1 , e2 )�}�5 + L (e1 , e3 )�id + + L (e2 , el )�i�� + L (e2 , e2 )�?�i + L (e2 , e3 )�?�� + + L (e3 , el )�r�� + L (e3 , e2 )a�5 + L ( e3 , e3 ) �r�� L (e1 , e2 )(�i�i - �?�� ) + L (e1 , e3 )(�r�� - �r�i ) + + L (e2 , e3 )(�?�� - �r�i ) l 'Sil2:::
=
=
=
=
=
• • •
=
=
=
=
=
=
=
(12.16)
200
12 Surfaces and Differential Forms in JR n
Then, in accordance with formula ( 12.12 ) this last equality can be rewrit ten as Thus, any skew-symmetric form can be represented as a linear combination ( 12.17 ) 1r i 1 1\ · · 1· 1\ 7r i k , which are the exterior product formed from the ofelementary the k-forms1-forms 1r , . . . , 1rn in ]Rn . Now suppose that a differential k-form w is defined in some domain D C JRn along with a curvilinear coordinate system x 1 , . . . , xn . At each point x E D we fix the basis e 1 ( x), . . . , en ( x) of the space TD formed from the unit vectors along the coordinate axes. ( For example, if x 1 , . . . , xn are Cartesian coordinatesparallel in JRn , then e 1 ( x), . . . , en ( x) is simply the frame e 1 , . . . , en in JRn translated to itself to x.) Then at each point x E D we find by formulas ( 12.14 ) from and (the12 . 16origin ) that L
L =
x,
or
w(x)( � l , . . . , � k ) = L w (e ;, (x), . . . , e ; k (x)) dx i ' A · · · /\ dx i k (� 1 , . . . , � k ) l :=;i , <··-
12.18) Thus, every differential k-form is a combination of the elementary k-forms dxi ' 1\ · · · 1\ dxi k formed from the differentials of the coordinates. As a matter of fact, that is the reason... for the term "differential form. " The coefficients ; k (x) of the linear combination ( 12.18 ) generally de pend x, that is, they are functions defined in the domain in whiInch onparticular, thetheformpoint w k is given. we have long known the expansion of the differential of (x) dx 1 + · · · + of df(x) ox ( 12. 1 9 ) 1 oxn (x) dxn , and, as can be seen from the equalities w(x)
(
=
a; ,
=
-
(F, � ) (Fi ' e ;, (x), � i 2e; 2 (x) ) = (e;, (x), e ; 2 (x) ) F i ' (x)� i 2 g; 1 ; 2 (x)Fi ' (x)� i 2 = g;,; 2 (x)Fi ' (x) dxi2 (� ) , =
the expansion
=
=
=
12.5 Elementary Facts about Differential Forms
w�(x) = ( F (x) , · ) = (gi,i (x)Fi ' (x)) dx i = ai (x) dx i
201
(12.20)
also holds. In Cartesian coordinates this expansion looks especially simple: n w�(x) = ( F (x) , · ) = L Fi (x) dx i . (12.21) i= 1 Next, the following equality holds in JR3 : V 1 (x) V 2 (x) V 3 (x) w� (x)(€ 1 , € 2 ) = � f �� ��§r ���r �2 = V 1 (x) I ��� �� I + v2 ( x) I �� d1 ' 2 from which it follows that Similarly, expanding the determinant of order n for the form w�- 1 by minors along the first row, we obtain the expansion n- 1 w�- 1 = L ( - 1) i + 1 V i (x) dx 1 I\ · · · I\ dxi I\ · · · I\ dxn , i1
(12.23) = where the sign stands over the differential that is to be omitted in the
indicated term.
�
12.5.3 The Exterior Differential of a Form
All thatindividual has beenpoint said xupoftothenowdomain about differential forms essentially involved each of definition of the form and hadsucha purely algebraic character. The operation of (exterior) differentiation of formsLetisusspecific to analysis. agree from to define the 0-forms in a domain to be functions f : -+ lR defined innow thatondomain. Definition 2 . The (exterior) differential of a 0-form f, when f is a differ entiable function, is the usual differential df of that function. If a differential p-form (p 1) defined in a domain JRn D
2':
D C
w (x) = a;, ... ; v (x) dx i ' /\ · · · 1\ dx i v has differentiable coefficients a;, ···iv (x), then its (exterior) differential is the
form
202
12 Surfaces and Differential Forms in IR n
Using the expansion ( 12.19 ) for the differential of a function, and relying onrelation the distributivity of the exterior product of 1-forms, which follows from ( 12.11 ) , we conclude that dw(x) aai,ax'···i. v (x) dx '. 1\ dx '. ' 1\ =
·
·
·
. =
1\ dx ' v
that is, the (exterior) differential of a p-form (p 0) is always a form of degree pnote+ 1.that Definition 1 given above for a differential p-form in a domain We D IR. n , as one can now understand, is too general, since it does not in any way connect the forms w(x) corresponding to different points of the domain D. In actuality, the only forms used in analysis are those whose coordinates ·i v ( x) in a coordinate representation are sufficiently regular ( most often infinitely differentiable) functions in the domain D . The order of smoothness of the form w in the domain D IR.n is customarily characterized by the smallest order0 wiof tsmoothness of ofitsclass coefficients. The totality of all forms of degree p h coefficients C ( 00 l ( D, IR) is most often denoted f!P ( D, IR) or f!P. Thus the operation of differentiation of forms that we have defined effects f!P --+ j]P+ 1 . a mapping d : Let us consider several useful specific examples. For a 0-form w inExample a domain D IR3 , we obtainf(x, y, z ) � a differentiable function � defined dw afax dx + afay dy + afaz dz . Example Let w(x, y) P(x, y) dx + Q(x, y) dy beAssuming a differential 1-form i n a domain D of IR 2 endowed with coordinates (x, y). that P and Q are differentiable in D, by Definition 2 we obtain dw(x, y) dP 1\ dx + dQ 1\ dy ( �= dx + : dy) 1\ dx + ( �; dx + �� dy) 1\ dy aP dy 1\ dx + aQ dx 1\ dy ( aQ - aP ) (x, y) dx 1\ dy . ax ax ay ay Example For a 1-form w P dx + Q dy + R dz defined in a domain D in IR3 we obtain aQ - aP aP - oR aR - aQ ) dy 1\ dz + ( dw = ( ay 8z ax ay ) dx 1\ dy . 8z ax ) dz 1\ dx + ( �
C
ah
. .
c
�
9.
C
1 0.
=
=
=
=
=
=
=
=
=
11.
=
-
12.5 Elementary Facts about Differential Forms
Example 1 2.
Computing the differential of the 2-form
203
where P, Q, and R are differentiable in the domain D �3, leads to the relation 8Q oR ) dw = ( 8P ax + ay + az dx dy dz . 1 , x2 , x3) are Cartesian If ( x coordinates in the Euclidean space �3 and xsmooth H f(x), x H F(x) = (F 1 , F 2 , F3)(x), and x H V = (V l , V 2 , V3)(x) are vectorthefieldsrespecti in thevedomain fields, wescalar oftenandconsider vector Dfields�3, then along with these grad f = ( axofl ' 8xof2 ' 8xof3 ) -the gradient of f , (12. 24) 8F2 , 8F 1 - 8F3l , 8F12 - 8F 1 ) -the curl of F , (12.25) curlF = ( 8F3 ax2 ax3 ax3 ax ax ax2 and the scalar field 1 + 8V2 = 8V3 -the divergence of div = 8V (12.26) ax l 8x2 8x3 We haveon the already mentioned theof gradient ofanda scalar field earlier. Without dwelling physical content the curl divergence of a vector field athavethewithmoment, we note only the connections that these classical operators the operation of differentiating forms.is a one-to-one correspondence In the oriented Euclidean space �3 there between vector fields and 1- and 2-forms: F w� = (F, ) , V w�(V, , ) We 3-formng this in thecircumstance domain D into �3account, has theoneform p(x l remark , x2 , x:1) dxals1o 1\that dx 2 1\every dx3. Taki can introduce the following definitions for gradf, curlF, and divV: f H w0 (= f) H dw0 (= df) = w� H g := gradf , (12.24') (12. 2 5') F H w� H dw� = w; H r := curl. F , (12. 26') V H Wy2 H dwy2 = WP3 H p := d1vV . Examples 9, 11, and 12 show that when we do this in Cartesian coor dinates, we arrive atdivtheV. Thus expressions (12.24), (12. 2 5),theory and (12.26) above for grad f, curl F, and these operators i n field can be regarded as concrete manifestations of themanner operationon offorms differentiation of exterior forms, which is carried out in a single of any degree. More details on the gradient, curl, and divergence will be given in Chap. 14. (\
C
(\
c
v
v .
+-+
·
+-+
· ·
.
c
204
12 Surfaces and Differential Forms in JRn
1 2 . 5 . 4 Transformation of Vectors and Forms under Mappings
Let us consider in more detail what happens with functions (0-forms) under a mapping of their domains. Let
C
(--+
---+
Now let us consider the general case of transformation of forms of any degree. Let .w) = >.
C
r1 ,
C
:
, T
P
:
Now let us consider how to carry out the transformation of forms practice.
m
12.5 Elementary Facts about Differential Forms
205
Example 1 3. In the domain V IR� let us take the 2-form w dx i 1 1\ dxi 2 • i i 1 m Let x
=
=
r
, . . . , r
matrix via the formulas t: i - ox i
C
r
j
=
C
=
, . . . , r
i
j
i ox ( t ) T , i � utJ ( t ) Tl , �2 � ut J 2 ( The summation on j runs from 1 to m . ) '> 1 -
=
=
1, . . . , n .
Thus,
Consequently, we have shown that If we useandproperties 28 ) andof( 12.the2 9last ) for the operation of transformation ofequality: forms9 repeat the( 12.reasoning example, we obtain the following 9 If (12.29) is used pointwise, one can see that
206
12 Surfaces and Differential Forms in IR.n
Weformremark thattheifargument we make ofthecp*formal change of variablethe xdifferentials x(t) in the that is on the left, express 1 , . . . , dtm , and gather like terms dxin 1the, . . resulting . , dxn in terms of the differentials dt expression, usingsidetheofproperties of the exterior product, we obtain precisely the right-hand Eq. (12.31). Indeed, for each fixed choice of indices i 1 , . . . , ip we have =
Summingthe such equalitiessideover all ordered sets 1 :::; i 1 < < ip :::; n, we obtain right-hand of (12.31). tance.Thus we have proved the following propostion, of great technical impor · · ·
C
If a differential form w is defined in a domain V �" and cp U is a smooth mapping of a domain U �m into V, then the coordinate expression of the form cp*w can be obtained from the coordinate expression
Proposition. : --+ V
C
of the form w by the direct change of variable x = cp(t) (with subsequent transformations in accordance with the properties of the exterior product). Example 1 4 . In particular, if = n = relation (12.31) reduces to the equality (12.32) Hence, if we write f(x ) dx 1 1\ 1\ dxn in a multiple integral instead of 1 n f ( x ) dx dx , the formula J f (x ) dx = J f (cp(t)) det cp'(t) dt p,
m
· · ·
· · ·
V =
U
207
12.5 Elementary Facts about Differential Forms
change of variable inwhen a multiple integral via an orientation-preserving dibyforffthe eomorphism (that is, det cp ( t ) > 0) could be obtained automatically substitution ( t ) , just as happened in the one-dimensional case, andformal it could be givenxthe= cpfollowing form: (12. 3 3) J w = J cp*w . We remark in conclusion that if the degree p of the form w in the domain V IR� is larger than the dimension m of the domain U !R m that is mapped into V via cp : U ---+ V, then the formP cp*w onPU corresponding to w isinjecti obviously zero. Thus the mapping cp* : .f.? (V) ---+ .f.? (U) is not necessarily v e in general. On the other hand, if cp : U ---+ V has a smooth inverse cp - 1 V ---+ U, then by (12. 30) and the equalities cp - 1 cp = e u, cp cp - 1 = ev, we - 1 )* eb and (cp- 1 )* cp* ey. And, since eb and ev find that (cp)* (cp are :the.f.?P(V) identity mappings on( cp-.f.?P(U) respectively, the mappings and .f.?P(V) 1 ) * : .f.?P(U) ---+ .f.?P (V) , as one would expect, and cp*turn out to ---+be .f.?P(U) inverses of each other. That is, in this case, the mapping ---+ .f.?P (U) is bij ective. cp* :We.f.?P(V) alongaswith the map ping cp*note thatfinally transfersthatforms, one thecanproperties verify, also(12.28)-(12. satisfies the3 0)relation (12. 34) cp*(dw) = d(cp*w) . important equality shows in particular that the oper atiis actually oThis n of theoretically differentiation of forms, whi c h we defined in coordinate notation, independent of thebe coordinate in whichin Chapt. the differentiable form w is written. This will discussed insystem more detail 15. '
U
C
C
o
=
o
o
o
=
:
1 2 . 5 . 5 Forms o n Surfaces
We say that a differential p-form w is defined on a smooth surface S !Rn if a p-form w(x) is defined on the vectors of the tangent
Definition 3. C
plane TSx to S at each point x E S. Example If the smooth surface S is contained in the domain D !Rn in which a form woneis can defined, then,thesincerestriction the inclusion TBxto TSTD.x Inholdsthisatwayeacha poi n t x E S, consider of w(x) form w l 8 arises, which it is natural to call the restriction ofxw to S. As weLetknow,cp : aU surface can be defined parametrically, either locally or globally. ---+ S = cp (U) D be a parametrized smooth surface in thedomain domainU ofDparameters and w a form can transfer form w to the and onwriteD. cp*Then w in wecoordinate formthein accordance 1 5.
C
C
C
12 Surfaces and Differential Forms in IR n
208
with the algorithm given above. It is clear that the form cp*w in U obtained in this way coincides with thecp'(t)formTUtcp* (w-+I 8 ).TB is an isomorphism between We remark that, since x TUt and TSx at every point t E U, we can transfer forms both from S to U and fromlocally U to S, and so just as the smooth surfaces themselves are usually defined or globally forms onof local them,charts. in the final analysis, are usually definedbyin parameters, the parameterthedomains Example Let w� be the flux form considered in Example 8, generated by the velocity field V of a flow in the domain D of the oriented Euclidean space JR3. If S is a smooth oriented surface in D, one may consider the restriction w� l s of the form w� to S. The form w� l s so obtained characterizes the flux acrossIf eachI -+element of the surface S. S is a local chart of the surface S, then, making the change of variablethex coordinate cp(t) in the coordinate expression (12. 2 2) for the form w�, we obtain for thecoordinates form cp*w�of thecp*surface. (w� I s ), which is defined on the square I,expression in these local Let w}F acting be theinwork form considered in Example 7, generated byExample the force field a domain D of Euclidean space. Let I -+ cp(I) D be a smooth path ( is not necessarily a homeomorphism). Then, accordance with the general principle of restriction and transfer of forms, aina(t)form cp*w} arises on the closed interval I, whose coordinate representation dt can (12.21) be obtained the change of variable x cp(t) in the coordinate expression for thebyform w}. :
1 6.
'P
:
=
=
1 7.
C
'P
'P
:
=
1 2 . 5 . 6 Problems and Exercises 1. Compute the values of the differential forms w in IR n given below on the indicated sets of vectors: a) w = x 2 dx 1 on the vector e = ( 1 , 2, 3) E TIR (3 , 2 , 1 ) · b) w = dx 1 /\dx 3 +x 1 dx 2 Adx4 on the ordered pair of vectors e 1 ' e 2 E TIR[1 ,0,0,0 ) . c) w = df, where f = x 1 + 2x 2 + · · · + n x n , and e = (1, -, 1 , . . . , ( - 1 ) n 1 ) E -
TIR n( 1 , 1 , . . . , 1 ) .
2 . a) Verify that the form dx i1 1\ · · · A dx i k is identically zero if the indices i 1 , . . . , i k are not all distinct . b) Explain why there are no nonzero skew-symmetric forms of degree p > n on an n-dimensional vector space. c) Simplify the expression for the form
2 dx 1 1\ dx 3 1\ dx 2 + 3 dx 3 1\ dx 1 1\ dx 2 - dx 2 1\ dx 3 1\ dx 1 . d) Remove the parentheses and gather like terms:
(x 1 dx 2 + x 2 dx 1 ) 1\ (x 3 dx 1 1\ d x 2 + x 2 dx 1 1\ dx 3 + x 1 dx 2 1\ dx3 ) .
12.5 Elementary Facts about Differential Forms
209
e) Write the form df 1\ dg, where f = In ( 1 + l x l 2 ) , g = sin l x l , and x l ( x , x 2 , x 3 ) as a linear combination of the forms dx i 1 1\ d x i 2 , 1 ::::; i 1 < i 2 ::::; 3.
=
f) Verify that in IR n
dt 1\ · · · 1\ d r ( x ) = det
( ��;) (x) dx 1 1\ · · · 1\ dxn .
g) Carry out all the computations and show that for 1 ::::; k ::::; n dt 1\ . . . 1\ df k = 3. a
a) Show that a form a of even degree commutes with any form (3, that is, 1\ (3 = (3 1\ a. n b) Let w = L dp; 1\ dq i and w n = w 1\ · · · 1\ w ( n factors) . Verify that
i= l
n ( n -1) W n = n! dp 1 1\ dq 1 1\ · · · 1\ dpn /\ dq n = (- 1) 2 dp 1 1\ · · · l\ dpn l\ dq 1 1\ · · · 1\ dq n . 4. a) Write the form w = df, where f ( x ) = ( x 1 ) 2 + ( x 2 ) 2 + · · · + (x n ) 2 , as a combination of the forms dx 1 , . . . , dx n and find the differential dw of w. b) Verify that d 2 f 0 for any function f E c C 2 ) (D , IR ) , where d 2 = d o d, and d is exterior differentiation. c) Show that if the coefficients a; 1 , ... ,ik of the form w = a; 1 , ... ,ik ( x ) dx i 1 1\ · · · I\ ik dx belongs to the class cC 2) (D, IR) , then d 2 w 0 in the domain D. x d) Find the exterior differential of the form d'2 X � 2dy in its domain of definition. =
=
Y
5.
y
If the product dx 1 · · · dx n in the multiple integral J f ( x ) d x 1 · · · dx n is interD
preted as the form dx 1 1\ · · 1\ dx n , then, by the result of Example 14, we have the possibility of formally obtaining the integrand in the formula for change of variable in a multiple integral. Using this recommendation, carry out the following changes of variable from Cartesian coordinates: a) to polar coordinates in IR 2 , b) to cylindrical coordinates in JR 3 , c) to spherical coordinates in JR3 . ·
6. Find the restriction of the following forms: a) dxi to the hyperplane x i = 1 . b ) dx 1\ dy t o the curve x = x (t ) , y = y (t ) , a < t < b. c) dx 1\ dy to the plane in JR 3 defined by the equation x = c. d) dy 1\ dz + dz 1\ dx + dx 1\ dy to the faces of the standard unit cube in IR 3 . e) w; = dx 1 1\ · · · 1\ dx i - 1 1\ dx i 1\dx i + l 1\ · · · 1\ dx n to the faces of the standard unit cube in IRn . The symbol stands over the differential dx i that is to be omitted in the product. �
210
1 2 Surfaces and Differential Forms in .IR. n
7. Express the restriction of the following forms to the sphere of radius R with center at the origin in spherical coordinates on .IR.3 : a) dx, b) dy, c) dy 1\ dz. 8. The mapping 'P : .IR. 2 -+ .IR. 2 is given in the form ( u , v ) 1--t ( u · v , 1) = (x, y) . Find: a) 'P* (dx) , b) 'P* (dy) , c) 'P* (y dx) .
Verify that the exterior differential d : Q P (D) -+ !] P +1 (D) has the following properties: a) d(w 1 + w 2 ) = dw 1 + dw 2 , b) d(w 1 1\ w2 ) = dw 1 1\ w 2 + ( - 1 ) deg w 1 w 1 1\ dw 2 , where deg w 1 is the degree of the form w 1 . c) Vw E W d(dw) = 0. d) Vf E il 0 df = f: *!r dx ; . t= l Show that there is only one mapping d : QP (D) -+ !]P +1 (D) having properties a) , b) , c) , and d) . 9.
1 0. Verify that the mapping 'P* : QP (V) -+ QP (U) corresponding to a mapping 'P : U -+ V has the following properties: a) 'P* (w l + w2 ) = '{J*w l + '{J*w 2 . b) 'P* (w l 1\ w 2 ) = '{J*w l 1\ '{J*w 2 . c) d'{J*w = 'P*dw. d) If there is a mapping 'ljJ : V -+ W , then ('1/J o '{)) * = 'P* o 'ljJ*.
1 1 . Show that a smooth k-dimensional surface is orientable if and only if there exists a k-form on it that is not degenerate at any point.
1 3 L ine and S urface Int egrals
1 3 . 1 The Integral of a Differential Form 13. 1 . 1 The Original Problems, Suggestive Considerations, Examples The Work of a Field Let F(x) be a continuous force field acting in aa.indomain G of the Euclidean space rn: n . The displacment of a test particle thebyfieldthe isfieldaccompanied byunitwork.testWeparticle ask how weacangivencompute the more work done in moving a along trajectory, precisely, a smooth path I ---+ "'((/) G. We have already touched on that this reason problemwewhen we studied thethe solution applica tions of the definite integral. For can merely recall ofwilthel beproblem atwhat this point, noting certain elements of the construction that useful in follows. It is known that in a constant field F the displacement by a vector e is associated with an amount of work (F, e). Let t H x(t) be a smooth mapping I ---+ G defined on the closed interval I = {t E IR I a :::; t :::; b } . We take a sufficiently fine partition of the closed interval [a, b]. Then onequality each x(interval Ii = { t E II ti� 1 :::; t :::; ti } of the partition we have the t) - x( ti) ;::::; x' ( t) ( ti - ti� 1 ) up to infinitesimals of higher order. To the displacement vector Ti ti+l - ti from the point ti (Fig. 13.1) there 'Y
C
:
'Y
=
Fig. 1 3 . 1 .
:
13 Line and Surface Integrals
212
corresponds a displacement of x(ti ) in !Rn by the vector Llxi = xi +l - Xi , which can atbex(tregarded as equal to the tangent vector �i = x(ti )Ti to the trajectory )i with the same precision. Since the field F(x) is continuous, it canLlA be regarded a locallytoconstant, for thatJ with reasonsmall we canrelative compute work corresponding the (time)andinterval errortheas i
i
or Hence,
A = L LlA i L(F (x(t i ) ) , x(t i ))Llt i i i �
weandfindso, passing that to the limit as the partition of the closed interval I is refined, (13.1) A '= / (F ( x(t) ) , x(t)) dt . If thetheexpression (F ( x(t) ) , x(t)) dt is rewritten as (F(x), dx), then, as n suming coordinates coordinates, we can give this n dxn , after which we can write (13.1) as expression the form F 1 dxin1 +!R are+ FCartesian A = J F 1 dx 1 + + Fn dxn (13. 2 ) or as (13. 2') Formula the precise meaning the integrals 1-form along(13.1) the pathprovides "'( written in formulas (13. 2of) and (13. 2'). of the work Example Consider the force field F = ( - x2 ty2 , x2�y2 ) defined at all the origin. Let us compute the work of this pointsalong of thetheplane IR2"Y except field curve l defined as x = cost, y = sin t, 0 t 2n, and along curve defined by x(13.= 22'),+ coswe t,findy = sin t, 0 t 2n. According to formulasthe (13.1), (13. 2 ), and J WF1 = J - X2 +Y y2 dx + X2 +X y2 dy = 211" ( sin t (- sin t) cos t cos t ) =J cos2 t + sin2 t + cos2 t + sin2 t dt = 27r b
a
· · ·
'"'(
· · ·
1.
::;
'"'11
'"'11
0
·
::;
::;
·
::;
and
13. 1 The Integral of a Differential Form
J
72
21T
213
� -ydx + xdy - � - sint(- sint) + (2 + cost)(cost) dt WF1 -2 2 2 2 X + y sin (2 + cos t) t + 72 0 1T 0
_
k
2cost dt = J 1 + 2cost dt + J 1 + 2cos(27r - u) du = = J 51 ++ 4cost 0 0 5 + 4cost 1T 5 + 4cos(27r - u) 1T 1 + 2cost r 1 + 2cosu = J 5 + 4cost dt - }0 5 + 4cosu du = O . 0 Example Let r be the radius vector of a point (x, y, z) E JR3 and r = l r l . Suppose a force field F = f(r)r is defined everywhere in JR 3 except at the origin. is a so-called central force field. Let us find the work of F on a path This [0, 1 ] JR3 \ 0. Using (13. 2 ), we find J f(r)(xdx + ydy + zdz) = � j f (r)d(x2 + y2 + z2 ) = 2.
'Y :
7
--+
1
= �J 0
7
f(r(t) ) dr2 (t) =
1
� J0 f ( vfuW) du(t) = = � J f ( v'U) du = 4i(ro, r 1 ) . r1
2
r
5
Here, as one can see, we have set x2 (t)+y2 (t)+z2 (t) = r2 (t), r2 (t) = u(t),
ro = r(O), and r 1 = r(1). Thus in any centralr fieldandther work onbeginning a path and has turned out path to depend only on the distances of the end of the from 0 the center 0 of the field. 1 'Y
for the gravitational field � r of a unit point mass located at theIn particular, origin, we obtain
b. The Flux Across a Surface Suppose there is a steady flow of liquid (or gas) in a domain G of the oriented Euclidean space JR 3 and that x V(x) issurface the velocity field of this flow. In addition, suppose that a smooth oriented S has been chosen in G. For definiteness we shall suppose that the orientation of S is given by a field of normal vectors. We ask how to determine H
214
13 Line and Surface Integrals
the ) outflow or flux of fluid across the surface S. More precisely, weunitask(volumetric how to finddirection the volume of fluid thatorienting flows across the surface toS per time in the indicated by the field of normals the surface. To solveandtheequalproblem, we remark that if the velocity field of the flow is constant to V, then the flow per unit time across a parallelogram spanned onby thevectors e 1 and e 2 equals the volume of the parallelepiped constructed vectors V, e 1 , e 2 . If is normal to and we seek the flux across i n the direction of it equals the scalar triple product (V, e 1 , e2 ), provided and the frame e 1 , e 2 give the same orientation ( that is, if , e 2 is a frame having the given orientation in JR3 ) . If the frame e l , e 2 githeveesldirection the orientation opposite to the one given by in then the flow in of is - (V, e 1 , e 2 ) . We nowthatreturnthe toentire the original statement of the problem. For simplicity let usS assume surface S admits a smooth parametrization
II
TJ,
TJ,
TJ
TJ
II
TJ
II
TJ
II,
--+
c
=
=
=
Fig. 1 3 . 2 .
TJ,
13. 1 The Integral of a Differential Form
Summing the elementary fluxes, we obtain
215
where w�(x) (V(x) , ) is the flux 2-form ( studied in Example 8 of Sect. 12. pass totoassume the limit, interval I,5 ) it. Ifis wenatural thattaking ever finer partitions P of the ( 13. 3 ) This last symbol is the integral of the 2-form w� over the oriented surface S. ( 12. 2 2 ) of Sect. 12. 5 ) the coordinate expression for the fluxRecalling form w� (informula Cartesian coordinates, we may now also write ( 13. 4 ) :F J V 1 dx 2 1\ dx3 + V2 dx 3 1\ dx 1 + V 3 dx 1 1\ dx 2 . We havealldicussed here only thegivegeneral principle for solving this problem. Inandessence we have done is to the precise definition ( 13. 3 ) of the flux :F introducedcomputational certain notation ( 13. 3 ) and ( 13. 4 ) ; we have still not obtained anyWeeffective formula similar to formula ( 13.1 ) for the work. remark that formula ( 13.1 ) can be obtained from ( 13. 2 ) by replacing x\ . . . , x n with the functions (x\ . . . , x n ) (t) = x(t) that define the path "'(. We recallof (theSect.form12.w5 ) defined that suchin Ga tosubstitution can be Iinterpreted as the transfer the closed interval = [a, b]. In a completely analogous way, ofa the computational formula forofthethefluxsurface can beintoobtained by direct substitution parametric equations In( 13.fact,4) . ·, ·
=
=
s
and The form cp* w� is defined on a two-dimensional interval I IR 2 . Any 2-form in I has the form f(t) dt 1 1\ dt 2 , where f is a function on I depending on the form. Therefore c
But dt1 1\ dt 2 (r 1 , r 2 ) Tf · Ti is the area of the rectangle Ii spanned by the orthogonal vectors r 1 , r2 . =
216
13 Line and Surface Integrals
Thus,
As the partition is refined we obtain in the limit
J f (t) dt1 I
(\
dt 2 =
J f (t) dt 1 dt2 ' I
(13.5)
where, waccording to (13.3), the left-hand side contains the integral of the 2 = f(t) dt 1 1\ dt 2 over the elementary oriented surface I, and the 2-form right-hand sideonlythe tointegral ofthatthethefunction f over the rectangle I. It remains recall coordinate representation f(t) dt 1 1\ dt2 ofby thetheform cp*w� is obtained from the coordinate expression for the form w� direct substitution x = cp(t) , where cp : I -+ G is a chart of the surface s. Carrying out this change of variable, we obtain from (13.4)
last integral,I. as Eq. (13.5) shows, is the ordinary Riemann integral overThis the rectangle Thus we have found that (13.6)
where x = cp( t) = ( cp l , cp2 , cp:�) ( t 1 , t 2 ) is a chart of the surface S defining the same orientation as the field of normals we have given. If the chart cp : I -+ S gives S the opposite orientation, Eq. (13.6) does not generally hold. But, as follows from thesides considerations at thein beginning of case. this subsection, the left andThe right-hand will di ff er only si g n in that finalfluxes formulaL1:F(13.6) is obviously merely the limit of the sums of the elementary ;::::; (V(x i ) , e 1 , e 2 ) familiar to us, written accurately in i the Wecoordinates t 1 and t 2 . have considered thecancasebe decomposed of a surface into definedsmooth by apieces single Schart. In general a smooth surface i having essentially another,theandpiecesthenS we can find the flux through S asno theintersections sum of thewithfluxesonethough i.
13. 1 The Integral of a Differential Form
217
Example 3 . Suppose a medium is advancing with constant velocity V = (1, 0, 0). If we take any closed surface in the domain of the flow, then, since
the density of the medium does not change, the amount of matter in the volume bounded by this surface must remain constant. Hence the total flux of the medium through such a surface must be zero. In this case, let us check formula (13.6) by taking S to be the sphere x 2 + y 2 + z 2 = R2 . Up to a set of area zero, which is therefore negligible, this sphere can be defined parametrically
x = R cos 'ljJ cos
0
where <
j I
V = (1,
0, 0) are substituted in
8x 1 8x 2 7r / 2 2 7r 8t 1 8t 1 d
j
j
0.
Since the integral equals zero, we have not even bothered to consider whether it was the inward or outward flow we were computing. Suppose the velocity field of a medium moving in IR. 3 is defined in Cartesian coordinates x, y, z by the equality V(x, y, z) (V l , V 2 , V 3 ) (x, y, z) = (x, y , z) . Let us find the flux through the sphere x 2 + y 2 + z 2 = R2 into the ball that it bounds (that is, in the direction of the inward normal) in this case. Taking the parametrization of the sphere given in the last example, and carrying out the substitution in the right-hand side of (13.6), we find that
Example 4.
R cos 'ljJ cos
0
2 7r
=
7r / 2
j d
- 7r / 2
We now check to see whether the orientation of the sphere given by the curvilinear coordinates (
13 Line and Surface Integrals
218
In this case the result is easy to verify: the velocity vector V of the flow has magnitude equal to R at each point of the sphere, is orthogonal to the sphere, and points outward. Therefore the outward flux from the inside equals the area of the sphere 47r R 2 multiplied by R. The flux in the opposite direction is then -47r R3 . 13. 1 . 2 Definition of the Integral of a Form over an Oriented Surface
13.1.1
The solution of the problems considered in Subsect. leads to the defi nition of the integral of a k-form over a k-dimensional surface. First let S be a smooth k-dimensional surface in �n , defined by one stan dard chart
13. 2 ).
i
the mesh A(P) of the partition tends to zero. Thus we adopt the following definition: Definition 1 . (Integral of a k-form w over a given chart smooth k-dimensional surface.)
Jw s
:=
lim
.\ ( P ) --t O
---+
S of a
L(
z
(13. 7)
If we apply this definition to the k-form f (t) dt 1 1\ · · · 1\ dt k on I (when
J f (t) dt 1 1\ . . . 1\ dtk I It thus follows from (13. 7 ) that
J=
S (I)
(13. 8 )
JI f(t) dt 1 . . . dtk .
(13 . 8)
J
(13. 9 )
=
w=
I
and the last integral, as Eq. shows, reduces to the ordinary multiple integral over the interval I of the function f corresponding to the form
13.1 The Integral of a Differential Form
219
We have derived the important relations (13. 8 ) and (13. 9 ) from Definition 1,In but they themselves could have been adopted as the original definitions. particular, if D is an arbitrary domain in !R n (not necessarily an interval) ,
then, so as not to repeat the summation procedure, we set
J
f(t) dt 1 A · · · A dt k :=
D
J
j (t) dt 1 · · · dt k .
(13. 8 ')
D
and for a smooth surface given in the form cp : D --+ S and a k-form w on it we set
J
w :=
J
cp *w .
(13. 9 ')
D
S = cp ( D )
If S is an arbitrary piecewise-smooth k-dimensional surface and w is a k-form defined on the smooth pieces of S, then, representing S as the union Ui Si of smooth parametrized surfaces that intersect only in sets of lower dimension, we set w := � w . )
J
2
S
J
(13.10
S;
In the absence of substantive physical or other problems that can be solved using ( ) such a definition raises the question whether the magni tude of the integral of the partition u si is independent of the choice of the i parametrization of its pieces. Let us verify that this definition is unambiguous.
13. 1 0 ,
Proof. We begin by considering the simplest case in which S is a domain Dx in JR k and cp : D t --+ Dx is a diffeomorphism of a domain D t C JR k onto D x · In Dx = S the k-form w has the form f(x) dx 1 A · · · A dx k . Then, on the one hand implies
(13. 8 )
J
f(x) dx 1 A · · · A dx k =
Dx
J
f(x) dx 1 · · · dx k .
Dx
13. 9' ) and (13. 8 '), j w : = J cp*w = j f (cp(t)) det cp' (t) dt 1 · · · dt k .
On the other hand, by ( Dx
D,
D,
But if det cp' (t) > 0 in D t , then by the theorem on change of variable in a multiple integral we have
J
D x = cp ( D, )
f(x) dx 1 · · · dx k =
/ J (cp(t) ) det cp' (t) dt 1 · · · dt k .
D,
220
13 Line and Surface Integrals
Hence, assuming that there were coordinates x 1 , . . . , x k in S = Dx and curvilinear coordinates t 1 , . . . , t k of the same orientation class, we have shown that the value of the integral J w is the same, no matter which of these two s
coordinate systems is used to compute it . We note that if the curvilinear coordinates t 1 , . . . , t k had defined the op posite orientation on S, that is, det 0 and differ in sign if det
i
follows from the additivity of the ordinary multiple integral (it suffices to consider a finer partition obtained by superimposing two partitions and ver ify that the value of the integral over the finer partition equals the value over each of the two original partitions) . D
On the basis of these considerations, it now makes sense to adopt the following chain of formal definitions corresponding to the construction of the integral of a form explained in Definition 1 . Definition 1 ' . (Integral of a form over an oriented surface S
a ) I f the form f(x)dx 1 1\
J
D
· · ·
1\ dx k i s defined i n a domain
J(x) dx 1 1\ · · · 1\ dx k :=
J
D
c
D
f(x) dx 1 · · · dx k .
b) If S c !Rn is a smooth k-dimensional oriented surface,
JRn . ) c JR k , then
: D --+ S is a
13.1 The Integral of a Differential Form
J w J �.p*w , :=
S
±
221
D
where the + sign is taken if the parametrization i.p agrees with the given orientation of S and the - sign in the opposite case. c ) If S is a piecewise-smooth k-dimensional oriented surface in JR n and w is a k-form on S ( defined where S has a tangent plane ) , then
where 51 , . . . , Sm , . . . is a decomposition of S into smooth parametrizable k-dimensional pieces intersecting at most in piecewise-smooth surfaces of smaller dimension. We see in particular that changing the orientation of a surface leads to a change in the sign of the integral. 13. 1 . 3 Problems and Exercises
a) Let x, y be Cartesian coordinates on the plane JR 2 . Exhibit the vector field whose work form is w = - x2t y2 dx + x2: y 2 dy. b) Find the integral of the form w in a) along the following paths 1; : [0, JT] 3 t � (cos t, sin t) E JR 2 ; [O, JT] 3 t � (cos t, - sin t) E JR 2 ; 1.
13
consists of a motion along the closed intervals joining the points ( 1 , 0), ( 1 , 1 ) , ( - 1 , 1 ) , ( - 1 , 0) i n that order; 14 consists of a motion along the closed intervals joining ( 1 , 0) , ( 1 , - 1 ) , ( - 1 , - 1 ) , ( - 1 , 0) in that order. 2. Let f be a smooth function in the domain D C R n and 1 a smooth path in D with initial point Po E D and terminal point PI E D. Find the integral of the form w = df over I · 3. a) Find the integral of the form w = dy 1\ dz + dz 1\ dx over the boundary of the standard unit cube in JR3 oriented by an outward-pointing normal. b) Exhibit a velocity field for which the form w in a) is the flux form. 4. a) Let x, y, z be Cartesian coordinates in JR n . Exhibit a velocity field for which the flux form is x dy 1\ dz + y dz 1\ dx + z dx 1\ dy w = (x 2 + y 2 + z 2 ) 3 / 2 b) Find the integral of the form w in a) over the sphere x 2 + y 2 + z 2 = R 2 oriented by the outward normal. x c) Show that the flux of the field X 2 +(y � y+, z ;)31 2 across the sphere (x - 2) 2 + y 2 + z C z 2 = 1 is zero. d) Verify that the flux of the field in c) across the torus whose parametric equations are given in Example 4 of Sect. 12. 1 is also zero.
222
13 Line and Surface Integrals
5. It is known that the pressure P, volume V, and temperature T of a given quantity of a substance are connected by an equation f (P, V, T) = 0, called the equation of state in thermodynamics. For example, for one mole of an ideal gas the equation of state is given by Clapeyron ' s formula Pi - R = 0, where R is the universal gas constant. Since P, V, T are connected by the equation of state, knowing any pair of them, one can theoretically determine the remaining one. Hence the state of any system can be characterized, for example, by points (V, P) of the plane IR. 2 with coordinates V, P. Then the evolution of the state of the system as a function of time will correspond to some path "( in this plane. Suppose the gas is located in a cylinder in which a frictionless piston can move. By changing the position of the piston, we can change the state of the gas enclosed by the piston and the cylinder walls at the cost of doing mechanical work. Con versely, by changing the state of the gas (heating it, for example) we can force the gas to do mechanical work (lifting a weight by expanding, for example) . In this problem and in Problems 6, 7, and 8 below, all processes are assumed to take place so slowly that the temperature and pressure are able to average out at each partic ular instant of time; thus at each instant of time the system satisfies the equation of state. These are the so-called quasi-static processes. a) Let "( be a path in the V P-plane corresponding to a quasi-static transition of the gas enclosed by the piston and the cylinder walls from state Vo , Po to V1 , P1 . Show that the quantity A of mechanical work performed on this path is defined by the line integral A = J P d V.
b) Find the mechanical work performed by one mole of an ideal gas in passing from the state V0 , Po to state V1 , H along each of the following paths (Fig. 13.3) : "(OL I , consisting of the isobar OL (P = Po ) followed by the isochore LI (V = V1 ) ; "(o K 1 , consisting o f the isochore 0 K (V = Vo ) followed by the isobar K I ( P = H ) ; "(OJ , consisting o f the isotherm T = canst (assuming that Po Vo = P1 V1 ) . p Po
0
L
E:JJ
Vo
Fig. 1 3 . 3 .
c) Show that the formula obtained in a) for the mechanical work performed by the gas enclosed by the piston and the cylinder walls is actually general, that is, it remains valid for the work of a gas enclosed in any deformable container. 6. The quantity of heat acquired by a system in some process of varying its states, like the mechanical work performed by the system (see Problem 5) , depends not only on the initial and final states of the system, but also on the transition path. An
13.1 The Integral of a Differential Form
223
important characteristic of a substance and the thermodynamic process performed by ( or on ) it is its heat capacity, the ratio of the heat acquired by the substance to the change in its temperature. A precise definition of heat capacity can be given as follows. Let x be a point in the plane of states F ( with coordinates V, P or V, T or P, T) and e E TFx a vector indicating the direction of displacement from the point x. Let t be a small parameter. Let us consider the displacment from the state x to the state x + te along the closed interval in the plane F whose endpoints are these states. Let LlQ(x, te) be the heat acquired by the substance in this process and LlT(x, te) the change in the temperature of the substance. The heat capacity C = C(x, e) of the substance ( or system ) corresponding to the state x and the direction e of displacement from that state is
C ( x, te )
=
te) . . LlQ(x, " hm L.l T( x, te )
t --+ O
In particular, if the system is thermally insulated, its evolution takes place with out any exchange of heat with the surrounding medium. This is a so-called adiabatic process. The curve in the plane of states F corresponding to such a process is called an adiabatic. Hence, zero heat capacity of the system corresponds to displacement from a given state x along an adiabatic. Infinite heat capacity corresponds to displacement along an isotherm T = canst . The heat capacities at constant volume Cv = C(x, ev ) and at constant pressure Cp = C(x, ep ) , which correspond respectively to displacement along an isochore V = canst and an isobar P = canst, are used particularly often. Experiment shows that in a rather wide range of states of a given mass of substance, each of the quantities Cv and Cp can be considered practically constant. The heat capacity corresponding to one mole of a given substance is customarily called the molecular heat capacity and is denoted ( in contrast to the others ) by upper case letters rather than lower case. We shall assume that we are dealing with one mole of a substance. Between the quantity LlQ of heat acquired by the substance in the process, the change LlU in its internal energy, and the mechanical work LlA it performs, the law of conservation of energy provides the connection LlQ = LlU + LlA. Thus, under a small displacement te from state x E F the heat acquired can be found as the value of the form r5Q := dU + P dV at the point x on the vector te E TFx ( for the formula P dV for the work see Problem 5c ) ) . Hence if T and V are regarded as the coordinates of the state and the displacement parameter ( in a nonisothermal direction ) is taken as T, then we can write The derivative �� determines the direction of displacment from the state x E F in the plane of states with coordinates T and V. In particular, if �� = 0 then the displacement is in the direction of the isochore V = canst, and we find that . ( In the general case V = V ( P, T) Cv = �� . If P = canst, then �� = ( ��) P=COnst is the equation of state f( P, V, T) = 0 solved for V. ) Hence
224
13 Line and Surface Integrals
where the subscripts P, V, and T on the right-hand side indicate the parameter of state that is fixed when the partial derivative is taken. Comparing the resulting expressions for Cv and Cp , we see that
By experiments on gases ( the Joule 1 -Thomson experiments ) it was established and then postulated in the model of an ideal gas that its internal energy depends only on the temperature, that is, ( ��) = 0. Thus for an ideal gas Cp - Cv = T
P ( ��) Taking account of the equation PV = RT for one mole of an ideal gas, we obtain the relation Cp - Cv = R from this, known as Mayer 's equation 2 in thermodynamics. The fact that the internal energy of a mole of gas depends only on temperature makes it possible to write the form 8Q as 8Q = au aT dT + P dV = Cv dT + P dV . To compute the quantity of heat acquired by a mole of gas when its state varies over the path "( one must consequently find the integral of the form Cv dT + P d V over 'Y · It is sometimes convenient to have this form in terms of the variables V and P. If we use the equation of state PV = RT and the relation Cp - Cv = R, we obtain p 8Q = Cp R dV + Cv v dP . R a ) Write the formula for the quantity Q of heat acquired by a mole of gas as its state varies along the path 'Y in the plane of states F. b ) Assuming the quantities Cp and Cv are constant, find the quantity Q cor responding to the paths "(O L I , "(O KI , and "(O I in Problem 5b ) . c ) Find ( following Poisson ) the equation of the adiabatic passing through the point (Vo , Po) in the plane of states F with coordinates V and P. ( Poisson found that PV C p / C v = const on an adiabatic. The quantity CpfCv is the adiabatic constant of the gas. For air CpfCv ::::: 1 .4.) Now compute the work one must do in order to confine a thermally isolated mole of air in the state (Vo , Po) to the volume V1 = � Vo . 7 . We recall that a Carnot cycle 3 of variation i n the state of the working body of a heat engine ( for example, the gas under the piston in a cylinder ) consists of the following ( Fig. 13.4) . There are two energy-storing bodies, a heater and a cooler ( for 1 G.P. Joule (1818-1889) - British physicist who discovered the law of thermal action of a current and also determined, independently of Mayer, the mechanical equivalent of heat. 2 J.P. Mayer (1814-1878) - German scholar, a physician by training; he stated the law of conservation and transformation of energy and found the mechanical equivalent of heat. 3 N.L.S. Carnot ( 1796-1832) - French engineer, one of the founders of thermody namics. P
.
13. 1 The Integral of a Differential Form p
225
1
v
Fig. 13.4.
example, a steam boiler and the atmosphere ) maintained at constant temperatures T1 and T2 respectively (T1 > T2 ) . The working body ( gas ) of this heat engine, having temperature T1 in State 1, is brought into contact with the heater, and by decreasing the external pressure along an isotherm, expands quasi-statically and moves to State 2. In the process the engine borrows a quantity of heat Q 1 from the heater and performs mechanical work A 1 2 against the external pressure. In State 2 the gas is thermally insulated and forced to expand quasi-statically to state 3, until its temperature reaches T2 , the temperature of the cooler. In this process the engine also performs a certain quantity of work A 23 against the external pressure. In State 3 the gas is brought into contact with the cooler and compressed isothermically to State 4 by increasing the pressure. In this process work is done on the gas ( the gas itself performs negative work A 3 4 ) , and the gas gives up a certain quantity of heat Q 2 to the cooler. State 4 is chosen so that it is possible to return from it to State 1 by a quasi-static compression along an adiabatic. Thus the gas is returned to State 1. In the process it is necessary to perform some work on the gas ( and the gas itself performs negative work A 41 ) . As a result of this cyclic process ( a Carnot cycle ) the internal energy of the gas ( the working body of the engine ) obviously does not change ( after all, we have returned to the initial state ) . Therefore the work performed by the engine is A = A 1 2 + A 23 + A 3 4 + A 41 = Q 1 - Q 2 . The heat Q 1 acquired from the heater went only partly to perform the work A. It is natural to call the quantity TJ = = Q'(J1Q2 the efficiency of the heat engine under consideration. a) Using the results obtained in a) and c ) of Problem 6, show that the equality � = � holds for a Carnot cycle. b ) Now prove the following theorem, the first of Carnot's two famous theorems. The efficiency of a heat engine working along a Carnot cycle depends only on the temperatures T1 and T2 of the heater and cooler. ( It is independent of the structure of the engine or the form of its working body. )
r!}-1
8. Let 'Y be a closed path in the plane of states F of the working body of an arbitrary heat engine ( see Problem 7) corresponding to a closed cycle of work performed by it. The quantity of heat that the working body ( a gas, for example ) exchanges with the surrounding medium and the temperature at which the heat exchange takes place are connected by the Clausius inequality J §!1- ::; 0. Here 8Q is the heat exchange ' form mentioned in Problem 6.
226
13 Line and Surface Integrals
a) Show that for a Carnot cycle (see Problem 7) , the Clausius inequality becomes equality. b) Show that if the work cycle "( can be run in reverse, then the Clausius inequality becomes equality. c) Let "( 1 and "(2 be the parts of the path "( on which the working body of a heat engine acquires heat from without and imparts it to the surrounding medium respectively. Let T1 be the maximal temperature of the working body on "( 1 and T2 its minimal temperature on "(2 · Finally, let Q 1 be the heat acquired on "(1 and Q 2 the heat given up on 'Y2 · Based on Clausius' inequality, show that � :::; R · d) Obtain the estimate TJ :::; T,n_T2 for the efficiency of any heat engine (see Problem 7) . This is Carnot 's second theorem. (Estimate separately the efficiency of a steam engine in which the maximal temperature of the steam is at most 150° C, that is, T1 = 423 K, and the temperature of the cooler - the surrounding medium - is of the order 20° C, that is T2 = 291 K.) e) Compare the results of Problems 7b) and 8d) and verify that a heat engine working in a Carnot cycle has the maximum possible efficiency for given values of T1 and T2 . The differential equation � = ��:? is said to have variables separable. It is usually rewritten in the form g(y) dy = f(x) dx, in which "the variables are sepa rated," then "solved" by equating the primitives I g(y) dy = I f(x) dx. Using the language of differential forms, now give a detailed mathematical explanation for this algorithm. 9.
1 3 . 2 The Volume Element . Integrals of First and Second Kind 1 3 . 2 . 1 The Mass of a Lamina
S be a lamina in Euclidean space JR.n . Assume that we know the density p (x) ( per unit area ) of the mass distribution on S. We ask how one can determine the total mass of S. Let
In order to solve this problem it is necessary first of all to take account of the fact that the surface density p (x) is the limit of the ratio .1m of the quantity of mass on a portion of the surface in a neighborhood of x to the area ..1 a of that same portion of the surface, as the neighborhood is contracted to x. By breaking S into small pieces Si and assuming that p is continuous on S, we can find the mass of Si , neglecting the variation of p within each small piece, from the relation
..1 m i ::::::: p (x i ) ..d ai , in which ..d ai is the area of the surface Si and X i E Si .
13.2 The Volume Element. Integrals of First and Second Kind
227
Summing these approximate equalities and passing to the limit as the partition is refined, we find that
m=
j s
p da .
(13. 12)
The symbol for integration over the surface S here obviously requires some clarification so that computational formulas can be derived from it . We note that the statement of the problem itself shows that the left-hand side of Eq. ( 13. 12) is independent of the orientation of S, so that the integral on the right-hand side must have the same property. At first glance this appears to contrast with the concept of an integral over a surface, which was discussed in detail in Sect . 13. 1 . The answer to the question that thus arises is concealed in the definition of the surface element da, to whose analysis we now turn. 13. 2 . 2 The Area of a Surface as the Integral of a Form
Comparing Definition 1 of Sect . 13. 1 for the integral of a form with the construction that led us to the definition of the area of a surface (Sect . 12.4) , we see that the area of a smooth k-dimensional surface S embedded in the Euclidean space JR n and given parametrically by 'P : D --+ S, is the integral of a form D, which we shall provisionally call the volume element on the surface S. It follows from relation (12. 10) of Sect . 12.4 that [l (more precisely
J
in the curvilinear coordinates 'P : D --+ S (that is, when transferred to the domain D) . Here % (t) = ( � , � ) , i,j = 1 , · :..._ · , k. To compute the area of S over a domain D in a second parametrization 0 : i5 --+ S, one must correspondingly integrate the form w
=
Z1
vdet ( g;j ) ( t)- dt A · · · A dt-k ,
( 13. 14)
- (t-) - I\ !!.f.. where g,J !!..<£._ ) . - 1 . . . ' k . at' ' aiJ ' z , J - ' We denote by 7/J the diffeomorphism
.
(13. 15) At the same time, it is obvious that (13. 16)
228
13 Line and Surface Integrals
Comparing the equalities (13. 13)-( 13. 16) , we see that 'ljJ*w = w if det '1/J' ( i) > 0 and '1/J* w = -w if det '1/J' ( i) < 0. If the forms w and w were obtained from the same form [l on S through the transfers r.p* and 0* , then we must always have the equality '1/J* ( r.p* fl) = :p• [l or, what is the same,
'1/J*w
=
w.
We thus conclude that the forms on the parametrized surface S that one must integrate in order to obtain the areas of the surface are different: they differ in sign if the parametrizations define different orientations on S; these forms are equal for parametrizations that belong to the same orientation class for the surface S. Thus the volume element [l on S must be determined not only by the surface S embedded in !R n , but also by the orienation of S. This might apprear paradoxical: in our intuition, the area of a surface should not depend on its orientation! But after all, we arrived at the definition of the area of a parametrized surface via an integral, the integral of a certain form. Hence, if the result of our computations is to be independent of the orientation of the surface, it follows that we must integrate different forms when the orientation is different. Let us now turn these considerations into precise definitions. 1 3 . 2 . 3 The Volume Element Definition 1 . If IR k is an oriented Euclidean space with inner product ( , ) , the on JR k corresponding to a particular orientation and the
volume element
inner product ( , ) is the skew-symmetric k-form that assumes the value 1 on an orthonormal frame of some orientation class.
The value of the k-form on the frame e 1 , . . . , e k obviously determines this form. We remark also that the form [l is determined not by an individual or thonormal frame, but only by its orientation class.
Proof. In fact, if e 1 , . . . , e k and e 1 , . . . , e k are two such frames in the same orientation class, then the transition matrix 0 from the second basis to the first is an orthogonal matrix with det 0 = 1 . Hence If an orthonormal basis e 1 , . . . , e k is fixed in JR k and n 1 , . . . , n k are the projections of JR k on the corresponding coordinate axes, obviously n 1 1\ · · · 1\ n k ( e1 , . . . , e k ) = 1 and
13.2 The Volume Element. Integrals of First and Second Kind
229
Thus,
��
�t
This is the oriented volume of the parallelepiped spanned by the ordered set of vectors e l ' ' e k . 0
0
0
Definition 2 . If the smooth k-dimensional oriented surface S is embedded in a Euclidean space IR.n , then each tangent plane T Bx to S has an orientation consistent with the orientation of S and an inner product induced by the the inner product in IR.n ; hence there is a volume element .f.?(x). The k-form n that arises on S in this way is the volume element on S induced by the embedding of S in IR.n . Definition 3. The area of an oriented smooth surface is the integral over the surface of the volume element corresponding to the orientation chosen for the surface.
This definition of area, stated in the language of forms and made precise, is of course in agreement with Definition 1 of Sect . 12.4, which we arrived at by consideration of a smooth k-dimensional surface S C IR.n defined in parametric form.
Proof. Indeed, the parametrization orients the surface and all its tan gent planes T Bx . If e 1 , . . . , e k is a frame of a fixed orientation class in TSx , it follows from Definitions 2 and 3 for the volume element n that .f.?(x)(e 1 , . . . , e k ) > 0. But then ( see Eq. ( 12.7) of Sect . 12.4 )
J
n(x) (e l , . . . , e k ) = det ((e i , e1 )) .
o
( 13 . 1 7)
We note that the form n(x) itself is defined on any set e 1 , . . . , e k of vectors TSx , but Eq. ( 13. 17) holds only on frames of a given orientation class in TSx . in
We further note that the volume element is defined only on an oriented surface, so that it makes no sense, for example, to talk about the volume element on a Mobius band in IR. 3 , although it does make sense to talk about the volume element of each orientable piece of this surface. Definition 4. Let S be a k-dimensional piecewise-smooth surface ( ori entable or not ) in IR.n , and 81 , . . . , Sm , . . . a finite or countable number of smooth parametrized pieces of it intersecting at most in surfaces of dimen sion not larger than k 1 and such that s = u si
The area ( or surfaces si 0
-
i
0
k-dimensional volume ) of S is the sum of the areas of the
13 Line and Surface Integrals
230
In this sense we can speak of the area of a Mobius band in JR. 3 or, what is the same, try to find its mass if it is a material surface with matter having unit density. The fact that Definition 4 is unambiguous (that the area obtained is independent of the partition 8 1 , . . . , Bm , . . . of the surface) can be verified by traditional reasoning. 1 3 . 2 . 4 Expression of the Volume Element in Cartesian Coordinates
Let S be a smooth hypersurface (of dimension n - 1) in an oriented Euclidean space JR. n endowed with a continuous field of unit normal vectors 11(x), x E S, which orients it. Let V be the n-dimensional volume in JR. n and [l the ( n - 1 ) dimensional volume element on S. If we take a frame e 1 , . . . , e n - 1 in the tangent space TBx from the orien tation class determined by the unit normal n(x) to TBx , we can obviously write the following equality:
(13. 1 8 ) Proof This fact follows from the fact that under the given hypotheses both sides are nonnegative and equal in magnitude because the volume of the parallelepiped spanned by 11, e 1 ' . . . ' e n - 1 is the area of the base il(x) (e 1 , . . . , e n _ 1 ) multiplied by the height 1 11 1 = 1 . D But,
=
n
'L) -1) i - 1 1/ (x) dx 1 i=1
1\
·
·
·
I\ dx i 1\
·
·
·
I\ dx n ( e 1 , . . . , e n - 1 )
·
Here the variables x 1 , . . . , x n are Cartesian coordinates in the orthonor mal basis e 1 , . . . , e n that defines the orientation, and the frown over the differential dx i indicates that it is to be omitted. Thus we obtain the following coordinate expression for the volume element on the oriented hypersurface S C JR. n : [l
=
n
�) -1) i - 1 1Ji (x) dx 1 . . I\ dxi i=1
1\
·
1\ . . . 1\
dx n (e u . . . ' e n - 1 ) .
(13. 1 9)
13.2 The Volume Element. Integrals of First and Second Kind
231
At this point it is worthwhile to remark that when the orientation of the susrface is reversed, the direction of the normal 'fl(x) reverses, that is, the form n is replaced by the new form - n. It follows from the same geometric considerations that for a fixed value of i E { 1 , . . . , n} (13.20) This last equality means that ( 1 3.21) For a two-dimensional surface S in JR n the volume element is most often denoted da or dS. These symbols should not be interpreted as the differen tials of some forms a and S; they are only symbols. If x, y, z are Cartesian coordinates on JR 3 , then in this notation relations (13. 19) and (13.21) can be written as follows: da = cos o: 1 dy 1\ dz + cos o: 2 dz 1\ dx + cos o: 3 dx 1\ dy , cos 0: 1 da = dy 1\ dz , (oriented areas of the projections cos 0: 2 da = dz 1\ dx , on the coordinate planes) . cos 0: 3 da = dx 1\ dy , Here (cos o: 1 , cos o: 2 , cos o: 3 ) ( x) are the direction cosines (coordinates) of the unit normal vector 'fl(x) to S at the point x E S. In these equalities (as also in (13. 19) and (13.21)) it would of course have been more correct to place the restriction sign I s on the right-hand side so as to avoid misunderstanding. But , in order not to make the formulas cumbersome, we confine ourselves to this remark. 13.2.5 Integrals of First and Second Kind
Integrals of type ( 13. 12) arise in a number of problems, a typical represen tative of which is the problem consisdered above of determining the mass of a surface whose density is known. These integrals are often called integrals over a surface or integrals of first kind. Definition 5. The
integral
integral of a function p over an oriented surface S is the
Js pD
(13.22)
of the differential form pD, where D is the volume element on S (correspond ing to the orientation of S chosen in the computation of the integral) .
232
13 Line and Surface Integrals
It is clear that the integral (13.22) so defined is independent of the orienta tion of S, since a reversal of the orientation is accompanied by a corresponding replacement of the volume element. We emphasize that it is not really a matter of integrating a function, but rather integrating a form pfl of special type over the surface S with the volume element defined on it.
Definition
6 . If S is a piecewise-smooth ( orientable or non-orientable) sur face and p is a function on S, then the integral (13.22) of p over the surface S is the sum L pfl of the integrals of p over the parametrized pieces
j
' s ,
81 , . . . , Sm , . . . of the partition of S described in Definition 4. The integral (13.22) is usually called a surface integral of first For example, the integral
kind.
(13.12), which expresses the mass of the surface
S in terms of the density p of the mass distribution over the surface, is such
an integral. To distinguish integrals of first kind, which are independent of the ori entation of the surface, we often refer to integrals of forms over an oriented surface as surface integrals of second kind. We remark that, since all skew-symmetric forms on a vector space whose degrees are equal to the dimension of the space are multiples of one an other, there is a connection w = pfl between any k-form w defined on a k-dimensional orientable surface S and the volume element [l on S. Here p is some function on S depending on w . Hence
That is, every integral of second kind can be written as a suitable integral of first kind.
Example 1.
the path
s
1
(13.2' ) of Sect . 13. 1 , which expresses the work on }Rn , can be written as the integral of first kind
The integral :
[a, b]
-+
s
j (F,e)ds , I
(13.23)
e
where is arc length on 1, d is the element of length (a 1-form) , and is a unit velocity vector containing all the information about the orientation of f. From the point of view of the physical meaning of the problem solved by the integral (13.23), it is just as informative as the integral (13.1) of Sect . 13. 1 .
13.2 The Volume Element. Integrals of First and Second Kind
n(x) j(v,n)dCJ .
233
Example 2. The flux (13.3) of Sect . 13.1 of the velocity field V across a surface S c lRn oriented by unit normals can be written as the surface integral of first kind (13.24)
'Y
n.
The information about the orientation of S here is contained in the direction of the field of normals The geometric and physical content of the integrand in (13.24) is just as transparent as the corresponding meaning of the integrand in the final computational formula (13.6) of Sect . 13.1. For the reader ' s information we note that quite frequently one encounters the notation ds := e ds and du := which introduce a vector element of length and a vector element of area. In this notation the integrals (13.23) and (13.24) have the form
ndCJ,
j (F,
ds)
j( , ) ,
and
V du
'Y
'Y
which are very convenient from the point of view of physical interpretation. For brevity the inner product (A, B ) of the vectors A and B is often written
A · B. Example 3. Faraday 's law 4
asserts that the electromotive force arising in a closed conductor r in a variable magnetic field B is proportional to the rate of variation of the flux of the magnetic field across a surface S bounded by r . Let E be the electric field intensity. A precise statement of Faraday ' s law can be given as the equality
f E · ds
['
=
-
:t J B · s
du .
The circle in the integration sign over r is an additional reminder that the integral is being taken over a closed curve. The work of the field over a closed curve is often called the circulation of the field along this curve. Thus by Faraday ' s law the circulation of the electric field intensity generated in a closed conductor by a variable magnetic field equals the rate of variation of the flux of the magnetic field across a surface S bounded by r , taken with a suirable sign.
4 M. Faraday (1791-1867) - outstanding British physicist, creator of the concept of an electromagnetic field.
234
13 Line and Surface Integrals
Example 4. Ampere 's law5
f B · ds
r
B
=
1
c: o c
2
--
/ j · du s
(where is the magnetic field intensity, j is the current density vector, and and c are dimensioning constants) asserts that the circulation of the intensity of a magnetic field generated by an electric current along a contour r is proportional to the strength of the current flowing across the surface S bounded by the contour.
c:0
We have studied integrals of first and second kind. The reader might have noticed that this terminological distinction is very artificial. In reality we know how to integrate, and we do integrate, only differential forms. No integral is ever taken of anything else (if the integral is to claim independence of the choice of the coordinate system used to compute it) . 1 3 . 2 . 6 Problems and Exercises 1.
Give a formal proof of Eqs. ( 13. 18) and (13.20) .
2.
Let "( be a smooth curve and ds the element of arc length on "Y · a) Show that
II
f(s) ds
'Y
l$I 'Y
l f(s) l ds
for any function f on "( for which both integrals are defined. b) Verify that if lf(s) l $ M on "( and l is the length of "(, then
II
l
f(x) ds $ Ml .
'Y
c) State and prove assertions analogous to a) and b) in the general case for an integral of first kind taken over a k-dimensional smooth surface. 3. a) Show that the coordinates ( x 6 , x 6 , x �) of the center of masses distributed with linear density p ( x ) along the curve "Y should be given by the relations x
�
I
p x
( ) ds =
Ii
x p x
( ) ds ,
i
=
1 , 2, 3 .
b) Write the equation of a helix in JR 3 and find the coordinates of the center of mass of a piece of this curve, assuming that the mass is distributed along the curve with constant density equal to 1 . 5
A.M. Ampere ( 1775-1836) French physicist and mathematician, one of the founders of modern electrodynamics. -
13.2 The Volume Element. Integrals of First and Second Kind
235
c) Exhibit formulas for the center of masses distributed over a surface S with surface density p and find the center of masses that are uniformly distributed over the surface of a hemisphere. d) Exhibit the formulas for the moment of inertia of a mass distributed with density p over the surface S. e) The tire on a wheel has mass 30 kg and the shape of a torus of outer diameter 1 m and inner diameter 0.5 m. When the wheel is being balanced, it is placed on a balancing lathe and rotated to a velocity corresponding to a speed of the order of 100 km/hr, then stopped by brake pads rubbing against a steel disk of diameter 40 em and width 2 em. Estimate the temperature to which the disk would be heated if all the kinetic energy of the the spinning tire went into heating the disk when the wheel was stopped. Assume that the heat capacity of steel is c = 420 Jf (kg-K) . a) Show that the gravitational force acting o n a point mass mo located at ( Xo, Yo, zo) due to a material curve "{ having linear density p is given by the formula
4.
F = Gmo
J � �3 'Y
where G is the gravitational constant and xo, y - yo, z - zo).
r
r
ds
,
is the vector with coordinates
(x -
b ) Write the corresponding formula i n the case when the mass i s distributed over a surface S. c) Find the gravitational field of a homogeneous material line. d) Find the gravitational field of a homogeneous material sphere. (Exhibit the field both outside the ball bounded by the sphere and inside the ball. ) e ) Find the gravitational field created i n space by a homogeneous material ball (consider both exterior and interior points of the ball) . f) Regarding the Earth as a liquid ball, find the pressure in it as a function of the distance from the center. (The radius of the Earth is 6400 km, and its average density is 6 g/cm3 .) 5. Let "{1 and "{2 be two closed conductors along which currents J1 and h re spectively are flowing. Let ds 1 and ds 2 be the vector elements of these conductors corresponding to the directions of current in them. Let the vector R 1 2 be directed from ds 1 to ds 2 , and R2 1 = -R 1 2 . According to the Biot-Savart law6 the force dF 1 2 with which the first element acts on the second is
where the brackets denote the vector product of the vectors and co is a dimensioning constant. a) Show that, on the level of an abstract differential form, it could happen that dF 1 2 # -dF 2 1 in the differential Biot-Savart formula, that is, "the reaction is not equal and opposite to the action."
6 Biot (1774-1862), Savart (1791-1841) - French physicists.
13 Line and Surface Integrals
236
b) Write the (integral) formulas for the total forces F 12 and F 21 for the inter action of the conductors /' 1 and /'2 and show that F 12 = -F 2 1 · 6. The co-area formula (the Kronrod-Federer formula}. Let M m and N n be smooth surfaces of dimensions m and n respectively, em bedded in a Euclidean space of high dimension ( M m and N n may also be abstract Riemannian manifolds, but that is not important at the moment) . Suppose that
m
2:
n.
Let f : M m --+ N n be a smooth mapping. When m > n , the mapping df(x) : TxM m --+ Tf (x ) N n has a nonempty kernel ker df(x) . Let us denote by Tf M m the orthogonal complement of ker df(x) , and by J(f, x) the Jacobian of the mapping df(x) : Tf M m --+ Tf ( x ) N n . If m = n, then J( J, x) is the usual Jacobian. T.l. M m Let d v k (p) denote the volume element on a k-dimensional surface at the point p. We shall assume that vo (E) = card E, where v k (E) is the k-volume of E. a) Using Fubini's theorem and the rank theorem (on the local canonical form of a smooth mapping) if necessary, prove the following formula of Kronrod and Federer: I J( J , x) dv m (x) = I Vm -n ! - 1 (y) dv n (y) .
i
Mm
(
Nn
)
b) Show that if A is a measurable subset of M m , then
I J(J, x) dvm(x) = I Vm -n (A n r 1 (y) ) dvn (y) . Nn
A
This is the general Kronrod-Federer formula. c) Prove the following strengthening of Sard's theorem (which in its simplest version asserts that the image of the set of critical points of a smooth mapping has measure zero) . (See Problem 8 of Sect. 1 1 .5.) Suppose as before that f : M m --+ N n is a smooth mapping and K is a compact set in M m on which rank df(x) < n for all x E K . Then J. Vm - n K n f - 1 (y) dv n ( Y ) = 0. Use this result to obtain in addition the simplest version of Sard 's theorem stated above. d) Verify that if f : D --+ lR and u : D --+ lR are smooth functions in a regular domain D C JR n and u has no critical points in D, then
(
)
e) Let VJ (t) be the measure (volume) of the set {x E D l f(x) function f be nonnegative and bounded in the domain D. Show that I f dv = - I t dVJ (t) = I VJ (t) dt. D
R
>
t}, and let the
00
0
f) Let r.p E c < 1 l (JR, JR + ) and r.p(O) = 0, while f E C ( 1) (D, JR) and Vi11 (t) is the measure of the set {x E D l f(x) l > t } . Verify that I r.p o f dv = J r.p' (t)VjJI (t) dt.
i
D
0
237
13.3 The Fundamental Integral Formulas of Analysis
13.3 The Fundamental Integral Formulas of Analysis The most important formula of analysis is the Newton-Leibniz formula ( fun damental theorem of calculus ) . In the present section we shall obtain the formulas of Green, Gauss-Ostrogradskii, and Stokes, which on the one hand are an extension of the Newton-Leibniz formula, and on the other hand, taken together, constitute the most-used part of the machinery of integral calculus. In the first three subsections of this section, without striving for generality in our statements, we shall obtain the three classical integral formulas of anal ysis using visualizable material. They will be reduced to one general Stokes formula in the fourth subsection, which can be read formally independently of the others. 13.3. 1 Green's Theorem
Green ' s7 theorem is the following.
Let JR2 be the plane with a fixed coordinate grid x, y, and let D be a compact domain in this plane bounded by piecewise-smooth curves. Let P and Q be smooth functions in the closed domain D. Then the following relation holds: ( - �: ) dx dy = P dx + Q Dy , ( 1 3.25)
Proposition 1.
JJ ��
J
D
aD
in which the right-hand side contains the integral over the boundary aD of the domain D oriented consistently with the orientation of the domain D itself We shall first consider the simplest version of ( 13.25) in which D is the square I = { (x, y) E JR 2 1 0 :::; x :::; 1 , 0 :::; y :::; 1 } and Q 0 in /. Then Green ' s =
theorem reduces to the equality
JJ �: I
dx d y =
-
JP
dx ,
(13.26)
81
which we shall prove.
Proof Reducing the double integral to an iterated integral and applying the fundamental theorem of calculus, we obtain 7 G. Green (1793-1841) - British mathematician and mathematical physicist. Newton's grave in Westminster Abbey is framed by five smaller gravestones with brilliant names: Faraday, Thomson ( Lord Kelvin ) , Green, Maxwell, and Dirac.
238
13 Line and Surface Integrals 1 1 dx dy = dx dy = 0 0
jj �: I5
j 1
j j �:
j
j
1
1
(P(x, 1 ) - P(x, 2)) dx = - P(x, 0) dx + P(x, 1 ) dx . 0 0 0 The proof is now finished. What remains is a matter of definitions and interpretation of the relation just obtained. The point is that the difference of the last two integrals is precisely the right-hand side of relation (13.26) . =
/'3 1 1----� /'4
/'2
0
/'1
1
Fig. 1 3 . 5 .
Indeed, the piecewise-smooth curve 81 breaks into four pieces (Fig. 13.5) , which can be regarded as parametrized curves 1'1 : [0, 1] -+ where x r2.4 (x, 0), : [0, 1 ] -+ where y A ( 1 , y) , 1'3 : [0, 1] -+ where x A (x, 1 ) , 1'4 : [0, 1] -+ where y � (0, y) , By definition of the integral of the 1-form w = P dx over a curve 1 'Y ; (P(x, y) dx) : = P(x, O) dx , P(x, y) dx : = 0 ')'1 [0 , 1 ] 1 P(x, y) dx : = ')'HP(x, y) dx) : = O dy = 0 , 0 0 ')'2 [ , 1] 1 P(x, y) dx : = 1'3 (P(x, y) dx) : = P(x, 1 ) dx , 0 0 1 ')'3 [,] 1 P(x, y) dx : = 1H P(x , y) dx) : = O dy = 0 , 0 ')'4 [0 , 1]
')'2
IR2 , IR2 , IR2 , IR2 ,
j
j
j
j
j
j
j
j
j
j
j
j
13.3 The Fundamental Integral Formulas of Analysis
239
and, in addition, by the choice of the orientation of the boundary of the domain, taking account of the orientations of 1'2 , 1'3 , 1'4 , it is obvious that
/'l ,
jw = jw + jw + J w + J w = jw + jw - jw - jw ,
BI
/1
/1
/4
/2
where -l'i is the curve l'i taken with the orientation opposite to the one defined by l'i . Thus Eq. (13.26) is now verified. D It can be verified similarly that
JJ ��
dx dy
I
Adding
=J
(13.27)
Q dy ,
BI
(13.26) and (13.27), we obtain Green ' s formula
JJ (�� - �:)
dx dy
I
=J + P dx
Q dy
(13.25')
BI
for the square I . We remark that the asymmetry of P and Q in Green ' s formula (13.25) and in Eqs. (13.26) and (13.27) comes from the asymmetry of x and y: after all, x and y are ordered, and it is that ordering that gives the orientation in IR2 and in I. In the language of forms, the relation (13.25') just proved can be rewritten as
(13.25 11 )
w
I
BI is an arbitrary smooth form on I. The integrand on the right-hand
where side here is the restriction of the form to the boundary oi of the the square I. The proof of relation (13.26) just given admits an obvious generalization: If Dy is not a square, but a "curvilinear quadrilateral" whose lateral sides are vertical closed intervals (possibly degenerating to a point) and whose other two sides are the graphs of piecewise-smooth functions cp 1 ( x) :S cp 2 ( x) over the closed interval [a, b] of the x-axis, then
w
jj �: = - j dx dy
Dy
P dx .
(13.26')
BDy
Similarly, if there is such a "quadrilateral" Dx with respect to the y-axis, that is, having two horizontal sides, then for it we have the equality
jj ��
dx dy
=j
Q dy .
(13.27')
240
13 Line and Surface Integrals
Now let us assume that the domain D can be cut into a finite number of domains of type D y (Fig. 13.6). Then a formula of the form (13.26 1 ) also holds for that region D . y
Fig. 1 3 . 6 .
Proof. In fact, by additivity, the double integral over the domain D is the sum of the integrals over the pieces of type D y into which D is divided. Formula ( 13.26') holds for each such piece, that is, the double integral over that piece equals the integral of P dx over the oriented boundary of the piece. But adjacent pieces induce opposite orientations on their common boundary, so that when the integrals over the boundaries are added, all that remains after cancellation is the integral over the boundary 8D of the domain D itself. 0
Similarly, if D admits a partition into domains of type Dx , an equality of type (13.27') holds for it . We agree to call domains that can be cut both into pieces of type Dx and into pieces of type D y elementary domains. In fact, this class is sufficiently rich for all practical applications. By writing both relations (13.26 1 ) and (13.27') for a simple domain, we obtain (13.25) by adding them. Thus, Green ' s theorem is proved for simple domains. We shall not undertake any further sharpenings of Green ' s formula at this point (on this account see Problem 2 below) , but rather demonstrate a second, very fruitful line of reasoning that one may pursue after establishing Eqs. (13.25') and (13.25 11 ). Suppose the domain C has been obtained by a smooth mapping cp : ---+ C of the square If w is a smooth 1-form on C, then
I
I.
J
c
dw :=
J I
cp * dw =
J I
dcp*w �
J
EJI
cp*w =:
J
w.
(13.28)
EJC
The exclamation point here distinguishes the equality we have already proved (see ( 13.25")); the extreme terms in these equalities are definitions
13.3 The Fundamental Integral Formulas of Analysis
241
or direct consequences of them; the remaining equality, the second from the left, results from the fact that exterior differentiation is independent of the coordinate system. Hence Green ' s formula also holds for the domain C. Finally, if it is possible to cut any oriented domain D into a finite num ber of domains of the same type as C, the considerations already described involving the mutual cancellation of the integrals over the portions of the boundaries of the Ci inside D imply that
(13.29) that is, Green ' s formula also holds for D. It can b e shown that every domain with a piecewise-smooth boundary belongs to this last class of domains, but we shall not do so, since we shall describe below (Chap. 15) a useful technical device that makes it possible to avoid such geometric complications, replacing them by an analytic problem that is comparatively easy to solve. Let us consider some examples of the use of Green ' s formula.
Example 1.
- y, Q = x in (13.25). We then obtain -y dx + x dy = 2 dx dy = 2 a( D) ,
Let us set P =
J
J
D
8D
where a (D) is the area of D. Using Green ' s formula one can thus obtain the following expression for the area of a domain on the plane in terms of line integrals over the oriented boundary of the domain:
a(D) =
� J -y dx + x dy = - J y dx = J x dy . 8D
8D
It follows in particular from this that the work
8D
A = J P dV performed "Y
by a heat engine in changing the state of its working substance over a closed cycle 'Y equals the area of the domain bounded by the curve 'Y in the PV-plane of states (see Problem 5 of Sect . 13.1).
Example 2 . Let B = { (x, y) E �2 1 x 2 + y 2
1} be the closed disk in the plane. We shall show that any smooth mapping f : B ---+ B of the closed disk into itself has at least one fixed point (that is, a point p E B such that f(p) = p). Proof. Assume that the mapping f has no fixed points. Then for every point p E B the ray with initial point f(p) passing through the point p and the point cp(p) E aB where this ray intersects the circle bounding B are uniquely �
242
13 Line and Surface Integrals
determined. Thus a mapping r.p : B ---+ 8B would arise, and it is obvious that the restriction of this mapping to the boundary would be the identity mapping. Moreover, it would have the same smoothness as the mapping f itself. We shall show that no such mapping r.p can exist . In the domain JR 2 \ 0 ( the plane with the origin omitted ) let us consider the form w = - yx���� d y that we encountered in Sect . 13.1. It can be verified immediately that dw = 0. Since 8B C 1R 2 \0 , given the mapping r.p : B ---+ 8B, one could obtain a form r.p*w on B , and dr.p*w = r.p* ( dw ) = r.p*O = 0. Hence by Green ' s formula r.p*w = dr.p*w = 0 . &B B But the restriction of r.p to 8B is the identity mapping, and so
J
J
J
r.p * w
=
J
w.
&B &B This last integral, as was verified in Example 1 of Sect . 13.1, is nonzero. This contradiction completes the proof of the assertion. 0 This assertion is of course valid for a ball of any dimension ( see Example 5 below ) . It also holds not only for smooth mappings, but for all continuous mappings f : lB\ ---+ lll\ . In this general form it is called the Brouwer fixed-point
theorem. 8
1 3 . 3 . 2 The Gauss-Ostrogradskii Formula
Just as Green's formula connects the integral over the boundary of a plane domain with a corresponding integral over the domain itself, the Gauss Ostrogradskii formula given below connects the integral over the boundary of a three-dimensional domain with an integral over the domain itself. Proposition 2. Let JR 3 be three-dimensional space with a fixed coordinate system x, y, z and D a compact domain in JR3 bounded by piecewise-smooth surfaces. Let P, Q, and R be smooth functions in the closed domain D .
Then the following relation holds: ( aP + aQ + &R ) dx dy Dz =
JJJ
ax
ay
=
az
JJ
P dy I\ dz + Q dz I\ dx + R dx I\ dy .
(13.30)
&75
8
L.E.J. Brouwer (1881-1966) - Well-known Dutch mathematician. A number of fundamental theorems of topology are associated with his name, as well as an analysis of the foundations of mathematics that leads to the philosophico mathematical concepts called intuitionism.
13.3 The Fundamental Integral Formulas of Analysis
243
The Gauss-Ostrogradskii formula (13.30) can be derived by repeating the derivation of Green ' s formula step by step with obvious modifications. So as not to do a verbatim repetition, let us begin by considering not a cube in IR3 , but the domain Dx shown in Fig. 13. 7, which is bounded by a lateral cylindrical surface 8 with generator parallel to the z-axis and two caps 81 and 82 which are the graphs of piecewise-smooth functions r.p 1 and r.p 2 defined in the same domain G C IR; Y . We shall verify that the relation
R dx 1\ d y jjj �� dx dy dz = jj av.
(13.31)
holds for Dz . z
X
Fig.
13.7.
Proof.
dx d y dz = jj d x d y j �� dz = �� jjj Dz 'Pl ( x ,y ) = jj (R(x, y, r.p 2 (x, y)) - R(x, y, r.p 1 (x, y))) dx dy = = - jj (R(x, y, r.p 1 (x, y)) dx dy + jj (R(x, y, r.p 2 (x, y)) dx dy . G
G
G
The surfaces
G
81 and 82 have the following parametrizations: 81 : (x, y ) t----+ (x, y, r.p l (x, y)) ' 82 : (x, y) t----+ (x, y, r.p 2 (x, y )) .
244
13 Line and Surface Integrals
The curvilinear coordinates (x, y) define the same orientation on S2 that is induced by the orientation of the domain D z , and the opposite orientation on S1 . Hence if S1 and S2 are regarded as pieces of the boundary of D z oriented as indicated in Prosition 2, these last two integrals can be interpreted as integrals of the form R dx 1\ dy over S1 and S2. The cylindrical surface S has a parametric representation ( t , z) H ( x( t ) , y( t ) , z ) , so that the restriction of the form R dx 1\ dy to S equals zero, and so consequently, its integral over S is also zero. Thus relation (13.31) does indeed hold for the domain D z . D If the oriented domain D can be cut into a finite number of domains of the type Dz , then, since adjacent pieces induce opposite orientations on their common boundary, the integrals over these pieces will cancel out, leaving only the integral over the boundary 8 D . Consequently, formula (13.31) also holds for domains that admit this kind of partition into domains of type Dz . Similarly, one can introduce domains D y and Dx whose cylindrical sur faces have generators parallel to the y-axis or x-axis respectively and show that if a domain D can be divided into domains of type Dy or Dx , then the relations
JJJ �� dx dy dz = JJ Q dz 1\ dx ,
(13.32) (13.33)
Thus, if D is a simple domain, that is, a domain that admits each of the three types of partitions just described into domains of types Dx , D y , and D z , then, by adding (13.31), (13.32), and (13.33), we obtain (13.30) for D . For the reasons given in the derivation of Green ' s theorem, we shall not undertake the description of the conditions for a domain to be simple or any further sharpening of what has been proved (in this connection see Problem 8 below or Example 12 in Sect. 17.5). We note, however, that i n the language of forms, the Gauss-Ostrogradskii formula can be written in coordinate-free form as follows:
(13.30') where w is a smooth 2-form in D . 0� = { (x, y, z) E Since formula (13.30') holds for the cube I = 1 ::; 1, 0 ::; y � 1, 0 � z � 1}, as we have shown, its extension to more general classes of domains can of course be carried out using the standard computations (13.28) and (13.29).
13
JR3 I
13.3 The Fundamental Integral Formulas of Analysis
245
Example 3. The law of Archimedes. Let us compute the buoyant force of a homogeneous liquid on a body D immersed in it . We choose the Cartesian coordinates x, y, z in JR 3 so that the xy-plane is the surface of the liquid and the z-axis is directed out of the liquid. A force pgzn da is acting on an element da of the surface S of D located at depth z, where p is the density of the liquid, g is the acceleration of gravity, and n is a unit outward normal to the surface at the point of the surface corresponding to da. Hence the resultant force can be expressed by the integral F
=
JJs pgzn da .
If n = e x COS O! x + e y COS O! y + ez COS O!z , then n da = ex dy 1\ dz + e y dz 1\ dx + e z dx 1\ dy ( see Subsect . 13.2.4). Using the Gauss-Ostrogradskii formula (13.30), we thus find that
F = expg JJs zdy/\dz +eypg JJs zdz/\dx +ezpg JJs z dx /\ dy = p Iff = ex pg Iff ezpg JJJ = O dx dy dz +
O dx dy dz + e y g
D
D
dx dy dz
+
pgV ez ,
D
where V is the volume of the body D. Hence P = pgV is the weight of a volume of the liquid equal to the volume occupied by the body. We have arrived at Archimedes ' law: F = Pez .
Example 4. Using the Gauss-Ostrogradskii formula (13.30), one can give the following formulas for the volume V(D) of a body D bounded by a surface aD.
V(D) =
� JJ
x dy 1\ dz + y dz 1\ dx + z dx 1\ dy =
aD
=
JJ
x dy 1\ dz
aD
=
JJ aD
y dz 1\ dx =
JJ
z dx 1\ dy .
aD
13.3.3 Stokes' Formula in JR3
Let S be an oriented piecewise-smooth compact two dimensional surface with boundary 88 embedded in a domain G IR3 , in which a smooth 1-form w = P dx + Q dy + R dz is defined. Then the following relation holds: Proposition 3.
c
246
13 Line and Surface Integrals
Jas
JsJ ( ��
��) d y A d z + 8Q 8P 8P oR - - dx A dy , + ( - - - ) dz A dx + ( {)z ox ox oy )
P dx + Q dy + R dz =
-
(13.34) where the orientation of the boundary as is chosen consistently with the orientation of the surface S. In other notation, this means that dw = w . (13.34') -
j j s
ow
Proof. If C is a standard parametrized surface i.p : I -+ C in �3 , where I is a square in � 2 , relation (13.34) folios from Eqs. (13.28) taking account of what has been proved for the square and Green ' s formula. If the orientable surface S can be cut into elementary surfaces of this type, then relation (13.34) is also valid for it, as follows from Eqs. (13.29) with D replaced by S. D As in the preceding cases, we shall not prove at this point that, for exam ple, a piecewise-smooth surface admits such a partition. Let us show what this proof of formula (13.34) would look like in coordi nate notation. To avoid expressions that are really too cumbersome, we shall write out only the first, main part of its two expressions, and with some sim plifications even in that. To be specific, let us introduce the notation x 1 , x 2 , x 3 for the coordinates of a point x E � 3 and verify only that 8P 8P 2 dx A dx 1 + -3 dx 3 A dx 1 , P(x) dx 1 = 2 8x 8x
J
as
!!s
since the other two terms on the left-hand side of (13.34) can be studied s imilarly. For simplicity we shall assume that S can be obtained by a smooth
t
D
x
=
T
Fig.
13.8.
x(t)
{3
13.3 The Fundamental Integral Formulas of Analysis
247
mapping x = x ( t ) of a domain D in the plane IR 2 of the variables t 1 , t 2 and bounded by a smooth curve "( = 8D parametrized via a mapping t = t ( T ) by the points of the closed interval a :S; T :S; (3 ( Fig. 13.8 ) . Then the boundary T = OS of the surface S can be written as X = x (t ( T )) , where T ranges OVer the closed interval [a, (3]. Using the definition of the integral over a curve, Green ' s formula for a plane domain D, and the definition of the integral over a parametrized surface, we find successively
J
r
P ( x ) dx 1 :=
=
J {3
a
P ( x ( t ( T )))
!! ( I tt:D
8P 8x 2
( 8x 1 dt 1
f)t l d T
+
8x 1 dt 2 ) dT = ot 2 dT
8x 1
8t1
=:
!! ( s
8P 2 dx A dx 1 ox 2
+
8P 3 ) dx A dx 1 . ox 3
The colon here denotes equality by definition, and the exclamation point denotes a transition that uses the Green ' s formula already proved. The rest consists of identities. Using the basic idea of the proof of formula ( 13.34' ) , we have thus verified directly ( without invoking the relation cp* d = dcp* , but essentially proving it for the case under consideration ) that formula ( 13.34 ) does indeed hold for a simple parametrized surface. We have carried out the reasoning formally only for the term P dx, but it is clear that the same thing could also be done for the other two terms in the 1-form in the integrand on the left-hand side of ( 13.34 ) .
248
13 Line and Surface Integrals
13.3.4 The General Stokes Formula
Despite the differences in the external appearance of formulas (13.25), (13.30), and (13.34), their coordinate-free expressions (13.25"), (13.29), (13.30'), and (13.34') turn out to be identical. This gives grounds for sup posing that we have been dealing with particular manifestations of a general rule, which one can now easily guess.
Let S be an oriented piecewise smooth k-dimensional com pact surface with boundary aS in the domain G �n , in which a smooth ( k - 1) -form w is defined. Then the following relation holds:
Proposition 4.
c
(13.35) in which the orientation of the boundary aS is that induced by the orientation of S. Proof. Formula (13.35) can obviously be proved by the same general com putations (13.28) and (13.29) as Stokes ' formula (13.34') provided it holds k k k for a standard k-dimensional interval J = { x = ( x 1 , . . . , x ) E �k I 0 ::; xi ::; 1, i = 1, . . . , k }. Let us verify that (13.35) does indeed hold for J . k Since a (k - I ) -form on J has the form w = L i ai (x)dx 1 1\ · · · I\ dx i 1\ · · · 1\ dx k ( summation over i = 1, . . . , k, with the differential dxi omitted ) , it suffices to prove (13.35) for each individual term. Let w = a( x) dx 1 /\ · · · 1\ dx i /\ · · · I\ dx k . Then dw = (- l) i - l g:i ( x ) dx 1 1\ · · · I\ dxi 1\ · · · I\ dx k . We now
carry out the computation:
1
dx 1 · · · dx' · · · dx k
J J ox'oa (x) dx". = - ( - l ) i - 1 J ( a ( x 1 , . . . , x i - 1 , 1 , x i +l , . . . , x k ) •-
= ( - 1) . 1
�
Jk - l
.
.
0
Jk - l
- a ( x 1 , . . . , X i - 1 , 0 , Xi +l , . . . X k ) ) dX 1 · · · dXi · · · dX k = = ( - l) i - 1 a(t l , . . . , t i - l , l, t i , . . . , t k - 1 ) dt l . . . dt k - 1 +
j
Jk - l
+
( l) i -
J a (t l , . . . , ti - l 0 ti , . . . , tk - 1 ) t l . . . tk- 1 .
Jk - l
'
'
d
d
13.3 The Fundamental Integral Formulas of Analysis
249
Here J k - 1 is the same as J k in R k , only it is a (k - I)-dimensional interval in R k - 1 . In addition, we have relabeled the variables x 1 = t 1 , . . . , x i - 1 = ti - l ,
xi + 1 = t i , . . . , x k = t k - 1 . The mappings
l k - 1 3 t = ( t 1 , . . . , t k - 1 ) ( t l , . . . , t i - l , l, ti , . . . , t k - 1 ) E l k , [ k - 1 3 t = ( t l , . . . ' t k - 1 ) t---+ ( t l , . . . ' t i - l , 0, t i ' . . . ' t k - 1 ) E [ k are parametrizations of the upper and lower faces ri l and rio of the inter val J k respectively orthogonal to the x i axis. These coordinates define the same frame e 1 . . . . , e i - 1 , e i + 1 , . . . , e k orienting the faces and differing from the frame e 1 , . . . , e k of R k in the absence of e i . On ri 1 the vector ei is the exterior normal to J k , as the vector -e i is for the face riO . The frame ei , e 1 , . . . , ei - 1 , ei + 1 , . . . , e k becomes the frame e 1 , . . . , e k after i - 1 inter changes of adjacent vectors, that is, the agreement or disagreement of the orientations of these frames is determined by the sign of ( - l) i - l . Thus, this parametrization defines an orientation on ri l consistent with the orientation of J k if taken with the corrective coefficient ( - l) i - l (that is, not changing the orientation when i is odd, but changing it when i is even) . Analogous reasoning shows that for the face rio it is necessary to take a corrective coefficient ( - 1 ) i to the orientation defined by this parametrization of the face riO . t---+
Thus, the last two integrals (together with the coefficients in front of them) can be interpreted respectively as the integrals of the form w over the faces ri l and riO of [ k with the orientation induced by the orientation of [ k . We now remark that on each of the remaining faces of J k one of the co ordinates x 1 , . . . , x i - 1 , x i + 1 , . . . , x k is constant. Hence the differential corre sponding to it is equal to zero on such a face. Thus, the form dw is identically equal to zero and its integral equals zero over all faces except riO and ri l · Hence we can interpret the sum of the integrals over these two faces as the integral of the form w over the entire boundary aJ k of the interval [ k oriented in consistency with the orientation of the interval J k itself. The formula
(13.35), is now proved. 0 formula (13.35) is a corollary of the
and along with it formula
As one can see, Newton-Leibniz formula (fundamental theorem of calculus) , Fubini's theorem, and a series of definitions of such concepts as surface, boundary of a surface, orientation, differential form, differentiation of a differential form, and transference of forms. Formulas (13.25), (13.30), and (13.34), the formulas of Green, Gauss Ostrogradskii, and Stokes respectively, are special cases of the general formula
250
13 Line and Surface Integrals
( 13.35 ) . Moreover, if we interpret a function f defined on a closed interval [a, b] C 1ft as a 0-form w , and the integral of a 0-form over an oriented point as the value of the function at that point taken with the sign of the orien tation of the point , then the Newton-Leibniz formula itself can be regarded as an elementary ( but independent ) version of ( 13.35) . Consequently, the fundamental relation (13.35) holds in all dimensions k :::: 1 . Formula ( 1 3.35) is often called the general Stokes formula. As historical information, we quote here some lines from the preface of M. Spivak to his book ( cited in the bibliography below ) : The first statement of the Theorem9 appears as a postscript to a letter, dated July 2, 1850, from Sir William Thomson ( Lord Kelvin ) to Stokes. It appeared publicly as question 8 on the Smith ' s Prize Ex amination for 1854. This competitive examination, which was taken annually by the best mathematics students at Cambridge University, was set from 1849 to 1882 by Professor Stokes; by the time of his death the result was known universally as Stokes ' theorem. At least three proofs were given by his contemporaries: Thomson published one, another appeared in Thomson and Tait ' s Treatise on Natural Philosophy, and Maxwell provided another in Electricity and Mag netism. Since this time the name of Stokes has been applied to much more general results, which have figured so prominently in the devel opment of certain parts of mathematics that Stokes ' theorem may be considered a case study in the value of generalization. We note that the modern language of differential forms originates with
E lie Cartan , 10 but the form (13 .35) for the general Stokes' formula for surfaces
in lft n seems to have been first proposed by Poincare. For domains in n dimensional space lft n Ostrogradskii already knew the formula, and Leibniz wrote down the first differential forms. Thus it is not an accident that the general Stokes formula ( 13.35) is sometimes called the Newton-Leibniz-Green-Gauss-Ostrogradski-Stokes Poincare formula. One can conclude from what has been said that this is by no means its full name. Let us use this formula to generalize the result of Example 2.
Example 5 .
ball B
C
Let us show that every smooth mapping
lftm into itself has at least one fixed point .
f:B
---+
B of a closed
Proof. If the mapping f had no fixed points, then, as in Example 2, one could construct a smooth mapping cp : B ---+ aB that is the identity on the sphere a B. In the domain lftm \ 0, we consider the vector field lrlm , where r is the radius-vector of the point = ( , m ) E lftm \ 0, and the flux form
x xl , . . . x
9 The classical Stokes theorem ( 13.34) is meant. E lie Cartan ( 1869-1951 ) - outstanding French geometer.
10
13.3 The Fundamental Integral Formulas of Analysis
251
corresponding to this field (see formula (13. 19) of Sect. 13.2) . The flux of such a field across the boundary of the ball B = { x E IR I I x I = 1 } in the direction of the outward normal to the sphere 8B is obviously equal to the area of the sphere 8B, that is, J w -j. 0. But , as one can easily verify by DB direct computation, dw = 0 in !Rm \ 0, from which, by using the general Stokes formula, as in Example 2, we find that
J J w=
r.p*w =
J
dr.p*w =
DB DB This contradiction finishes the proof. D
J
r.p* dw
=
J
r.p* O = 0 .
13.3.5 Problems and Exercises 1. a) Does Green ' s formula (13.25) change if we pass from the coordinate system x, y to the system y, x? b) Does formula (13.25") change in this case? 2. a) Prove that formula (13.25) remains valid if the functions P and Q are con tinuous in a closed square I, their partial derivatives �� and !ff: are continuous at interior points of I, and the double integrals exist, even if as improper integrals (13.25') . b) Verify that if the boundary of a compact domain D consists of piecewise smooth curves, then under assumptions analogous to those in a) , formula (13.25) remains valid.
a) Verify the proof of (13.26') in detail. b) Show that if the boundary of a compact domain D C JR. 2 consists of a finite number of smooth curves having only a finite number of points of inflection, then D is a simple domain with respect to any pair of coordinate axes. c) Is it true that if the boundary of a plane domain consists of smooth curves, then one can choose the coordinate axes in JR. 2 such that it is a simple domain relative to them? 3.
a) Show that if the functions P and Q in Green's formula are such that �� �� = 1 , then the area a ( D) of the domain D can be found using the formula
4. a
( D) = I P dx + Q dy. aD
b) Explain the geometric meaning of the integral I y dx over some (possibly 'I
nonclosed) curve in the plane with Cartesian coordinates x, y. Starting from this, give a new interpretation of the formula a ( D) = - I y dx. aD
252
13 Line and Surface Integrals c) As a check on the preceding formula, use it to find the area of the domain
{
1
2 2 D = (x, y) E lR2 xa2 + yb2
:::;
}
1 .
5 . a) Let x = x(t) be a diffeomorphism of the domain Dt C lRz onto the do main D x C JR; . Using the results of Problem 4 and the fact that a line integral is independent of the admissible change in the parametrization of the path, prove that
I dx = I lx' (t) l dt ,
where dx = dx 1 dx 2 , dt = d e d e , lx' (t) l = det x' (t) . b) From a) derive the formula
I f(x) dx = I t (x(t) ) l det x' (t) l dt
Dx
Dt
for change of variable in a double integral.
( *) 2
6. Let f(x, y, t) be a smooth function satisfying the condition � ) + =I 0 in its domain of definition. Then for each fixed value of the parameter t the equation f(x, y, t) = 0 defines a curve It in the plane JR 2 . Then a family of curves {It } depending on the parameter t arises in the plane. A smooth curve r C JR 2 defined by parametric equations x = x(t) , y = y(t) , is the envelope of the family of curves {It } if the point x(to ) , y(to ) lies on the corresponding curve Ito and the curves r and Ito are tangent at that point , for every value of to in the common domain of definition of {It } and the functions x(t) , y(t) . a) Assuming that x, y are Cartesian coordinates in the plane, show that the functions x(t) , y(t) that define the envelope must satisfy the system of equations
(
{
2
f(x, y, t) = 0 ,
%f (x, y, t) = O , and from the geometric point of view the envelope itself is the boundary of the projection (shadow) of the surface f(x, y, t) = 0 of lR(x , y , t) on the plane lR(x , y ) . b) A family of lines x cos a + y sin a p( a ) = 0 is given in the plane with Cartesian coordinates x and y. The role of the parameter is played here by the polar angle a . Give the geometric meaning of the quantity p ( a ) , and find the envelope of this family if p( a ) = c + a cos a + b sin a, where a , b, and c are constants. c) Desribe the accessible zone of a shell that can be fired from an adjustable cannon making any angle a E [0, /2] to the horizon. d) Show that if the function p ( a ) of b) is 2rr-periodic, then the corresponding envelope r is a closed curve. -
1r
13.3 The Fundamental Integral Formulas of Analysis
253
e) Using Problem 4, show that the length L of the closed curve r obtained in d) can be found by the formula 2 7r L = p (a ) da .
I 0
(Assume that p E c < 2 l .) f) Show also that the area u of the region bounded by the closed curve r obtained in d) can be computed as 2 7r p(a) = u= (p2 - P2 ) ( a ) d a ' (a) .
�I 0
��
Consider the integral I cos � . n ) ds, in which "Y is a smooth curve in ne , r is the
7.
"'
radius-vector of the point (x, y) E '"f, r = lrl = y'x 2 + y 2 , n is the unit normal vector to "Y at ( x, y) varying continuously along "Y, and ds is arc length on the curve. This integral is called Gauss ' integml. a) Write Gauss' integral in the form of a flux I (V, n ) ds of the plane vector field V across the curve "Y . b) Show that in Cartesian coordinates x and y Gauss' integral has the form ± I -��'"+.�2d y familiar to us from Example 1 of Sect. 13. 1 , where the choice of sign "'
is determined by the choice of the field of normals n . c) Compute Gauss' integral for a closed curve "Y that encircles the origin once and for a curve "Y bounding a domain that does not contain the origin. d) Show that cos � , n ) ds = dcp, where cp is the polar angle of the radius-vector r , and give the geometric meaning of the value of Gauss' integral for a closed curve and for an arbitrary curve "Y C ll�? .
In deriving the Gauss-Ostrogradskii formula we assumed that D is a simple domain and the functions P, Q, R belong to C ( ll (D, IR) . Show by improving the reasoning that formula (13.30) holds if D is a compact domain with piecewise smooth boundary, P, Q, R E C(D, IR) , �= , � ' �� E C(D, IR) , and the triple integral converges, even if it is an improper integral. 8.
9. a) If the functions P, Q, and R in formula (13.30) are such that �= + � + �� = 1 , then the volume V(D) of the domain D can b e found by the formula
V(D) =
II P dy !\ dz + Q dz !\ dx + R dx !\ dy . av
b) Let f ( x, t) be a smooth function of the variables x E D x C IR� , t E Dt C IRt' and � = ( -§!r , . . . , lfn) =/= 0. Write the system of equations that must be satisfied by the ( n - 1 )-dimensional surface in IR� that is the envelope of the family of surfaces {St } defined by the conditions f(x, t) = 0, t E Dt (see Problem 6).
13 Line and Surface Integrals
254
c ) Choosing a point on the unit sphere as the parameter t, exhibit a fmaily of planes in JR 3 depending on the parameter t whose envelope is the ellipsoid 5+ a 2
2
� + � = 1.
d ) Show that if a closed surface S is the envelope of a family of planes cos cl: l ( t ) x + cos a ( t ) y + cos a 3 ( t ) z - p ( t ) = 0 , where a1 , a 2 , a 3 are the angles formed by the normal to the plane and the coordinate axes and the parameter t is a variable point of the unit sphere S 2 C JR3 , then the area rJ of the surface S can be found by the formula rJ = J p ( t ) drJ. s2
e ) Show that the volume of the body bounded by the surface S considered in d ) can be found by the formula V = t J p ( t ) drJ. s
z2
f ) Test the formula given in e ) by finding the volume of the ellipsoid � + � +
� :::; 1 .
g ) What does the n-dimensional analogue of the formulas in d ) and e ) look like? 1 0. a ) Using the Gauss-Ostrogradskii formula, verify that the flux of the field r j r 3 ( where r is the radius-vector of the point x E JR 3 and r = l r l ) across a smooth surface S enclosing the origin and homeomorphic to a sphere equals the flux of the same field across an arbitrarily small sphere l x l = E . b ) Show that the flux in a) is 47r. c ) Interpret Gauss' integral J cos(; ,n) ds in JR 3 as the flux of the field r / r 3 across s the surface S. d ) Compute Gauss' integral over the boundary of a compact domain D C JR3 , considering both the case when D contains the origin in its interior and the case when the origin lies outside D. e ) Comparing Problems 7 and lOa) -d ) , give an n-dimensional version of Gauss ' integral and the corresponding vector field. Give an n-dimensional statement of problems a) -d ) and verify it. 1 1 . a ) Show that a closed rigid surface S C JR 3 remains in equilibrium under the action of a uniformly distributed pressure on it. ( By the principles of statics the problem reduces to verifying the equalities JJ n drJ = 0, JJ [r, nj drJ = 0, where n s s is a unit normal vector, r is the radius-vector, and [r, n] is the vector product of r and n. ) b ) A solid body of volume V is completely immersed in a liquid having specific gravity 1 . Show that the complete static effect of the pressure of the liquid on the body reduces to a single force F of magnitude V directed vertically upward and attached to the center of mass C of the solid domain occupied by the body. 1 2 . Let r : I k ---+ D be a smooth ( not necessarily homeomorphic ) mapping of an interval I k C lR k into a domain D of lR n , in which a k-form w is defined. By analogy with the one-dimensional case, we shall call a mapping r a k- cell or k-pathand by definition set J w = J r*w. Study the proof of the general Stokes formula and Jk r verify that it holds not only for k-dimensional surfaces but also for k-cells.
13.3 The Fundamental Integral Formulas of Analysis
255
13. Using the generalized Stokes formula, prove by induction the formula for change of variable in a multiple integral (the idea of the proof is shown in Problem 5a) ) .
Integration b y parts in a multiple integral. Let D be a bounded domain in JR m with a regular (smooth or piecewise smooth) boundary 8D oriented by the outward unit normal n = (n 1 , . . . , n m ) . Let J, g be smooth functions in D. a) Show that 14.
1 o;f dv 1 fni da . =
D
8D
b) Prove the following formula for integration by parts:
I (8d)g dv I fgni da - I f(oi g) dv .
D
=
8D
D
1 4 Elements of Vector Analysis
and Field Theory
14. 1 The Differential Operations of Vector Analysis 14. 1 . 1 Scalar and Vector Fields
In field theory we consider functions x t-+ T(x) that assign to each point x of a given domain D a special object T(x) called a tensor. If such a function is defined in a domain D, we say that a tensor field is defined in D. We do not intend to give the definition of a tensor at this point: that concept will be studied in algebra and differential geometry. We shall say only that numerical functions D 3 x t-+ f (x ) E R and vector-valued functions Rn :::> D 3 x t-+ V ( x) E TR� � Rn are special cases of tensor fields and are called scalar fields and vector fields respectively in D (we have used this terminology earlier) . A differential p-form w in D is a function Rn :::> D 3 x t-+ w ( x ) E C ( (Rn)P , R) which can be called a field of forms of degree p in D. This also is a special case of a tensor field. At present we are primarily interested in scalar and vector fields in do mains of the oriented Euclidean space Rn . These fields play a major role in many applications of analysis in natural science. 14. 1 . 2 Vector Fields and Forms in R3
We recall that in the Euclidean vector space R3 with inner product ( , ) there is a correspondence between linear functionals A : R3 ---+ R and vectors A E R3 consisting of the following: Each such functional has the form A( e ) = (A, e) , where A is a completely definite vector in R3 . If the space is also oriented, each skew-symmetric bilinear functional B : R3 x R3 ---+ R can be uniquely written in the form B (e 1 , e 2 ) = (B, e 1 , e 2 ) , where B is a completely definite vector in R3 and (B, e 1 , e 2 ) , as always, is the scalar triple product of the vectors B , e 1 , and e 2 , or what is the same, the value of the volume element on these vectors. Thus, in the oriented Euclidean vector space R3 one can associate with each vector a linear or bilinear form, and defining the linear or bilinear form is equivalent to defining the corresponding vector in R3 .
258
14 Elements of Vector Analysis and Field Theory
If there is an inner product in IR3 , it also arises naturally in each tan gent space TIR� consisting of the vectors attached to the point JR3 , and the orientation of IR 3 orients each space TIR� . Hence defining a 1-form w 1 (x) or a 2-form w 2 (x) in TIR� under the con ditions just listed is equivalent to defining some vector A(x) E TIR� corre sponding to the form w 1 (x) or a vector B(x) E TIR� corresponding to the form w 2 (x) . Consequently, defining a 1-form w 1 or a 2-form w 2 in a domain D of the oriented Euclidean space IR3 is equivalent to defining the vector field A or B in D corresponding to the form. In explicit form, this correspondence amounts to the following:
wl (x)( e ) ( A(x), e/ , (14.1) w� (x) (e l , e 2 ) (B(x) , e l , e 2 ) , ( 14.2) where A(x) , B(x), e, e 1 , and e 2 belong to TDx . Here we see the work form w 1 wl of the vector field A and the flux form w 2 = w� of the vector field B, which are already familiar to us. =
To a scalar field follows:
f : D --+ IR, we can assign a 0-form and a 3-form in D as w� w}
f, = f dV , =
( 14.3) ( 14.4)
where dV is the volume element in the oriented Euclidean space IR3 . In view of the correspondences ( 14. 1 )-( 14.4) , definite operations on vector and scalar fields correspond to operations on forms. This observation, as we shall soon verify, is very useful technically.
To a linear combination of forms of the same degree there corresponds a linear combination of the vector and scalar fields corresponding to them. Proof. Proposition 1 is of course obvious. However, let us write out the full
Proposition 1 .
proof, as an example, for 1-forms:
a 1 (A 1 , · ) + a (A2 , · ) = (a 1 A 1 + a 2 A 2 , · ) w;,A, +u 2 A 2 • D It is clear from the proof that a 1 and a 2 can be regarded as functions a 1 wl , + a 2 wl 2
=
=
=
(not necessarily constant) in the domain D in which the forms and fields are defined. As an abbreviation, let us agree to use, along with the symbols ( , ) and [ , ] for the inner product and the vector product of vectors A and B in IR3 , the alternative notation A · B and A x B wherever convenient .
14.1 The Differential Operations of Vector Analysis
259
If A, B, A 1 , and B 1 are vector fields in the oriented Eu clidean space JR3 , then
Proposition 2 .
1 /\ wA1 = wA, 2 xA ' ( 14.5 ) wA, 2 2 ( 14.6 ) wl l\ w� = w_i.B . In other words, the vector product A 1 x A 2 of fields A 1 and A 2 that
generate 1-forms corresponds to the exterior product of the 1-forms they generate, since it generates the 2-form that results from the product . In the same sense the inner product of the vector fields A and B that generate a 1-form wl and a 2-form w� corresponds to the exterior product of these forms.
Proof. To prove these assertions, fix an orthornormal basis in JR 3 Cartesian coordinates x 1 , x 2 , x 3 corresponding to it . In Cartesian coordinates
and the
3 3 wl (x) ( e ) = A(x) . e = L A i (x) � i = L A i (x) dx i ( e ) ' i=1 i= l that is,
( 14. 7 )
and
that is,
( 14.8 )
Therefore in Cartesian coordinates, taking account of expressions ( 14.7 ) and ( 14.8 ) , we obtain
wl , 1\ wt = (At dx1 + Ai dx 2 + Ai dx 3 ) 1\ (A� dx 1 + A� dx 2 + A� dx 3 ) = = (Ai A� - Ai A�) dx 2 1\ dx 3 + (Ai A� - At A�) dx 3 1\ dx 1 + + (At A� - AiA�) dx 1 /\ dx 2 = w� , where B = A 1 x A 2 .
Coordinates were used in this proof only to make it easier to find the vector B of the corresponding 2-form. The equality ( 14.5 ) itself, of course, is independent of the coordinate system. Similarly, multiplying Eqs. ( 14.7 ) and ( 14.8 ) , we obtain
wl l\ w� = (A 1 B 1 + A 2 B 2 + A3 B 3 ) dx1 1\ dx 2 1\ dx 3 = w2 .
14 Elements of Vector Analysis and Field Theory
260
In Cartesian coordinates dx i A dx 2 A dx 3 is the volume element in JR.3 , and the sum of the pairwise products of the coordinates of the vectors A and B , which appears in parentheses just before the 3-form, is the inner product of these vectors at the corresponding points of the domain, from which it follows that p (x) = A ( x ) · B ( x ) . D 14. 1 .3 The Differential Operators grad, curl, div, and V Definition 1 . To the exterior differentiation of 0-forms (functions) , 1-forms,
and 2-forms in oriented Euclidean space JR.3 there correspond respectively the operations of finding the gradient (grad) of a scalar field and the curl and divergence ( div) of a vector field. These operations are defined by the relations
_. dw fO -. -· dwAi -. I _ .. d wB -
I
Wgrad f ' 2 wcuri A' wd3 i v B ·
( 14.9) (14.10) ( 14. 1 1 )
B y virtue of the correspondence between forms and scalar and vector fields in JR. 3 established by Eqs. ( 14. 1)-( 14.4) , relations ( 14.9)-( 14. 1 1 ) are unambiguous definitions of the operations grad, curl, and div, performed on scalar and vector fields respectively. These operations, the operators of field theory as they are called, correspond to the single operation of exterior differentiation of forms, but applied to forms of different degree. Let us give right away the explicit form of these operators in Cartesian coordinates x i , x 2 , x 3 in JR. 3 . As we have explained, in this case
w� = f , wl = A I dx i + A 2 dx 2 + A 3 dx 3 , w� = B I dx 2 A dx 3 + B 2 dx 3 A dx i + B 3 dx i A dx 2 , w� = p dx i A dx 2 A dx3 .
( 14.3') ( 14. 7') ( 14.8') ( 14.4')
Since
I f := dwf0 = df = wgrad
8] dx I + 8] dx 2 + 8] dx3 ' 8x 2 8x 3 8x i
it follows from ( 14.7') that in these coordinates ( 14.9') where
e i , e2 , e3 is a fixed orthonormal basis of JR.3 .
14.1 The Differential Operations of Vector Analysis
261
Since
as
As an aid to memory this last relation is often written in symbolic form
( 14. 10")
curi A = Next , since
w� iv B : = dw� =
d(B 1 dx 2 A. dx3 B 2 dx 3 A. dx 1 B 3 dx 1 A. dx 2 ) = I aB 2 aB 3 l 2 3 = oB ox l ox 2 ox 3 dx A. dx A. dx ,
+
(
+
+ +
)
it follows from ( 14.4') that in Cartesian coordinates div
l
f)B 2 aB 3 B = f)B l ox + ox 2 + ox 3 .
( 14. 1 1')
One can see from the formulas ( 14.9' ) , ( 14 . 10') , and ( 14. 1 1') just obtained that grad, curl, and div are linear differential operations (operators) . The grad operator is defined on differentiable scalar fields and assigns vector fields to the scalar fields. The curl operator is also vector-valued, but is defined on differentiable vector fields. The div operator is defined on differentiable vector fields and assigns scalar fields to them. We note that in other coordinates these operators will have expressions that are in general different from those obtained above in Cartesian coordi nates. We shall discuss this point in Subsect . 14. 1 . 5 below. We remark also that the vector field curl A is sometimes called the rotation of A and written rot A.
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14 Elements of Vector Analysis and Field Theory
As an example of the use of these operators we write out the famous 1 system of equations of Maxwell, 2 which describe the state of the components of an electromagnetic field as functions of a point x = (x 1 , x 2 , x 3 ) in space and time t.
Example 1 .
uum.)
(The Maxwell equations for an electromagnetic field
1 . div E =
.!!._ .
Eo
2. div B
=
m
a vac
0.
(14. 12) 1 8E j . + -8t EoC2 2 C Here p(x, t) is the electric charge density (the quantity of charge per unit volume) , j (x, t) is the electrical current density vector (the rate at which charge is flowing across a unit area) , E(x, t) and B(x, t) are the electric and magnetic field intensities respectively, and Eo and c are dimensioning constants (and in fact c is the speed of light in a vacuum) . 3. curl E =
-
at
as
.
4. curl B
=
I n mathematical and especially i n physical literature, along with the op erators grad, curl, and div, wide use is made of the symbolic differential operator nabla proposed by Hamilton (the Hamilton operator) 3 ( 14. 13) where { e 1 , e 2 , e 3 } is an orthonormal basis of lR 3 and x 1 , x 2 , x 3 are the corre sponding Cartesian coordinates. By definition, applying the operator \7 to a scalar field f (that is, to a function) , gives the vector field
1 On this subject the famous American physicist and mathematician R. Feynman (1918-1988) writes, with his characteristic acerbity, "From a long view of the history of mankind - seen from, say, ten thousand years from now - there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scien tific event of the same decade." Richard R. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics: Mainly Electromagnetism and Matter, Reading, MA: Addison-Wesley, 1964. 2 J.C. Maxwell (1831-1879) - outstanding Scottish physicist; he created the math ematical theory of the electromagnetic field and is also famous for his researach in the kinetic theory of gases, optics and mechanics. 3 W.R. Hamilton (1805-1865) - famous Irish mathematician and specialist in mechanics; he stated the variational principle (Hamilton's principle) and con structed a phenomenological theory of optic phenomena; he was the creator of the theory of quaternions and the founder of vector analysis (in fact, the term "vector" is due to him) .
14. 1 The Differential Operations of Vector Analysis
263
of + e of + e of , 'Vf = e l ox l 2 ox 2 3 ox 3 which coincides with the field ( 14.9') , that is, the nabla operator is simply the grad operator written in a different notation. Using, however, the vector form in which V' is written, Hamilton pro posed a system of formal operations with it that imitates the corresponding algebraic operations with vectors. Before we illustrate these operations, we note that in dealing with V' one must adhere to the same principles and cautionary rules as in dealing with the usual differentiation operator D = d� . For example, r.pD f equals r.p¥x and not d� ( r.p f) or f � . Thus, the operator operates on whatever is placed to the right of it; left multiplication in this case plays the role of a coefficient , that is, r.pD is the new differential operator r.pd� ' not the function � · Moreover, D 2 = D · D, that is, D 2 f = D(D f ) = d� ( d� f) = � f. If we now, following Hamilton, deal with V' as if it were a vector field defined in Cartesian coordinates, then, comparing relations (14.13) , ( 14.9') , ( 14.10" ) , and ( 14. 1 1') , we obtain grad / = 'Vf curi A = V' x A , div B = V' · B .
( 14. 14) ( 14. 15) ( 14. 16)
In this way the operators grad, curl, and div, can be written in terms of the Hamilton operator and the vector operations in JR 3 .
Example 2. Only the curl and div operators occurred in writing out the Maxwell equations ( 14. 12) . Using the principles for dealing with V' = grad, we rewrite the Maxwell equations as follows, to compensate for the absence of grad in them: p
l . Y' · E = - .
eo
2. V' · B = 0 .
oB j 3. V' x E = - at . 4. V' X B = eoc2
oE + 21 """"!:>
c ut
(14.12') •
14. 1 .4 Some Differential Formulas of Vector Analysis
In the oriented Euclidean space JR 3 we have established the connection ( 14. 1 ) (14.4) between forms on the one hand and vector and scalar fields on the other. This connection enabled us to associate corresponding operators on fields with exterior differentiation ( see formulas ( 14.5) , ( 14.6) , and (14 .9) (14. 1 1 ) ) . This correspondence can b e used t o obtain a number of basic differential formulas of vector analysis.
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14 Elements of Vector Analysis and Field Theory
For example, the following relations hold: curl ( !A) = fcurlA - A x gradf , (14.17) div ( !A) = A · grad f + fdiv A , ( 14. 18) div (A x B) = B · curl A - A · curlB . ( 14. 19) Proof. We shall verify this last equality: w� i v A x B = dwi x a = d(w}. /\ w�) = dw}. l\ w� - w}. l\ dw� = 2 A 1\ Wa1 - WA1 1\ Wcuri 2 B = WB·curi A - WA·curi B = WB·curi A - A·CUrl B . = Wcuri first equalities two relations arealsoverified similarly. Ofdirect course,difftheerentiation verificationin ofcoordinates. allThethese can be carried out by 0 2 w = 0 for any form w, we can also If we take account of the relation d assert that the following equalities hold: curlgradf = 0 , ( 14.20) divcurlA = 0 . (14.21) Proof. Indeed: W�url grad f = dw!rad f = d(dw � ) = d 2 w � = 0 , W� i v curi A = dw�uri A = d(dw}.) = d 2 w}. = 0 . 0 In while formulas(14.20) (14. 1 7)-( 14. 19) the operators grad, curl, and div are applied once, and of(14.21) involve the second-order operatorsBesides obtained byrulessuccessive execution two of the three original operations. the in ( 14.20) and (14.21 ) , one can also consider other combinations of thesegivenoperators: grad divA, curl curl A, div grad f . ( 14.22) operator div(Delta) grad is applied, as onethecanLaplace see, tooperator a scalar4field. This op eratorThe is denoted and is called or Laplacian. It follows fromLl( 14.9') and (14. 1 1') that in Cartesian coordinates 3
3
3
( 14.23) 4 P.S. Laplace (1749-1827) - famous French astronomer, mathematician, and physicist; he made fundamental contributions to the development of celestial mechanics, the mathematical theory of probability, and experimental and math ematical physics.
265
14.1 The Differential Operations of Vector Analysis
Since the operator Ll acts on numerical functions, it can be applied com ponentwise to the coordinates of vector fields A = e 1 A 1 + e2 A 2 + e3 A 3 , where e l l e2 , and e3 are an orthonormal basis in JR 3 . In that case Taking account of this last equality, we can write the following relation for the triple of second-order operators ( 14.22) : (14.24)
curl curl A = grad div A - LlA ,
(
)
whose proof we shall not take the time to present see Problem 2 below . The equality ( 14.24) can serve as the definition of LlA in any coordinate system, not necessarily orthogonal. Using the language of vector algebra and formulas (14. 14)-( 14. 16) , we can write all the second-order operators ( 14.20)-( 14.22) in terms of the Hamilton operator V': curl grad f = V' x V' f = 0 , div curi A = V' · (V' x A = 0 , grad div A = Y' (Y' · A , curl curl A = V' x (Y' x A , div grad = Y' · 'Vf .
)
/
)
)
From the point of view of vector algebra the vanishing of the first two of these operators seems completely natural. The last equality means that the following relation holds between the Hamilton operator V' and the Laplacian Ll: Ll =
'\7 2 .
*Vector Operations in Curvilinear Coordinates a. Just as, for example, the sphere x 2 + y 2 + z 2 = a 2 has a particularly simple equation R = a in spherical coordinates, vector fields x >-+ A( x ) in JR 3 or 14. 1 . 5
(
Rn ) often assume a simpler expression in a coordinate system that is not Cartesian. For that reason we now wish to find explicit formulas from which one can find grad, curl, and div in a rather extensive class of curvilinear coordinates. But first it is necessary to be precise as to what is meant by the coordinate expression for a field A in a curvilinear coordinate system. We begin with two introductory examples of a descriptive character.
Suppose we have a fixed Cartesian coordinate system x 1 , x 2 in the Euclidean plane JR 2 . When we say that a vector field A 1 , A 2 ( x is defined
Example 3.
(
) )
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14 Elements of Vector Analysis and Field Theory
JR.2 , we mean that some vector A(x) E TJR.� is connected with each point x = (x 1 , x 2 ) E JR.2 , and in the basis of TJR.� consisting of the unit vectors e 1 ( x), e2 ( x) in the coordinate directions we have the expansion A ( x) = A 1 (x )e 1 (x) + A2 (x )e2 (x) ( see Fig. 14. 1 ) . In this case the basis { e 1 (x), e2 (x)} of TJR.� is essentially independent of x. in
A(x) X
e,
e 1 (x)
Fig.
A(x)
L
0
el
0 Fig.
14.2.
14. 1 .
Example 4. I n the case when polar coordinates (r, cp) are defined i n the same plane JR. 2 , at each point x E JR. 2 \ 0 one can also attach unit vectors e 1 (x) = er (x), e2 = e cp (x) ( Fig. 14.2) in the coordinate directions. They also form a basis in TJR.� with respect to which one can expand the vector A(x) of the field A attached to x: A(x) = A 1 (x)e 1 (x) + A 2 (x)e2 (x). It is then natural to regard the ordered pair of functions (A 1 , A 2 ) ( x) as the expression for the field A in polar coordinates. Thus, if (A 1 , A 2 ) (x) ( 1 , 0) , this is a field of unit vectors in JR. 2 pointing radially away from the center 0. The field (A 1 , A 2 ) ( x) ( 0, 1 ) can be obtained from the preceding field by rotating each vector in it counterclockwise by the angle /2. These are not constant fields in JR. 2 , although the components of their =
=
7f
coordinate representation are constant. The point is that the basis in which the expansion is taken varies synchronously with the vector of the field in a transition from one point to another. It is clear that the components of the coordinate representation of these fields in Cartesian coordinates would not be constant at all. On the other hand, a truly constant field ( consisting of a vector translated parallel to itself to all points of the plane ) which does have constant components in a Cartesian coordinate system, would have variable components in polar coordinates. b. After these introductory considerations, let us consider more formally the
problem of defining vector fields in curvilinear coordinate systems.
14. 1 The Differential Operations of Vector Analysis
267
We recall first of all that a curvilinear coordinate system t 1 , t 2 , t 3 in a domain D c JR 3 is a diffeomorphism
The quadratic form
( 14.25) whose coefficients are the pairwise inner products of the vectors in the canon ical basis determines the inner product on TJRr completely. If such a form is defined at each point of a domain Dt C lRt , then, as is known from geometry, one says that a Riemannian metric is defined in this domain. A Riemannian metric makes it possible to introduce a Euclidean structure in each tangent space TJRr ( t E Dt) within the context of rectilinear coordinates t 1 , t 2 , t 3 in
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14 Elements of Vector Analysis and Field Theory
JRr, corresponding to the "curved" embedding
;
what follows, for the sake of simplicity, we shall consider only triorthogonal curvilinear coordinate systems. For them, as has been noted, the quadratic form (14.25) has the following special form: ds 2 = E1 ( t ) ( dt 1 ) 2 + E ( t ) ( dt 2 ) 2 + E (t)(dt 3 ) 2 , ( 14.26)
2
Ei ( t ) = 9ii (t), i = 1 , 2, 3. Example 5. In Cartesian coordinates
3
where
( y , z ) , cylindrical coordinates (r,
x,
x
Thus, each of these coordinate systems is a triorthogonal system in its domain of definition. The vectors e 1 (t), e 2 (t), e3 (t) of the canonical basis ( 1 , 0, 0) , (0, 1 , 0) , (0, 0, 1 ) in TJR�, like the vectors e i ( ) E TJR� corre sponding to them, have the following norm5 : l e i I = ..;gii . Hence the unit vectors (in the sense of the square-norm of a vector) in the coordinate directions have the following coordinate representation for the triorthogonal system (14.26) :
x
1 ( t ) = (vk , o , o) , e2 ( t ) = (o , vk , o) , e3 ( t ) = (o , o , �) . ( 14.27) Example 6. It follows from formulas (14.27) and the results of Example 5 e
that for Cartesian, cylindrical, and spherical coordinates, the triples of unit vectors along the coordinate directions have respectively the following forms: er =
( 1 , 0, 0) , ( 1 , 0, 0) ,
eR =
(1, 0, 0) ,
ex =
(0, 1 , 0) , = o, , o ,
ey = e '"
e '" =
( � ) (o , R c�s () ' o) '
ez = ez = eo =
(0, 0, 1 ) ; (0, 0, 1 ) ;
(o , o , �) ;
( 14.27') ( 14.27") ( 14.27"')
5 In the triorthogonal system ( 14.26) we have le ; l = #; = H; , i = 1 , 2, 3. The quantities H1 , H2 , H3 are usually called the Lame coefficients or Lame parame ters. G. Lame ( 1795-1870) - French engineer, mathematician, and physicist.
14. 1 The Differential Operations of Vector Analysis
269
Examples 3 and 4 considered above assumed that the vector of the field was expanded in a basis consisting of unit vectors along the coordinate direc tions. Hence the vector A(t) E TlRt corresponding to the vector A(x) E TJR� of the field should be expanded in the basis e 1 ( t), e 2 ( t), e3 ( t) consisting of unit vectors in the coordinate directions, rather than in the canonical basis
e l (t), e z (t), e3 (t).
Thus, abstracting from the original space JR 3 , one can assume that a Riemannian metric ( 14.25) or ( 14.26) and a vector field t H A(t) are defined in the domain D t C lRt and that the coordinate representation (A 1 , A 2 , A 3 ) ( t) of A ( t) at each point t E Dt is obtained from the expansion of the vector A(t) = Ai (t)ei (t) of the field corresponding to this point with respect to unit vectors along the coordinate axes. d. Let us now investigate forms. Under the diffeomorphism ip : D t ---+ D every form in D automatically transfers to the domain Dt . This transfer, as we know, occurs at each point x E D from the space TJR� into the corresponding space TJRl. Since we have transferred the Euclidean structure into TJRl from TJR�, it follows from the definition of the transfer of vectors and forms that , for example, to a given form wl(x) = (A(x), ·) defined in TJR� there corresponds exactly the same kind of form w l(t) = (A(t), ·) in TJRl, where A(x) = ip1 (t)A(t). The same can be said of forms of the type w� and w� , to say nothing of forms w � that is, functions. After these clarifications, the rest of our study can be confined to the domain D t C Rt, abstracting from the original space JR 3 and assuming that a Riemannian metric ( 14.25) is defined in D t and that scalar fields J, p and vector fields A, B are defined in D t along with the forms w � , w l , w� , w� , which are defined at each point t E D t in accordance with the Euclidean structure on TJRt defined by the Riemannian metric. -
Example
7.
The volume element dV in curvilinear coordinates we know, has the form
t 1 , t 2 , t3 , as
For a triorthogonal system ( 14.28) In particular, in Cartesian, cylindrical, and spherical coordinates, respec tively, we obtain dV = dx 1\ dy 1\ dz = = r dr 1\ dip 1\ dz = = R 2 cos 8 dR 1\ dip 1\ d8 . What has just been said enables us to write the form w � different curvilinear coordinate systems.
( 14.28') ( 1 4.28") ( 14.28"' ) =
p
d V in
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14 Elements of Vector Analysis and Field Theory
e. Our main problem ( now easily solvable ) is, knowing the expansion A (t) = A i (t)ei(t) for a vector A(t) E TJK[ with respect to the unit vectors ei(t) E TlK[, i = 1 , 2 , 3, of the triorthogonal coordinate system determined by the Riemannian metric ( 14.26 ) , to find the expansion of the forms wl_(t) and w�(t) in terms of the canonical 1-forms dt i and the canonical 2-forms dti 1\ dt1 respectively. Since all the reasoning applies at every given point t, we shall abbreviate the notation by suppressing the letter t that shows that the vectors and forms are attached to the tangent space at t. Thus, e 1 , e 2 , e 3 is a basis in TJK[ consisting of the unit vectors ( 14.27 ) along the coordinate directions, and A = A 1 e 1 +A 2 e2 +A 3 e3 is the expansion of A E TJK[ in that basis. We remark first of all that formula ( 14.27 ) implies that where
6} =
where 6"� =
f. Thus, if
wl : = (A, · ) = a 1 dt 1
{ {
0 ' if i oF j ' 1 ' if i = j ' 0, if (i, j ) oF
( 14.29 )
(k, l) ' ( 14.30 ) if (i, j ) = (k, l) .
1' + a2 dt2 + a3 dt3 , then on the one hand
and on the other hand, as one can see from ( 14.29 ) ,
Consequently,
ai = A\(E;, and we have found the expansion
( 14.31 ) for the form wl_ corresponding to the expansion A = A 1 e 1 + A 2 e2 + A 3 e3 of
the vector A.
Example 8 .
respectively
Since in Cartesian, spherical, and cylindrical coordinates we have A = = =
Ax ex + A y ey + Azez = Ar er + A'Pe'P + Azez AReR + A'Pe'P + A e e e ,
as follows from the results of Example 6,
14. 1 The Differential Operations of Vector Analysis
271
( 14.31') wi_ = Ax dx + Ay dy + A z dz = ( 14.31") = Ar dr + Acpr dcp + A z dz = ( 14.3 1"' ) = AR dR + AcpR cos cp dcp + Ao R dO . Now let B = B 1 e 1 + B 2 e 2 + B 3 e 3 and w� = b 1 dt 2 1\ dt 3 + b 2 dt 3 1\ dt 1 + b3 dt 1 1\ dt2 . Then, on the one hand, w�(e2 , e3 ) : = dV ( B, e2 , e3 ) = 3 B' dV(ei , e2 , e3 ) = B 1 (e 1 , e2 , e3 ) = B 1 , =� L.., i= 1
g.
.
·
where dV is the volume element in TIR� ( see ( 14.28) and ( 14.27) ) . O n the other hand, by ( 14.30) we obtain
w� (e2 , e3 ) = (b 1 dt2 1\ dt 3 + b2 dt3 1\ dt 1 + b3 dt 1 1\ dt 2 )( e 2 , e3 ) = b1 . = b 1 dt 2 1\ dt 3 (e 2 , e3 ) = E2 E3 � Comparing these results, we conclude that b 1 = B 1 v'E2 E3 . Similarly, we verify that b2 = B 2 v'E1 E3 and b3 = B 3 v'E1 E2 . Thus we have found the representation
Example 9. Using the notation introduced in Example 8 and formulas (14.26') , ( 14.26") and ( 14.26"') , we obtain in Cartesian, cylindrical, and spherical coordinates respectively
� = � � /\ � + � � /\ � + � � /\ � = = Er r dcp 1\ dz + Bcp dz 1\ dr + Bz r dr 1\ dcp = = BRR2 cos O dcp 1\ dO + BcpR dO 1\ dR + BoR cos O dR 1\ dcp .
( 14.32') ( 14.32") ( 14.32"' )
h. We add further that on the basis of ( 14.28) we can write
( 14.33) Example 1 0. In particular, for Cartesian, cylindrical, and spherical coordi nates respectively, formula ( 14.33) has the following forms:
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14 Elements of Vector Analysis and Field Theory
w�
= p dx dy dz = = pr dr A dcp dz = = pR2 cos () dR A dcp A d() . !\
!\
(14.33') (14.3311 ) (14.33"')
!\
Now that we have obtained formulas (14.31) - (14.33), it is easy to find the coordinate representation of the operators grad, curl, and div in a triorthog onal curvilinear coordinate system using Definitions (14.9)-(14.11). Let grad f A 1 e 1 + A 2 e 2 + A 3 e 3 . Using the definitions, we write M 1 + M d 2 + 8f d 3 . 1 f dw a wgrad 82 83
=
: = t := df := Bt l dt t t t t (14.31), we conclude that 1 8f e + I"E' 1 1 8f e2 + � 8f e
From this, using formula grad f
�xample 1 1 .
= YI"E'�l 8t 1
1
Y �2 8t2
Y �3 8t3 3 .
(14.34)
In Cartesian, polar, and spherical coordinates respectively,
(14.34') f = ax8f ex + 8f8y ey + 8f8z ez = (14.3411 ) = 8!8r er + ;:1 88fcp e'P + 8f8z ez = 8f 1 8f 1 8f = 8R (14.34111 ) e R + R () cp e 'P + 2 e e . R 8() cos 8 Suppose given a field A (t) = (A 1 e 1 + A 2 e 2 + A 3 e3 ) (t) . Let us find the co ordinates B 1 , B 2 , B3 of the field curl A(t) = B(t) = (B 1 e 1 + B 2 e 2 + B 3 e 3 ) (t) . Based on the definition (14.10) and formula (14.31), we obtain W�url A : = dwl = d (A \/E\"dt 1 + A 2 .;E;"dt 2 + A 3 yiE;"dt 3 ) = ( 8A3 yE3 8A 2 � ) 2 d t dt 3 + 2 8t 3 8t ( 8A 1 v'FJ; - 8A3 yE3 ) dt3 A dt 1 + ( 8AB2 � 8A 1 v'FJ; ) 1 + d t A dt 2 . l 8t 1 8t 3 t 8t 2 grad
!\
On the basis of (14.32) we now conclude that
( 1 ( �� 1 (
8A 3 yE3 _ 8A 2 � at 3 Bt 2 J�2 �3 8A 1 y'FJ; 8A 3 yE3 8t 3 8t 1 J 3 1 8A 2 � _ 8A 1 v'FJ; 8t 1 8t 2 J�1 �2
1
_
) ) )
'
'
'
273
14. 1 The Differential Operations of Vector Analysis
that is,
�e 1 VE2e2 ,fEJe3 a a a curi A = JE1 E2 E3 at 1 l at 2 2 at3 �A VEJ2A ,f£3A 3 1
Example 1 2.
( 14.35)
In Cartesian, cylindrical, and spherical coordinates respectively
( aAz _ aAy ) ex + ( aAx aAz ) ey + ( aAy _ aAx ) ez = ( 14_35, ) ay az az ax ax ay � ( aAz aA ( aA arA"' ) aAz ) � ( arA'P _ _ r ) ez = (14 35,) r = _ r acp az er + az ar e'P + r ar acp 1_ ( aAe _ aA"' cos e ) e + _!_ ( aA n aRAe ) e"' + n R ae - R cos () acp ae aR 1 a ( A aRA"' _ n ) ee + _!_ ( 14_35,, )
curi A =
_
_
_
_
_
_
_
.
R aR cos () acp i. Now suppose given a field B ( t ) = (B 1 e 1 + B 2 e 2 + B 3 e 3 ) (t). Let us find an
expression for div B. Starting from the definition ( 14. 1 1 ) and formula ( 14.32) , we obtain
1 Wd i v B : = dw� = d (B VE2 E3 dt 2 A dt 3 + + B 2 )E3 E1 dt 3 A dt 1 + B 3 VEJf;dt 1 A dt 2 =
B 3 ) dt 1 _- ( aJEat2 El 3 B 1 + aJEat3 E2 1 B2 + a� at 3
A dt 2 A dt 3
.
On the basis of formula ( 14.33) we now conclude that
In Cartesian, cylindrical, and spherical coordinates respectively, we obtain
aBx + aBy + aBz = ( 14.36') az ay ax � ar Br + aB"' + aBz = ( 14_36,) = r ar acp az aR2 cos ()B n + aRB'P + aR cos ()Be . ( 14_36,,) 1 = 2 aR acp ae R cos e
div B =
(
(
)
)
j. Relations ( 14.34) and ( 14.36) can be used to obtain an expression for the Laplacian L1 = div grad in an arbitrary triorthogonal coordinate system:
274
14 Elements of Vector Analysis and Field Theory
Example 1 3. In particular, for Cartesian, cylindirical, and spherical coordi nates, we obtain respectively
14. 1 . 6 Problems and Exercises 1. The operators grad, curl, and div and the algebraic operations. Verify the following relations: for grad: a) 'il ( f + g ) = 'il f + 'il g, b) 'il ( f . g ) = f 'il g + g 'il J , c) 'il(A · B) = (B · 'il)A + (A · 'il)B + B x ('17 x A) + A x ('17 x B) . d) v O A2 ) = (A . v ) A + A x ('il x A) ;
for curl: e) 'il x { f A) = f 'il X A + 'il f X A, f) 'il X (A X B) = (B . 'il)A - (A . 'il)B + ('17 . B)A - ('17 . A)B; for div: g) 'il · ( f A) = 'il f · A + f 'il · A , h) 'il · (A x B) = B · ('il x A) - A · ('17 x B) and rewrite them in the symbols grad, curl, and div. (Hints. A · '17 = A 1 � + A 2 � + A 3 � ; B · 'il # 'il · B ; A x (B x C ) = B(A · C ) - C (A · B) . )
a) Write the operators (14.20)�( 14.22) in Cartesian coordinates. b) Verify relations ( 14.20) and ( 14.21) by direct computation. c) Verify formula ( 14.24) in Cartesian coordinates. d) Write formula ( 14.24) in terms of 'il and prove it, using the formulas of vector algebra. 2.
14. 1 The Differential Operations of Vector Analysis 3.
275
From the system of Maxwell equations in Example 2 deduce that 'V · j = !!If -
.
4. a) Exhibit the Lame parameters H1 , H2 , H3 of Cartesian, cylindrical, and spher ical coordinates in JR 3 . b) Rewrite formulas ( 14.28) , ( 14.34)-( 14.37) , using the Lame parameters. 5.
Write the field A = grad � ' where r = Jx 2 + y 2 + z 2 in a) Cartesian coordinates x, y, z; b) cylindrical coordinates; c) spherical coordinates. d) Find curl A and div A.
6. In cylindrical coordinates (r, cp, z) the function f has the form ln � . Write the field A = grad f in a) Cartesian coordinates; b) cylindrical coordinates; c) spherical coordinates. d) Find curl A and div A. 7. Write the formula for transformation of coordinates in a fixed tangent space TJR�, p E JR 3 , when passing from Cartesian coordinates in JR 3 to a) cylindrical coordinates; b) spherical coordinates; c) an arbitrary triorthogonal curvilinear coordinate system. d) Applying the formulas obtained in c) and formulas ( 14.34)-(14.37) , verify directly that the vector fields grad /, curl A, and the quantities div A and 11/ are invariant relative to the choice of the coordinate system in which they are computed.
The space JR 3 , being a rigid body, revolves about a certain axis with constant angular velocity w. Let v be the field of linear velocities of the points at a fixed instant of time. a) Write the field v in the corresponding cylindrical coordinates. b) Find curl v . c) Indicate how the field curl v is directed relative to the axis of rotation. d) Verify that I curl v i = 2w at each point of space. e) Interpret the geometric meaning of curl v and the geometric meaning of the constancy of this vector at all points of space for the situation in d). 8.
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14 Elements of Vector Analysis and Field Theory
1 4 . 2 The Integral Formulas of Field Theory 14. 2 . 1 The Classical Integral Formulas in Vector Notation
w }. and w � In the preceding chapter we noted ( see Sect. 13.2, formulas (13.23) and ( 13.24) ) that the restriction of the work form w� of a field F to an oriented smooth curve ( path ) 'Y or the restriction of the flux form w� of a field V to an oriented surface S can be written respectively in the following forms:
a. Vector Notation for the Forms
w
� l , = (F, e ) ds ,
w
� l s = ( V, n ) da ,
where e is the unit vector that orients "(, codirectional with the velocity vector of the motion along "(, ds is the element ( form ) of arc length on "(, n is the unit normal vector to S that orients the surface, and da is the element ( form ) of area on S. In vector analysis we often use the vector element of length of a curve d s : = e ds and the vector element of area on a surface do- := n da. Using this notation, we can now write:
i l, w� l s w
= (A, e ) ds = (A, d s ) = A · d s ,
= (B, n ) da = (B, do- ) = B · do- .
b. The Newton-Leibniz Formula Let
D be a path in the domain D.
f E C C l l (D, IR), and let
Applied to the 0-form w � , Stokes ' formula
( 14.38) ( 14.39) 'Y
:
[a, b]
--+
means, on the one hand, the equality
which agrees with the classical formula b
J('Y(b)) - J('Y(a)) = I df ('Y(t)) a
of Newton-Leibniz ( the fundamental theorem of calculus ) . On the other hand, by definition of the gradient, it means that
I W� = I W�rad f ·
a,
'
( 14.40)
14.2 The Integral Formulas of Field Theory as
277
Thus, using relation ( 14.38 ) , we can rewrite the Newton-Leibniz formula ( 14.40 ' )
J('r(b)) - J('r(a)) = / ( grad ! ) · ds .
,
In this form it means that
the increment of a function on a path equals the work done by the gradient of the function on the path. This is a very convenient and informative notation. In addition to the obvious deduction that the work of the field grad f along a path depends only on the endpoints of the path, the formula enables us to make a somewhat more subtle observation. To be specific, motion over a level surface f = c of f takes place without any work being done by the field grad f since in this case grad f · do- = 0. Then, as the left-hand side of the formula shows, the work of the field grad f depends not even on the initial and final points of the path but only on the level surfaces of f to which they belong. 1
c. Stokes' Formula We recall that the work of a field on a closed path is called the circulation of the field on that path. To indicate that the integral is taken over a closed path, we often write f F · ds rather than the traditional
notation
£ F · ds. If
1
,
is a curve in the plane, we often use the symbols
f ,
p, in which the direction of traversal of the curve is indicated. , The term circulation is also used when speaking of the integral over some 1
and
finite set of closed curves. For example, it might be the integral over the boundary of a compact surface with boundary. Let A be a smooth vector field in a domain D of the oriented Euclidean space JR 3 and S a ( piecewise ) smooth oriented compact surface with boundary in D. Applied to the 1-form wi_, taking account of the definition of the curl of a vector field, Stokes ' formula means the equality ( 14.41 )
Using relation ( 14.39 ) , we can rewrite ( 14.41 ) as the classical Stokes for mula
f A · ds = JJ ( curl A) · do- .
as
s
( 14.41' )
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14 Elements of Vector Analysis and Field Theory
In this notation it means that
the circulation of a vector field on the boundary of a surface equals the flux of the curl of the field across the surface.
As always, the orientation chosen on aS is the one induced by the orien tation of S. d. The Gauss-Ostrogradskii Formula Let V be a compact domain of
the oriented Euclidean space IR 3 bounded by a (piecewise-) smooth surface {)V, the boundary of V. If B is a smooth field in V, then in accordance with the definition of the divergence of a field, Stokes ' formula yields the equality ( 14.42) Using relation ( 14.39) and the notation p dV for the form w� in terms of the volume element dV in IR 3 , we can rewrite Eq. ( 14.42) as the classical Gauss-Ostrogradskii formula
!! av
B · dt7
In this form it means that
= !!! v
div B dV .
( 14.42')
the flux of a vector field across the boundary of a domain equals the integral of the divergence of the field over the domain itself.
e. Summary of the Classical Integral Formulas In sum, we have arrived
at the following vector notation for the three classical integral formulas of analysis:
J f = J ( V'!) J = J ( V' = J (V' 'Y
8-y
J
av
B · dt7
·
as
A · ds ·
v
s
d s (the Newton-Leibniz formula) , x A) · dt7
(Stokes ' formula) ,
B) dV (the Gauss-Ostrogradskii formula) .
( 14.40") ( 14.41") ( 14.42")
1 4 . 2 . 2 The Physical Interpretation of div, curl, and grad a. The Divergence Formula ( 14.42' ) can be used to explain the physical
x
x
meaning of div B ( ) - the divergence of the vector field B at a point in the domain V in which the field is defined. Let V( ) be a neighborhood of (for example, a ball) contained in V. We permit ourselves to denote the
x
x
14.2 The Integral Formulas of Field Theory
279
volume of this neighborhood by the same symbol V(x) and its diameter by the letter d. By the mean-value theorem and the formula ( 14.42') we obtain the fol lowing relation for the triple integral
JJ
B · du = div B (x ' ) V(x) ,
8V(x)
where x' is a point in the neighborhood V(x) . If d -+ 0, then x' -+ x, and since B is a smooth field, we also have div B (x') -+ div B (x) . Hence
JJ B · du av(x) . . d IV B( X ) = 1 Im d---t O -:..,V'=-,(,.---,X) -
( 14.43)
Let us regard B as the velocity field for a flow (of liquid or gas) . Then, by the law of conservation of mass, a flux of this field across the boundary of the domain V or, what is the same, a volume of the medium diverging across the boundary of the domain, can arise only when there are sinks or sources (including those associated with a change in the density of the medium) . The flux is equal to the total power of all these factors, which we shall collectively call "sources," in the domain V(x) . Hence the fraction on the right-hand side of ( 14.43) is the mean intensity (per unit volume) of sources in the domain V(x) , and the limit of that quantity, that is, div B (x) , is the specific intensity (per unit volume) of the source at the point x. But the limit of the ratio of the total amount of some quantity in the domain V(x) to the volume of that domain as d -+ 0 is customarily called the density of that quantity at x, and the density as a function of a point is usually called the density of the distribution of the given quantity in a portion of space. Thus, we can interpret the divergence div B of a vector field B as the density of the distribution of sources in the domain of the flow, that is, in the domain of definition of the field B .
Example 1 . If, in particular, div B = 0, that is, there are no sources, then the flux across the boundary of the region must be zero: the amount flowing in equals the amount flowing out . And, as formula ( 14.42' ) shows, this is indeed the case. Example 2. A point electric charge of magnitude q creates an electric field in space. Suppose the charge is located at the origin. By Coulomb 's law 6 the intensity E = E(x) of the field at the point x E JR. 3 (that is, the force acting 6 Ch.O. Coulomb (1736-1806) - French physicist. He discovered experimentally the law ( Coulomb's law ) of interaction of charges and magnetic fields using a torsion balance that he invented himself.
280
14 Elements of Vector Analysis and Field Theory
on a unit test charge at the point
x) can be written as
E-
q
--
4rrco
�
lrl3 ' and r
where c o is a dimensioning constant is the radius-vector of the point x. The field E is defined at all points different from the origin. In spherical coordinates E = 4f!eo Jb-eR, so that by formula ( 14.36111) of the preceding section, one can see immediately that div E = 0 everywhere in the domain of definition of the field E. Hence, if we take any domain V not containing the origin, then by formula ( 14.42') the flux of E across the boundary 8V of V is zero. Let us now take the sphere SR = {x E JR3 I I x l = R} of radius R with center at the origin and find the outward flux (relative to the ball bounded by the sphere) of E across this surface. Since the vector eR is itself the unit outward normal to the sphere, we find
I
Sn
E · du =
Jq
Sn
q
1
-- da = · 4rr R2 4rrco R2 4rrco R2
--
=
q
-
co
.
Thus, up to the dimensioning constant c o , which depends on the choice of the system of physical units, we have found the amount of charge in the volume bounded by the sphere. We remark that under the hypotheses of Example 2 just studied the left hand side of formula ( 14.42' ) is well-defined on the sphere 8V = SR, but the integrand on the right-hand side is defined and equal to zero everywhere in the ball V except at one point - the origin. Nevertheless, the computations show that the integral on the right-hand side of ( 14.42') cannot be interpreted as the integral of a function that is identically zero. From the formal point of view one could dismiss the need to study this situation by saying that the field E is not defined at the point 0 E V, and hence we do not have the right to speak about the equality ( 1 4.42' ) , which was proved for smooth fields defined in the entire domain V of integration. However, the physical interpretation of ( 1 4.42' ) as the law of conservation of mass shows that, when suitably interpreted, it ought to be valid always. Let us study the indeterminacy of div E at the origin in Example 2 more attentively to see what is causing it . Formally the original field E is not defined at the origin, but , if we seek div E from formula ( 14.43) , then, as Example 2 shows, we would have to assume that div E(O) = +oo. Hence the integrand on the right-hand side of ( 14.42) would be a "function" equal to zero everywhere except at one point, where it is equal to infinity. This corresponds to the fact that there are no charges at all outside the origin, and we somehow managed to put the entire charge into a space of volume zero - into the
q
14.2 The Integral Formulas of Field Theory
281
single point 0, at which the charge density naturally became infinite. Here we are encountering the so-called Dirac 7 8-function (delta-function) . The densities of physical quantities are needed ultimately so that one can find the values of the quantities themselves by integrating the density. For that reason there is no need to define the 8-function at each individual point; it is more important to define its integral. If we assume that physically the "function" 8x 0 (x) = 8(xo ; x) must correspond to the density of a distribution, for example the distribution of mass in space, for which the entire mass, equal to 1 in magnitude, is concentrated at the single point x 0 , it is natural to set
j
v
8(xo , x) dV =
{1'
when Xo E V ,
0 , when Xo tf. V .
Thus, from the point of view of a mathematical idealization of our ideas of the possible distribution of a physical quantity (mass, charge, and the like) in space, we must assume that its distribution density is the sum of an ordinary finite function corresponding to a continuous distribution of the quantity in space and a certain set of singular "functions" (of the same type as the Dirac 8-function) corresponding to a concentration of the quantity at individual points of space. Hence, starting from these positions, the results of the computations in Example 2 can be expressed as the single equality div E = !o 8(0; x) . Then, as applied to the field E, the integral on the right-hand side of (14.42') is indeed equal either to qfco or to 0, according as the domain V contains the origin (and the point charge concentrated there) or not. In this sense one can assert (following Gauss) that the flux of electric field intensity across the surface of a body equals (up to a factor depending on the units chosen) the sum of the electric charges contained in the body. In this same sense one must interpret the electric charge density p in the Maxwell equations considered in Sect. 14.1 (formula (14.12)). b.
The Curl
an example.
We begin our study of the physical meaning of the curl with
Example 3. Suppose the entire space, regarded as a rigid body, is rotating with constant angular speed w about a fixed axis (let it be the x-axis) . Let us find the curl of the field v of linear velocities of the points of space. (The field is being studied at any fixed instant of time.) In cylindrical coordinates (r, cp , z ) we have the simple expression v(r, cp , z ) = wrecp . Then by formula (14.35" ) of Sect. 14.1, we find imme diatly that curl v = 2wez. That is, curl v is a vector directed along the axis 7 P.A.M. Dirac (1902-1984) - British theoretical physicist, one of the founders of quantum mechanics. More details on the Dirac <5-function will be given in Subsects. 17.4.4 and 17.5.4.
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14 Elements of Vector Analysis and Field Theory
of rotation. Its magnitude 2w equals the angular velocity of the rotation, up to the coefficient 2 , and the direction of the vector, taking account of the orientation of the whole space JR 3 , completely determines the direction of rotation. The field described in Example 3 in the small resembles the velocity field of a funnel (sink) or the field of the vorticial motion of air in the neighborhood of a tornado (also a sink, but one that drains upward) . Thus, the curl of a vector field at a point characterizes the degree of vorticity of the field in a neighborhood of the point. We remark that the circulation of a field over a closed contour varies in direct proportion to the magnitude of the vectors in the field, and, as one can verify using the same Example 3, it can also be used to characterize the vorticity of the field. Only now, to describe completely the vorticity of the field in a neighborhood of a point, it is necessary to compute the circulation over contours lying in three different planes. Let us now carry out this program. We take a disk Si (x) with center at the point x and lying in a plane perpendicular to the ith coordinate axis, i = 1 , 2, 3. We orient Si(x) using a normal, which we take to be the unit vector e i along this coordinate axis. Let d be the diameter of Si (x). From formula ( 14.4 1 ) for a smooth field A we find that f A · ds 8S; (x) (curl A) · e i = lim ( 14.44) d--.0 Si ( X ) where Si(x) denotes the area of the disk under discussion. Thus the circula tion of the field A over the boundary 8Si per unit area in the plane orthogonal to the ith coordinate axis characterizes the ith component of curl A. To clarify still further the meaning of the curl of a vector field, we recall that every linear transformation of space is a composition of dilations in three mutually perpendicular directions, translation of the space as a rigid body, and rotation as a rigid body. Moreover, every rotation can be realized as a rotation about some axis. Every smooth deformation of the medium (flow of a liquid or gas, sliding of the ground, bending of a steel rod) is locally linear. Taking account of what has just been said and Example 3, we can conclude that if there is a vector field that describes the motion of a medium (the velocity field of the points in the medium) , then the curl of that field at each point gives the instantaneous axis of rotation of a neighborhood of the point , the magnitude of the instantaneous angular velocity, and the direction of rotation about the instantaneous axis. That is, the curl characterizes com pletely the rotational part of the motion of the medium. This will be made slightly more precise below, where it will be shown that the curl should be regarded as a sort of density for the distribution of local rotations of the medium.
14.2 The Integral Formulas of Field Theory
283
c. The Gradient We have already said quite a bit about the gradient of a
scalar field, that is, about the gradient of a function. Hence at this point we shall merely recall the main things. Since w�rad f (e) = ( grad f , e ) = d f ( e ) = Def, where Def is the derivative of the function f with respect to the vector e, it follows that grad f is orthog onal to the level surfaces of f, and at each point it points in the direction of most rapid increase in the values of the function. Its magnitude l grad !I gives the rate of that growth ( per unit o f length i n the space i n which the argument varies ) . The significance of the gradient as a density will be discussed below. 14.2.3 Other Integral Formulas a. Vector Versions of the Gauss-Ostrogradskii Formula The interpre tation of the curl and gradient as vector densities, analogous to the interpreta tion ( 14.43) of the divergence as a density, can be obtained from the following classical formulas of vector analysis, connected with the Gauss-Ostrogradskii formula.
V' · B dV =
vJ vJ vJ
V' x A dV = V' dV =
f
avJ avJ avJ
du · B
(the divergence theorem ) (the curl theorem )
du x A du f
( the gradient theorem )
( 14.45) ( 14.46) ( 14.47)
The first of these three relations coincides with ( 14.42') up to notation and is the Gauss-Ostrogradskii formula. The vector equalities ( 14.46) and ( 14.47) follow from ( 14.45) if we apply that formula to each component of the corresponding vector field. Retaining the notation V(x) and d used in Eq. (14.43) , we obtain from formulas ( 14.45)-( 14.47) in a unified manner, "v · B ( X ) =
8VJ(xV(x) )
du · B
. 1 liD -"--:,..,. ::'-: ,--.---, d-+0
' " v X A( X ) = l liD
d-+0
V'
lim f(x) = d-+O
8VJ(x)V(x) v_--J(V(x) �x-:: ):-_--:- f
( 14.43')
du x A
----,--...,.--
a
( 14.48)
du
( 14.49)
284
14 Elements of Vector Analysis and Field Theory
The right-hand sides of ( 14.45)-( 14.47) can be interpreted respectively as the scalar flux of the vector field B , the vector flux of the vector field A, and the vector flux of the scalar field f across the surface 8V bounding the domain V. Then the quantities div B, curl A, and grad f on the left hand sides of Eqs. ( 14.43' ) , ( 14.48) , and ( 14.49) can be interpreted as the corresponding source densities of these fields. We remark that the right-hand sides of Eqs. ( 14.43' ) , ( 14.48) , and ( 14.49) are independent of the coordinate system. From these we can once again derive the invariance of the gradient , curl, and divergence. b. Vector Versions of Stokes' Formula Just as formulas ( 14.45)-( 14.47) were the result of combining the Gauss-Ostrogradskii formula with the alge braic operations on vector and scalar fields, the following triple of formulas can be obtained by combining these same operations with the classical Stokes formula (which appears as the first of the three relations) . Let 8 b e a (piecewise-) smooth compact oriented surface with a consis tently oriented boundary 88, let du be the vector element of area on 8, and ds the vector element of length on 88. Then for smooth fields A, B , and J , the following relations hold:
J j s
s
du ·
(V x A)
(du x
j s
=
V) x B =
du x
Vf =
j j j dsf
as as
ds · A ,
( 14.50)
ds x B ,
( 14.51)
.
( 14.52)
as
Formulas ( 14.51) and ( 14.52) follow from Stokes ' formula ( 14.50) . We shall not take time to give the proofs. c. Green's Formulas If 8 is a surface and n a unit normal vector to 8, then the derivative Dnf of the function f with respect to n is usually denoted � in field theory. For example, (V J , du) = (V J , n) da = (grad /, n) da = Dnf da = � da. Thus, � da is the flux of grad f across the element of surface da. In this notation we can write the following formulas of Green, which are very widely used in analysis:
J Vf V JgV2 f J(gVf) J(gV2 f - fV2g) J(gVf - fVg) v
v
·
gdV +
dV =
v
dV =
av
av
· ·
( 14.53)
du
(
du =
J (g �� - ���)
av
)
da . ( 14.54)
14.2 The Integral Formulas of Field Theory
g f=g J l 'Vf l 2 dV + J fl1fdV = J f'Vf · du ( = J f :� ) V V 8V 8V j =j = j :� ) V l1fdV 8V \lf · ( 8V
In particular, if we set respectively,
in ( 14.53) and
=
285
1 in ( 14.54) , we find d a , (14.53' )
du
do- .
(14. 54' )
This last equality is often called Gauss ' theorem. Let us prove, for exam ple, the second of Eqs. ( 14.53) and ( 14.54) :
Proof.
j (g\lf - f\lg) · = j \l · (g\lf - f\lg) = V 8V = j (\lg 'Vf + g'V2 f - 'Vf · \lg - f\l2 g) = J (g\l2 f - f\l2g) = J (gl1f - fl1g) 'V · ('PA) = \lip . A + 1p\l . du
dV
dV =
·
v
dV
v
dV .
v
In this formula we have used the Gauss-Ostrogradskii formula and the relation A. 0 14.2.4 Problems and Exercises 1. Using the Gauss-Ostrogradskii formula (14.45) , prove relations (14.46) and (14.47) . 2.
Using Stokes' formula (14.50) , prove relations (14.51) and (14.52) .
a) Verify that formulas (14.45) , ( 14.46) , and (14.47) remaind valid for an un bounded domain V if the integrands in the surface integrals are of order 0 ( �) as r ---+ oo. ( Here r = l rl , and r is the radius-vector in IR. 3 .) b ) Determine whether formulas ( 14.50) , ( 14.51 ) , and ( 14.52) remain valid for a noncompact surface S C IR. 3 if the integrands in the line integrals are of order o (;\-) as r ---+ oo. c ) Give examples showing that for unbounded surfaces and domains Stokes' formula (14.41') and the Gauss-Ostrogradskii formula (14.42') are in general not true. 3.
286
14 Elements of Vector Analysis and Field Theory
4. a) Starting from the interpretation of the divergence as a source density, explain why the second of the Maxwell equations (formula ( 14.12) of Sect. 14. 1) implies that there are no point sources in the magnetic field (that is, there are no magnetic charges) . b ) Using the Gauss-Ostrogradskii formula and the Maxwell equations (formula (14.12) of Sect . 14. 1 ) , show that no rigid configuration of test charges (for example a single charge) can be in a stable equilibrium state in the domain of an electrostatic field that is free of the (other) charges that create the field. (It is assumed that no forces except those exerted by the field act on the system.) This fact is known as Earnshaw 's theorem. 5. If an electromagnetic field is steady, that is, independent of time, then the system of Maxwell equations (formula (14.12) of Sect. 14. 1) decomposes into two independent parts - the electrostatic equations \7 · E = � , \7 x E = 0, and the , \7 · B = 0. magnetostatic equations \7 x B = h eo c The equation \7 E = p/Eo , where p is the charge density, transforms via the Gauss-Ostrogradskii formula into J E · du = Q/so , where the left-hand side is the s flux of the electric field intensity across the closed surface S and the right-hand side is the sum Q of the charges in the domain bounded by S, divided by the dimensioning constant Eo . In electrostatics this relation is usually called Gauss ' law. Using Gauss ' law, find the electric field E a) created by a uniformly charged sphere, and verify that outside the sphere it is the same as the field of a point charge of the same magnitude located at the center of the sphere; b) of a uniformly charged line; c) of a uniformly charged plane; d) of a pair of parallel planes uniformly charged with charges of opposite sign; e) of a uniformly charged ball. ·
6. a) Prove Green's formula ( 14.53) . b) Let f be a harmonic function in the bounded domain V (that is, f satisfies Laplace's equation L1f = 0 in V). Show, starting from (14.54') that the flux of the gradient of this function across the boundary of the domain V is zero. c) Verify that a harmonic function in a bounded connected domain is determined up to an additive constant by the values of its normal derivative on the boundary of the domain. d) Starting from ( 14.53' ) , prove that if a harmonic function in a bounded domain vanishes on the boundary, it is identically zero throughout the domain. e) Show that if the values of two harmonic functions are the same on the bound ary of a bounded domain, then the functions are equal in the domain. f) Starting from ( 14.53) , verify the following principle of Dirichlet. Among all continuous differentiable functions in a domain assuming prescribed values on the boundary, a harmonic function in the region is the only one that minimizes the Dirichlet integral {that is, the integral of the squared-modulus of the gradient over the domain).
287
14.2 The Integral Formulas of Field Theory
7. a) Let r(p, q) = IP - q l be the distance between the points p and q in the Euclidean space JR 3 . By fixing p, we obtain a function rp (q) of q E JR 3 . Show that Llr; 1 (q) = 47r8(p; q), where 8 is the 8-function. b) Let g be harmonic in the domain V. Setting f = 1/rp in (14.54) and taking account of the preceding result, we obtain =
47rg(p )
I (gv_!_rp - _!_vg rp ) · du . s
Prove this equality precisely. c) Deduce from the preceding equality that if S is a sphere of radius R with center at p, then g (p) = 2 g do- .
�I
47r s This is the so-called mean-value theorem for harmonic functions. d) Starting from the preceding result, show that if B is the ball bounded by the sphere S considered in part c) and V(B) is its volume, then
g (p)
=
V(� )
I g dV .
B
e) If p and q are points of the Euclidean plane lR 2 , then along with the function ...!_ considered in a) above (corresponding to the potential of a charge located at Tp p) , we now take the function ln ...!_ (corresponding to the potential of a uniformly Tp charged line in space) . Show that Ll ln ...!_ = 21r8(p; q), where 8(p; q) is now the Tp 8-function in lR 2 . f) By repeating the reasoning in a) , b) , c), and d) , obtain the mean-value the orem for functions that are harmonic in plane regions. 8. Cauchy 's multi-dimensional mean-value theorem. The classical mean-value theorem for the integral ( "Lagrange's theorem" ) as serts that if the function f : D -+ lR is continuous on a compact, measurable, and connected set D c lR n (for example, in a domain) , then there exists a point e E D such that f(x) dx = f(e) · I D I ,
I
D
where I D I is the measure (volume) of D. a) Now let J , g E C(D, JR) , that is, f and g are continuous real-valued functions in D. Show that the following theorem ( "Cauchy's theorem" ) holds: There exists e E D such that g(e) f(x) dx = J(e) g (x) dx .
I
D
I
D
b) Let D be a compact domain with smooth boundary 8D and f and smooth vector fields in D. Show that there exists a point e E D such that div g(e) · Flux f = div f(e) · Flux g, aD aD where Flux is the flux of a vector field across the surface 8D. aD
g
two
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14 Elements of Vector Analysis and Field Theory
14.3 Potential Fields 14. 3 . 1 The Potential of a Vector Field :
--+
A
be a vector field in the domain D C Rn . A function R is called a potential of the field if grad U in D.
Definition 1. Let
U D
A A=
Definition 2 . A field that has a potential is called a
potential field.
Since the partial derivatives of a function determine the function up to an additive constant in a connected domain, the potential is unique in such a domain up to an additive constant. We briefly mentioned potentials in the first part of this course. Now we shall discuss this important concept in somewhat more detail. In connection with these definitions we note that when different force fields are studied in physics, the potential of a field F is usually defined as a function U such that F -grad U. This potential differs from the one given in Definition 1 only in sign.
=
Example 1 . At a point of space having radius-vector r the intensity F of the gravitational field due to a point mass M located at the origin can be computed from Newton ' s law as F=
r - GMr3 ,
(14.55)
where r = J r J . This is the force with which the field acts on a unit mass at this point of space. The gravitational field (14.55) is a potential field. Its potential in the sense of Definition 1 is the function
U = GM �r .
(14.56)
Example 2. At a point of space having radius-vector r the intensity E of the electric field due to a point charge q located at the origin can be computed from Coulomb ' s law q E - -- �
- 47rco r3 ·
Thus such an electrostatic field, like the gravitational field, is a potential field. Its potential cp in the sense of physical terminology is defined by the relation cp
= 47rqco r1 . -- -
14.3 Potential Fields
289
14.3.2 Necessary Condition for Existence of a Potential
In the language of differential forms the equality A = grad U means that wl_ = dw � = dU, from which it follows that dwl_ = 0 ,
( 14.57)
since d2 w � = 0. This is a necessary condition for the field A to be a potential field. In Cartesian coordinates this condition can be expressed very simply. If A = (A\ . . . , A n ) and A = grad U, then A i = g� , i = 1 , . . . , n, and if the potential U is sufficiently smooth (for example, if its second-order partial derivatives are continuous) , we must have 8A i
8A3 8x3 - 8xi
'
i, j = 1 , . . . , n ,
( 14.57' )
which simply means that the mixed partial derivatives are equal in both orders: n In Cartesian coordinates wl_ = L: A i dxi , and therefore the equalities
i=l ( 14.57) and ( 14.57') are indeed equivalent in this case.
In the case of JR3 we have dwl_ = w�uri A ' so that the necessary condition
( 14.57) can be rewritten as
curl A = O ,
which corresponds to the relation curl grad U = 0, which we already know. The field A = (x, xy, xyz) in Cartesian coordinates in JR3 cannot be a potential field, since, for example, � =f. g� .
Example 3.
Example 4.
Consider the field A = (Ax , Ay ) given by A=
( - x2 y y2 ' x2 x y2 ) ' +
+
( 14.5 8 )
defined in Cartesian coordinates at all points of the plane exc � t the origin. The necessary condition for a field to be a potential field 8�* = fixY is fulfilled in this case. However, as we shall soon verify, this field is not a potential field in its domain of definition. Thus the necessary condition ( 14. 57) , or, in Cartesian coordinates ( 14.57' ) , is in general not sufficient for a field to be a potential field.
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14 Elements of Vector Analysis and Field Theory
14.3.3 Criterion for a Field to be Potential C
A continuous vector field A in a domain D �n is a po tential field in D if and only if its circulation {work) around every closed curve "( contained in D is zero: (14.59)
Proposition 1.
Proof. N e c e s s i t y. Suppose A = grad U. Then by the Newton-Leibniz for mula (Formula (14.40') of Sect . 14.2),
f A · ds "'
=
U ('Y (b) ) - U ('Y (a) ) ,
where "( : [a, b] -+ D. If "f(a) = "f(b), that is, when the path "( is closed, it is obvious that the right-hand side of this last equality vanishes, and hence the left-hand side does also. S u ff i c i e n c y. Suppose condition (5) holds. Then the integral over any (not necessarily closed) path in D depends only on its initial and terminal points, not on the path joining them. Indeed, if "( 1 and "(2 are two paths having the same initial and terminal points, then, traversing first 'Y l , then - "(2 (that is, traversing "(2 in the opposite direction) , we obtain a closed path 'Y whose integral, by (14.59), equals zero, but is also the difference of the integrals over 'Y l and "(2 • Hence these last two integrals really are equal. We now fix some point x0 E D and set U(x)
=
X
j
A · ds ,
(14.60)
XQ
where the integral on the right is the integral over any path in D from x 0 to x. We shall verify that the function U so defined is the required potential for the field A. For convenience, we shall assume that a Cartesian coordinate system (x l , . . . , x n ) has been chosen in � n . Then A · ds = A 1 dx 1 + · · · + A n dx n . If we move away from x along a straight line in the direction h ei , where ei is the unit vector along the x i -axis, the function U receives an increment equal to xi +hi U(x + h ei ) - U(x) = A i (x 1 , . . . , x i - l , t, x i + 1 , . . . , x n ) dt ,
J
equal to the integral of the form A · ds over this path from x to x + hei . By the continuity of A and the mean-value theorem, this last equality can be written as
14.3 Potential Fields
where 0 we find
S () S
291
1. Dividing this last equality by h and letting h tend to zero,
that is, A = grad U. 0
. au 8 . (x) = A'(x) , x•
Remark 1 . As can be seen from the proof, a sufficient condition for a field to be a potential field is that ( 14.59) hold for smooth paths or, for example, for broken lines whose links are parallel to the coordinate axes.
We now return to Example 4. Earlier ( Example 1 of Sect . 8 . 1 ) we com puted that the circulation of the field ( 14.58) over the circle x 2 + y 2 = 1 traversed once in the counterclockwise direction was 27f (# 0). Thus, by Proposition 1 we can conclude that the field ( 14.58) is not a potential field in the domain � 2 \ 0. But surely, for example, grad arctan '!!... = X
( - X2 +y y 2 , X 2 +x y2 ) ,
and it would seem that the function arctan � is a potential for ( 14.58) . What is this, a contradiction?! There is no contradiction as yet , since the only correct conclusion that one can make in this situation is that the function arctan � is not defined in the entire domain � 2 \ 0. And that is indeed the case: Take for example, the points on the y-axis. But then, you may say, we could consider the function cp( x, y ) , the polar angular coordinate of the point ( x, y ) . That is practically the same thing as arctan � , but cp( x, y ) is also defined for x = 0, provided the point ( x, y ) is not at the origin. Throughout the domain � 2 \ 0 we have d cp =
- X 2 +y y2 dx + X 2 +X y2 dy .
However, there is still no contradiction, although the situation is now more delicate. Please note that in fact cp is not a continuous single-valued function of a point in the domain � 2 \ 0. As a point encircles the origin counterclock wise, its polar angle, varying continuously, will have increased by 27f when the point returns to its starting position. That is, we arrive at the original point with a new value of the function, different from the one we began with. Consequently, we must give up either the continuity or the single-valuedness of the function cp in the domain � 2 \ 0. In a small neighborhood ( not containing the origin ) of each point of the domain � 2 \ 0 one can distinguish a continuous single-valued branch of the function cp. All such branches differ from one another by an additive constant, a multiple of 27f. That is why they all have the same differential and can all serve locally as potentials of the field ( 14.58) . Nevertheless, the field ( 14.58) has no potential in the entire domain � 2 \ 0.
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14 Elements of Vector Analysis and Field Theory
The situation studied in Example 4 turns out to be typical in the sense that the necessary condition (14.57) or ( 14.57' ) for the field A to be a poten tial field is locally also sufficient . The following proposition holds.
If the necessary condition for a field to be a potential field holds in a ball, then the field has a potential in that ball. Proof. For the sake of intuitiveness we first carry out the proof in the case of a disk D { (x, y) E JR 2 1 x 2 + y 2 < r} in the plane JR 2 . One can arrive at the point (x, y) of the disk from the origin along two different two-link broken lines II and 12 with links parallel to the coordinate axes (see Fig. 14.3) . Since D is a convex domain, the entire rectangle I bounded by these lines is contained in D. Proposition 2.
=
B y Stokes ' formula, taking account o f condition ( 14.57) , we obtain
J wi = J dwi =
81
0.
I
Fig. 14.3. By the remark to Proposition 1 we can conclude from this that the field A is a potential field in D. Moreover, by the proof of sufficiency in Proposition 1 , the function ( 1 4.60) can again be taken as the potential, the integral being interpreted as the integral over a broken line from the center to the point in question with links parallel to the axes. In this case the independence of the choice of path 12 for such an integral followed immediately from Stokes ' formula for a rectangle. In higher dimensions it follows from Stokes ' formula for a two-dimensional rectangle that replacing two adjacent links of the broken line by two links forming the sides of a rectangle parallel to the original does not change the value of the integral over the path. Since one can pass from one broken-line path to any other broken-line path leading to the same point by a sequence of such reconstructions, the potential is unambiguously defined in the general case. D
Il l
14.3 Potential Fields
293
14.3.4 Topological Structure of a Domain and Potentials Comparing Example 4 and Proposition 2, one can conclude that when the necessary condition (14.57) for a field to be a potential field holds, the ques tion whether it is always a potential field depends on the (topological) struc ture of the domain in which the field is defined. The following considerations (here and in Subsect . 14.3.5 below) give an elementary idea as to exactly how the characteristics of the domain bring this about. It turns out that if the domain D is such that every closed path in D can be contracted to a point of the domain without going outside the domain, then the necessary condition (14.57) for a field to be a potential field in D is also sufficient . We shall call such domains simply connected below. A ball is a simply connected domain (and that is why Proposition 2 holds) . But the punctured plane JR 2 \ 0 is not simply connected, since a path that encircles the origin cannot be contracted to a point without going outside the region. This is why not every field in JR 2 \ 0 satisfying (14.57'), as we saw in Example 4, is necessarily a potential field in JR2 \ 0. We now turn from the general description to precise formulations. We begin by stating clearly what we mean we speak of deforming or contracting a path.
homotopy (or deformation) in D from a closed path 'Yo : [0,2 1] -+ D to a closed path "(1 : [0, 1] -+ D is a continuous mapping r : / -+ D of the square /2 = { W , t 2 ) E JR2 I O ::; t i ::; 1, i = 1 , 2} into D such that T(t l , O) = 'Yo W ) , T(t\ 1) = "(1 (t 1 ), and T( O , t 2 ) = T(1, t 2 ) for all t 1 , t2 E [0, 1]. Thus a homotopy is a mapping r / 2 -+ D (Fig. 14.4). If the variable t 2 is regarded as time, according to Definition 3 at each instant of time t = t 2 we have a closed path TW , t) = 'Yt (Fig. 14.4 ) . 8 The change in this path with time is such that at the initial instant t = t 2 = 0 it coincides with 'Yo and at time t = t 2 = 1 it becomes "(1 . Since the condition 'Yt (O) = T( O , t) = T(1, t) = 'Yt ( l ) , which means that the path 'Yt is closed, holds at all times t E [0, 1] , the mapping r : / 2 -+ D induces the same mappings /3o W ) : = T(t l , O) = T(t\ 1) = : !3 1 W ) on the vertical sides of the square / 2 . The mapping T is a formalization of our intuitive picture of gradually deforming 'Yo to "( 1 . It is clear that time can be allowed to run backwards, and then we obtain the path 'Yo from "(1 . Definition 3. A
:
8
Orienting arrows are shown along certain curves in Fig. 14.4. These arrows will be used a little later; for the time being the reader should not pay any attention to them.
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14 Elements of Vector Analysis and Field Theory
Definition 4. Two closed paths are homotopic in a domain if they can be obtained from each other by a homotopy in that domain, that is a homotopy can be constructed in that domain from one to the other.
Remark 2. Since the paths we have to deal with in analysis are as a rule paths of integration, we shall consider only smooth or piecewise-smooth paths and smooth or piecewise-smooth homotopies among them, without noting this explicitly. For domains in JR.n one can verify that the presence of a continuous ho motopy of (piecewise-) smooth paths guarantees the existence of (piecewise-) smooth homotopies of these paths.
wi
If the 1-form in the domain D is such that and the closed paths /o and /I are homotopic in D, then
Proposition 3.
/2
dwi = 0,
Let r : ---+ D be a homotopy from /o to 11 (see Fig. 14.4) . If Io and h are the bases of the square and J0 and J1 its vertical sides, then by definition of a homotopy of closed paths, the restrictions of r to 10 and h coincide with /o and 11 respectively, and the restrictions of r to J0 and J1 give some paths (30 and (31 in D. Since r( O , t 2 ) ( 1 , t 2 ), the paths (30 and (31 are the same. As a result of the change of variables ( t ) , the form transfers to the square as some 1-form In the process 0. Hence, by Stokes ' formula 0, since
Proof.
12
2 wi 1 F* dw = dF*wi = dwi = dwi = J w = J dw = O . i)[ 2
[2
Fig.
14.4.
=F x=F w = F*wi.
14.3 Potential Fields
295
But 8[ 2
Io
/1
h =
Jo
J l+J l J l J l J l J l w
�
�
w -
n
w -
�
w =
�
w -
n
w .
D
m
it
Definition 5. A domain is simply connected if every closed path homotopic to a point ( that is, a constant path ) .
IS
Thus simply connected domains are those in which every closed path can be contracted to a point .
A
If a field defined in a simply connected domain D satisfies the necessary condition (14.57) or (14.57') to be a potential field, then it is a potential field in D. Proof. By Proposition 1 and Remark 1 i t suffices t o verify that Eq. (14.59) holds for every smooth path in D. The path is by hypothesis homotopic Proposition 4.
'Y
'Y
to a constant path whose support consists of a single point. The integral over such a one-point path is obviously zero. But by Proposition 3 the integral does not change under a homotopy, and so Eq. (14.59) must hold for 'Y · D
Remark 3. Proposition 4 subsumes Proposition 2. However, since we had certain applications in mind, we considered it useful to give an independent constructive proof of Proposition 2.
Remark 4. Proposition 2 was proved without invoking the possibility of a smooth homotopy of smooth paths. 14.3.5 Vector Potential. Exact and Closed Forms if the relation B
=
AA
is a vector potential for a field B in a domain D curl holds in the domain.
Definition 6. A field
c
JR3
If we recall the connection between vector fields and forms in the oriented Euclidean space JR 3 and also the definition of the curl of a vector field, the relation B = curl can be rewritten as w� = dw l . It follows from this relation that w� i v B = dw� = d2 wi_ = 0. Thus we obtain the necessary condition div B = 0 , (14.61) which the field B must satisfy in D in order to have a vector potential, that is, in order to be the curl of a vector field in that domain. A field satisfying condition (14.61) is often, especially in physics, called a
A
A
solenoidal field.
14 Elements of Vector Analysis and Field Theory
296
Example 5.
In Sect . 14. 1 we wrote out the system of Maxwell equations. The second equation of this system is exactly Eq. ( 14.61 ) . Thus, the desire naturally arises to regard a magnetic field B as the curl of some vector field - the vector potential of B. When solving the Maxwell equations, one passes to exactly such a vector potential.
A
As can be seen from Definitions 1 and 6, the questions of the scalar and vector potential of vector fields ( the latter question being posed only in JR3 ) are special cases of the general question as to when a differential p-form wP is the differential dwP - 1 of some form wP - 1 .
exact in a domain D if there exists D such that wP = dwP - 1 . If the form wP is exact in D, then dwP = d 2 wP - 1 = 0. Thus the condition
Definition 7. A differential form wP is
a form wP - 1 in
dw
=
0
( 14.62 )
is a necessary condition for the form w to be exact. As we have already seen ( Example 4 ) , not every form satisfying this con dition is exact . For that reason we make the following definition. Definition 8 . The differential form w is condition ( 14.62 ) there.
The following theorem holds. Theorem. ( Poincare ' s lemma. ) If a form
there.
closed in a domain D if it satisfies is closed in a ball, then it is exact
Here we are talking about a ball in lR11 and a form of any order, so that Proposition 2 is an elementary special case of this theorem. The Poincare lemma can also be interpreted as follows: The necessary condition ( 14.62 ) for a form to be exact is also locally sufficient, that is, every point of a domain in which ( 14.62 ) holds has a neighborhood in which w is exact. In particular, if a vector field B satisfies condition ( 14.61 ) , it follows from the Poincare lemma that at least locally it is the curl of some vector field We shall not take the time at this point to prove this important theorem ( those who wish to do so can read it in Chap. 15 ) . We prefer to conclude by explaining in general outline the connection between the problem of the exactness of closed forms and the topology of their domains of definition ( based on information about 1-forms ) .
A.
Example 6.
JR2
Consider the plane with two points p 1 and p 2 removed ( Fig. 14.5 ) , and the paths /'o , /' 1 , and 1'2 whose supports are shown in the figure. The path 1'2 can be contracted to a point inside D, and therefore if a closed form w is given in D, its integral over 1'2 is zero. The path /'o cannot
14.3 Potential Fields
Fig.
297
14.5.
be contracted to a point, but without changing the value of the integral of the form, it can be homotopically converted into the path / 1 · The integral over 1 1 obviously reduces to the integral over one cycle en closing the point p 1 clockwise and the double of the integral over a cycle enclosing p2 counterclockwise. If T1 and T2 are the integrals of the form w over small circles enclosing the points p 1 and p2 and traversed, say, counter clockwise, one can see that the integral of the form w over any closed path in D will be equal to n 1 T1 + n2T2 , where n 1 and n2 are certain integers indi cating how many times we have encircled each of the holes P 1 and P2 in the plane JR. 2 and in which direction. Circles c 1 and c2 enclosing p 1 and p2 serve as a sort of basis in which every closed path r c D has the form r = n 1 c 1 + n2c2 , up to a homotopy, which has no effect on the integral. The quantities I w = Ti are called the cyclic c;
constants or the periods of the integral. If the domain is more complicated and there are k independent elementary cycles, then in agreement with the
n 1 c 1 + · · · + n k ck , it results that I w n 1 T1 + · · · + n k Tk · It turns out that for any set T1 , . . . , Tk of numbers in such a domain one can
expansion
r
=
'"Y
=
construct a closed 1-form that will have exactly that set of periods. ( This is a special case of de Rham ' s theorem - see Chap. 15. ) For the sake of visualization, we have resorted here to considering a plane domain, but everything that has been said can be repeated for any domain D c Rn .
In an anchor ring ( the solid domain in JR. 3 enclosed by a torus ) all closed paths are obviously homotopic to a circle that encircles the hole a certain number of times. This circle serves as the unique non-constant basic cycle c.
Example 7.
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14 Elements of Vector Analysis and Field Theory
Moreover, everything that has just been said can be repeated for paths of higher dimension. If instead of one-dimensional closed paths - mappings of a circle or, what is the same, mappings of the one-dimensional sphere - we take mappings of a k-dimensional sphere, introduce the concept of homotopy for them, and examine how many mutually nonhomotopic mappings of the k dimensional sphere into a given domain D C exist, the result is a certain characteristic of the domain D which is formalized in topology as the so called kth homotopy group of D and denoted n k (D). If all the mappings of the k-dimensional sphere into D are homotopic to a constant mapping, the group n k (D) is considered trivial. (It consists of the identity element alone.) It can happen that n 1 (D) is trivial and n2 (D) is not .
JRn
Example 8. If D
JR;3
0
is taken to be the space with the point removed, obviously every closed path in D can be contracted to a point, but a sphere enclosing the point cannot be homotopically converted to a point.
0
It turns out that the homotopy group n k (D) has less to do with the periods of a closed k-form than the so-called homology group Hk (D). (See Chap. 15.)
Example 9 . From what has been said we can conclude that, for example, in is a simply connected the domain every closed 1-form is exact domain) , but not very closed 2-form is exact. In the language of vector fields, this means that every irrotational field in is the gradient of a function, but not every source-free field B ( div B = is the curl of some field in this domain.
JR3 \ 0
(JR3 \ 0 A 0JR3) \0
Example 1 0.
To balance Example 9 we take the anchor ring. For the anchor ring the group n 1 (D) is not trivial (see Example 7) , but n2 (D) is trivial, since every mapping f : S 2 --t D of the two-sphere into D can be contracted to a constant mapping (any image of a sphere can be contracted to a point). In this domain not every irrotational field is a potential field, but every source-free field is the curl of some field. 14.3.6 Problems and Exercises 1.
Show that every central field A = f(r)r is a potential field.
2 . Let F = -grad U be a potential force field. Show that the stable equilibrium positions of a particle in such a field are the minima of the potential U of that field.
For an electrostatic field E the Maxwell equations (formula (14.12) of Sect. 14. 1 ) , already noted, reduce t o the pair o f equations \7 E = £; and \7 x E = 0 . The condition \7 x E = 0 means, at least locally, that E = -grad <.p. The field of a point charge is a potential field, and since every electric field is the sum (or integral) of such fields, it is always a potential field. Substituting E = -\l<.p in the first equation of the electrostatic field, we find that its potential satisfies 3.
as
·
14.3 Potential Fields
299
Poisson's equation 9 .1ip = ..e.. < . The potential 'P determines the field completely, so that describing E reduces too finding the function ip, the solution of the Poisson equation. Knowing the potential of a point charge (Example 2) , solve the following prob lem. a) Two charges +q and -q are located at the points (0, 0, - d/ 2 ) and (0, 0, d/2) in JR3 with Cartesian coordinates (x, y, z) . Show that at distances that are large relative to d the potential of the electrostatic field has the form 1 z 1 - 3 qd + o 3 ' 'P = 4 11"Eo r · r where r is the absolute value of the radius-vector r of the point (x, y, z). b) Moving very far away from the charges is equivalent to moving the charges together, that is, decreasing the distance d. If we now fix the quantity qd =: p and decrease d, then in the limit we obtain the function 'P = 4 1 -!:Jp in the domain 11'" € 0 r JR3 \ 0. It is convenient to introduce the vector p equal to p in absolute value and directed from - q to +q. We call the pair of charges -q and +q and the construction obtained by the limiting procedure just described a dipole , and the vector p the dipole moment. The function 'P obtained in the limit is called the dipole potential. Find the asymptotics of the dipole potential as one moves away from the dipole along a ray forming angle 8 with the direction of the dipole moment. c) Let 'Po be the potential of a unit point charge and ip 1 the dipole potential having dipole moment P 1 · Show that 4' 1 = - (p 1 · 'V )ipo . d ) We can repeat the construction with the limiting passage that we carried out for a pair of charges in obtaining the dipole for the case of four charges (more precisely, for two dipoles with moments P 1 and p 2 ) and obtain a quadrupole and a corresponding potential. In general we can obtain a multipole of order j with potential 'PJ = ( - 1 F ( PJ · Y') ( PJ - 1 · Y') · · · (p 1 · 'V)ipo = L Qi k l 8};y'f'koiJ z l , where Qi k l i+k+ l=j are the so-called components of the multipole moment. Carry out the computations and verify the formula for the potential of a multipole in the case of a quadrupole. e) Show that the main term in the asymptotics of the potential of a cluster of charges with increasing distance from the cluster is 4 1 , where Q is the total 7TCQ 9. r charge of the cluster. f) Show that the main term of the asymptotics of the potential of an electrically neutral body consisting of charges of opposite signs (for example, a molecule) at a distance that is large compared to the dimensions of the body is 4 1 . Here 1TCQ � r e r is a unit vector directed from the body to the observer; p = L q; d; , where q; is the magnitude of the ith charge and d; is its radius-vector. The origin is chosen at some point of the body. g) The potential of any cluster of charges at a great distance from the cluster can be expanded (asymptotically) in functions of multipole potential type. Show this using the example of the first two terms of such a potential (see d), e) , and f) ) . 9 S.D. Poisson (1781-1849) - French scientist , specializing i n mechanics and physics; his main work was on theoretical and celestial mechanics, mathematical physics, and probability theory. The Poisson equation arose in his research into gravitational potential and attraction by spheroids.
( )
300 4.
14 Elements of Vector Analysis and Field Theory
Determine whether the following domains are simply connected. a) the disk { (x, y) E IR 2 1 x 2 + y 2 < 1 } ;
1
b ) the disk with its center removed { (x, y ) E IR2 0 < x 2 + y2
< 1}.; c) a ball with its center removed { ( x , y, z) E IR3 1 0 < x 2 + y 2 + z 2 < 1 } ; d) a n annulus { (x, y) E IR2 1 � < x 2 + y2 < 1 }; e) a spherical annulus { (x, y, z) E JR3 1 � < x 2 + y 2 + z 2 < 1 }; f) an anchor ring i n JR3 .
a) Give the definition of homotopy of paths with endpoints fixed. b) Prove that a domain is simply connected if and only if every two paths in it having common initial and terminal points are homotopic in the sense of the definition given in part a) . 5.
6. Show that a) every continuous mapping f : 8 1 -+ 8 2 of a circle 8 1 (a one-dimensional sphere) into a two-dimensional sphere 8 2 can be contracted in 8 2 to a point (a constant mapping) ; b) every continuous mapping f : 8 2 -+ 8 1 is also homotopic to a single point; c) every mapping f : 8 1 -+ 8 1 is homotopic to a mapping -+ n
7. In the domain IR3 \ 0 (three-dimensional space with the point 0 removed) con struct: a) a closed but not exact 2-form; b) a source-free vector field that is not the curl of any vector field m that domain.
8 . a) Can there be closed, but not exact forms of degree p < n - 1 in the domain D = IR n \ 0 (the space IRn with the point 0 removed)? b) Construct a closed but not exact form of degree p = n - 1 in D = IRn \ 0. 9 . If a 1-form w is closed in a domain D C IR n , then by Proposition 2 every point x E D has a neighborhood U(x ) inside which w is exact. From now on w is assumed to be a closed form. a) Show that if two paths "{i : [0, 1] -+ D, i = 1, 2, have the same initial and terminal points and differ only on an interval [a, ,8] C [0, 1] whose image under either of the mappings "fi is contained inside the same neighborhood U(x) , then
I w = I w.
"1 1
"12
14.3 Potential Fields
301
b) Show that for every path [0, 1] 3 t >-+ 1(t) E D one can find a number 6 > 0 such that if the path ::Y has the same initial and terminal point as 1 and differs from 1 at most by 6, that is omax :o; t :o;1 I ::Y ( t ) - 1(t) l � 6, then I w = I w. ;y
1
c) Show that if two paths 1 1 and 12 with the same initial and terminal points are homotopic in D as paths with fixed endpoints, then I w = I w for any closed /1 form w in D.
12
a) It will be proved below that every continuous mapping r : I 2 -+ D of the square I 2 can be uniformly approximated with arbitrary accuracy by a smooth mapping (in fact by a mapping with polynomial components) . Deduce from this that if the paths 1 1 and 12 in the domain D are homotopic, then for every c > 0 there exist smooth mutually homotopic paths ::Y1 and ::Y2 such that omax :o; t :o;1 I::Yi ( t) - li ( t) I � c, i = 1 , 2. b) Using the results of Example 9, show now that if the integrals of a closed form in D over smooth homotopic paths are equal, then they are equal for any paths that are homotopic in this domain (regardless of the smoothness of the homotopy) . The paths themselves, of course, are assumed to be as regular as they need to be for integration over them. 10.
a) Show that if the forms w P , w P - 1 , and w P - 1 are such that w P dw P - 1 dwP 21 , then (at least locally) one can find a form wP - 2 such that wP - 1 wP - 1 + of a form have dwP - . (The fact that any two forms that differ by the differential the same differential obviously follows from the relation d 2 w 0.) 11.
=
=
=
=
b) Show that the potential t.p of an electrostatic field (Problem 3) is determined up to an additive constant, which is fixed if we require that the potential tend to zero at infinity.
12. The Maxwell equations (formula (14.12) of Sect. 14. 1) yield the following pair of magnetostatic equations: V · B = 0, V x B = - � . The first of these shows EQC that at least locally, B has a vector potential A, that is, B = V x A. a) Describe the amount of arbitrariness in the choice of the potential A of the magnetic field B (see Problem l la) ) . b ) Let x , y , z b e Cartesian coordinates i n JR3 . Find potentials A for a uniform magnetic field B directed along the z-axis, each satisfying one of the following additional requirements: the field A msut have the form (0, A y , 0) ; the field A must have the form (A x , O, O) ; the. field A must have the form (Ax , A y , O) ; the field A must be invariant under rotations about the z-axis. c) Show that the choice of the potential A satisfying the additional requirement V · A = 0 reduces to solving Poisson's equation; more precisely, to finding a scalar valued function 'lj; satisfying the equation LJ.'lj; = f for a given scalar-valued function
f.
d) Show that if the potential A of a static magnetic field B is chosen so that · A = 0, it will satisfy the vector Poisson equation LJ.A = - h . Thus, invoking EQC the potential makes it possible to reduce the problem of finding electrostatic and magnetostatic fields to solving Poisson's equation. V
14 Elements of Vector Analysis and Field Theory 1 3 . The following theorem of Helmholtz 1 0 is well known: Every smooth field F in a domain D of oriented Euclidean space JR(3 can be decomposed into a sum F = F 1 +F 2 of an irrotational field F 1 and a solenoidal field F 2 . Show that the construction of such a decomposition can be reduced to solving a certain Poisson equation. 302
14. Suppose a given mass of a certain substance passes from a state characterized thermodynamically by the parameters Vo , Po (To ) into the state V, P, (T) . Assume that the process takes place slowly (quasi-statically) and over a path 1 in the plane of states (with coordinates V, P) . It can be proved in thermodynamics that the quantity S = J 8:} , where tiQ is the heat exchange form, depends only on the "'
initial point (Vo , Po ) and the terminal point (V, P) of the path, that is, after one of these points is fixed, for example (Vo , Po) , S becomes a function of the state (V, P) of the system. This function is called the entropy of the system. a) Deduce from this that the form w = 8:} is exact, and that w = dS. b) Using the form of tiQ given in Problem 6 of Sect. 13.1 for an ideal gas, find the entropy of an ideal gas.
14.4 Examples of Applications To show the concepts we have introduced in action, and also to explain the physical meaning of the Gauss-Ostrogradskii-Stokes formula as a conserva tion law, we shall examine here some illustrative and important equations of mathematical physics. 14.4 . 1 The Heat Equation We are studying the scalar field T = T(x, y , z, t) of the temperature of a body being observed as a function of the point (x, y, z ) of the body and the time t. As a result of heat transfer between various parts of the body the field T may vary. However, this variation is not arbitrary; it is subject to a particular law which we now wish to write out explicitly. Let D be a certain three-dimensional part of the observed body bounded by a surface S. If there are no heat sources inside S, a change in the internal energy of the substance in D can occur only as the result of heat transfer, that is, in this case by the transfer of energy across the boundary S of D. By computing separately the variation in internal energy in the volume D and the flux of energy across the surface S, we can use the law of conservation of energy to equate these two quantities and obtain the needed relation. It is known that an increase in the temperature of a homogeneous mass m by LJ.T requires energy cmLJ.T, where c is the specific heat capacity of 1 0 H.L.F. Helmholtz (182 1-1894) - German physicist and mathematician; one of the first to discover the general law of conservation of energy. Actually, he was the first to make a clear distinction between the concepts of force and energy.
14.4 Examples of Applications
303
the substance under consideration. Hence if our field T changes by LJ.T = T(x, y, z, t + LJ.t) - T(x, y, z, t) over the time interval LJ.t, the internal energy in D will have changed by an amount
JJDJ
cpLJ.T dV ,
(14.63)
where p = p(x, y, z ) is the density of the substance. It is known from experiments that over a wide range of temperatures the quantity of heat flowing across a distinguished area du = n da per unit time as the result of heat transfer is proportional to the flux -grad T · du of the field -grad T across that area (the gradient is taken with respect to the spatial variables x, y, z ) . The coefficient of proportionality k depends on the substance and is called its coefficient of thermal conductivity. The negative sign in front of grad T corresponds to the fact that the energy flows from hotter parts of the body to cooler parts. Thus, the energy flux (up to terms of order o ( LJ.t)) -k grad T · du (14.64)
LJ.t JJ s
takes place across the boundary S of D in the direction of the external normal over the time interval LJ.t. Equating the quantity (14.63) to the negative of the quantity ( 14.64) , dividing by LJ.t, and passing to the limit as LJ.t -t 0, we obtain
JJJD
cp
k grad T · du . �� dV = JJ s
(14.65)
This equality is the equation for the function T. Assuming T is sufficiently smooth, we transform ( 14.65) using the Gauss-Ostrogradskii formula:
JJJD
cp
�� dV = JJJ div (k grad T) dV . D
Hence, since D is arbitrary, it follows obviously that ar cp 8t = div (k grad T) .
(14.66)
We have now obtained the differential version of the integral equation (14.65) . If there were heat sources (or sinks) in D whose intensities have density F(x, y, z, t), instead of ( 14.65) we would write the equality
JJDJ
cp
a:; dV = JsJ k grad T · du + JJDJ F d V ,
(14.65')
304
14 Elements of Vector Analysis and Field Theory
and then instead of ( 14.66) we would have the equation aT cp at
=
. (k grad T ) + F . d1v
(14.66')
If the body is assumed isotropic and homogeneous with respect to its heat conductivity, the coefficient k in (14.66) will be constant , and the equation will transform to the canonical form 8T (14.67) 8t t1T + f ,
= a2
where f = :;, and ckp is the coefficient of thermal diffusivity. The equation (14.67) is usually called the heat equation. In the case of steady-state heat transfer, in which the field T is indepen dent of time, this equation becomes Poisson 's equation
a2 =
t1T
=
'P ,
(14.68)
t1T
=
0.
(14.69)
where 'P - � f; and if in addition there are no heat sources in the body, the result is Laplace 's equation
=
The solutions of Laplace's equation, as already noted, are called har In the thermophysical interpretation, harmonic functions correspond to steady-state temperature fields in a body in which the heat flows occur without any sinks or sources inside the body itself, that is, all sources are located outside the body. For example, if we maintain a steady temperature distribution T l av = T over the boundary aV of a body, then the temperature field in the body V will eventually stabilize in the form of a harmonic function T. Such an interpretation of the solutions of the Laplace equation (14.69) enables us to predict a number of properties of harmonic functions. For example, one must presume that a harmonic function in V can not have local maxima inside the body; otherwise heat would only flow away from these hotter portions of the body, and they would cool off, contrary to the assumption that the field is stationary.
monic functions.
14.4.2 The Equation of Continuity Let p p(x, y, z, t ) be the density of a material medium that fills a space being observed and = v(x, y, z, t ) the velocity field of motion of the medium as function of the point of space (x, z ) and the time t.
=
v
y,
From the law of conservation of mass, using the Gauss-Ostrogradskii for mula, we can find an interconnection between these quantities.
14.4 Examples of Applications
305
Let D be a domain in the space being observed bounded by a surface S. Over the time interval Llt the quantity of matter in D varies by an amount
JJDJ (p(x,
y, z, t + Ll t)
- p(x, y, z, t) ) dV .
Over this small time interval Ll t, the flow of matter across the surface S in the direction of the outward normal to S is ( up to o(Llt))
Llt ·
JJ p s
If there were no sources or sinks in matter, we would have
or, in the limit as
v · du
.
D, then by the law of conservation of
JJDJ Llp
dV =
JJDJ a;:
dV = -
-
Llt
Llt -+ 0
JSJ p
JSJ p
v · du
v · du .
Applying the Gauss-Ostrogradskii formula to the right-hand side of this equality and taking account of the fact that D is an arbitrary domain, we con clude that the following relation must hold for sufficiently smooth functions p and v: (14.70) = -div (pv) ,
a;:
called the equation of continuity of a continuous medium. In vector notation the equation of continuity can be written as
ap + \7 . (pv) = 0 ' at
(14.70')
ap + v . v + p\7 . v = o , at
(14. 70")
or, in more expanded form,
P
If the medium is incompressible ( a liquid ) , the volumetric outflow of the medium across a closed surface S must be zero:
JJ s
v · du = O ,
from which ( again on the basis of the Gauss-Ostrogradskii formula ) it follows that for an incompressible medium div v = 0
.
(14.71)
306
14 Elements of Vector Analysis and Field Theory
Hence, for an incompressible medium of variable density ( a mixture of water and oil ) Eq. ( 14 . 70") becomes ( 14.72) If the medium is also homogeneous, then \7 p = 0 and therefore if
=
0.
14.4.3 The Basic Equations of the Dynamics of Continuous Media We shall now derive the equations of the dynamics of a continuous medium moving in space. Together with the functions p and v already considered, which will again denote the density and the velocity of the medium at a given point ( y , z ) of space and at a given instant t of time, we consider the pressure p = ( y, z, t) as a function of a point of space and time. In the space occupied by the medium we distinguish a domain D bounded by a surface S and consider the forces acting on the distinguished volume of the medium at a fixed instant of time. Certain force fields ( for example, gravitation ) are acting on each element p d V of mass of the medium. These fields create the so-called mass forces. Let F = ( y , z , t) be the density of the external fields of mass force. Then a force Fp dV acts on the element from the direction of these fields. If this element has an acceleration a at a given instant of time, then by Newton ' s second law, this is equivalent to the presence of another mass force called inertia, equal to -ap dV. Finally, on each element dO" = n da of the surface S there is a surface tension due to the pressure of the particles of the medium near those in D, and this surface force equals -p dlT ( where n is the outward normal to S) . By d ' Alembert ' s principle, at each instant during the motion of of any material system, all the forces applied to it , including inertia, are in mutual equilibrium, that is, the force required to balance them is zero. In our case, this means that (F - a )p dV p dlT = O . ( 14.73)
xp, x ,
F x,
JJJD
!!S
The first term in this sum is the equilibrant of the mass and inertial forces, and the second is the equilibrant of the pressure on the surface S bounding the volume. For simplicity we shall assume that we are dealing with an ideal ( nonviscous ) fluid or gas, in which the pressure on the surface dO" has the form p dlT , where the number p is independent of the orientation of the area in the space. Applying formula ( 14.47) from Sect . 14.2, we find by (14.73) that
!!!D
( F - a ) p dV -
JJDJ
grad p dv = 0 ,
14.4 Examples of Applications
307
from which, since the domain D is arbitrary, it follows that
pa = pF - grad p .
( 14.74)
In this local form the equation of motion of the medium corresponds perfectly to Newton ' s law of motion for a material particle. The acceleration a of a particle of the medium is the derivative �� of the velocity v of the particle. If x = x ( t ) , y = y ( t ) , z = z ( t ) is the law of motion of a particle in space and v = v ( x, y, z, t ) is the velocity field of the medium, then for each individual particle we obtain
av av dx av dy av dz a = dv dt = at + ax dt + ay dt + az dt or
a = av at + ( v V ) v . ·
Thus the equation of motion (14.74) assumes the following form dv 1 p = F - -grad dt
( 14.75)
p
or
av + ( v V ) v = F - -1 \7p . ( 14.76) at Equation ( 14. 76) is usually called Euler 's hydrodynamic equation . The vector equation (14.76) is equivalent to a system of three scalar equa tions for the three components of the vector v and the pair of functions p ·
P
and p. Thus, Euler ' s equation does not completely determine the motion of an ideal continuous medium. To be sure, it is natural to adjoin to it the equation of continuity ( 14.70) , but even then the system is underdetermined. To make the motion of the medium determinate one must also add to Eqs. ( 14.70) and ( 14.76) some information on the thermodynamic state of the medium (for example, the equation of state f(p, p , T) 0 and the equation for heat transfer. ) The reader may obtain some idea of what these relations can yield in the final subsection of this section. =
14.4.4
The Wave Equation
We now consider the motion of a medium corresponding to the propagation of an acoustic wave. It is clear that such a motion is also subject to Eq. ( 14.76) ; this equation can be simplified due to the specifics of the phenomenon. Sound is an alternating state of rarefaction and compression of a medium, the deviation of the pressure from its mean value in a sound wave being very
308
14 Elements of Vector Analysis and Field Theory
small - of the order of 1%. Therefore acoustic motion consists of small devi ations of the elements of volume of the medium from the equilibrium posi tion at small velocities. However, the rate of propagation of the disturbance ( wave ) through the medium is comparable with the mean velocity of motion of the molecules of the medium and usually exceeds the rate of heat trans fer between the different parts of the medium under consideration. Thus, an acoustic motion of a volume of gas can be regarded as small oscillations about the equilibrium position occurring without heat transfer ( an adiabatic process ) . Neglecting the term ( v · V' ) v in the equation of motion ( 14. 76) in view of the small size of the macroscopic velocities v, we obtain the equality P
8v 8t
=
pF - V'p .
If we neglect the term of the form 'tJi v for the same reason, the last equality reduces to the equation
[}
v ot ( p ) Applying the operator V' ( on
=
pF - V'p .
x,y, z coordinates) to it , we obtain
(V' v ) = V' . pF Llp . 8t . p Using the equation of continuity ( 14. 70') and introducing the notation V' · pF = - tl> , we arrive at the equation
[}
-
( 14.77) If we can neglect the influence of the exterior fields, Eq. ( 14. 77) reduces to the relation ( 14.78 ) between the density and pressure in the acoustic medium. Since the process is adiabatic, the equation of state f(p, p , T) = 0 reduces to a relation p = '1/J(p) , from which it follows that � = '1/J'(p) ftJ. + 'l/J"(p) ( '1Jf ( Since the pressure oscillations are small in an acoustic wave, one may assume that '1/J'(p) = 'l/J'(p0 ) , where Po is the equilibrium pressure. Then '1/J" = 0 and � � '1/J'(p) � . Taking this into account , from ( 14.78 ) we finally obtain
1
[}2 p 8t 2
= a
2 Llp ,
( 14.79)
where a = ('1/J'(po ) ) - 1 2 . This equation describes the variation in pressure in a medium in a state of acoustic motion. Equation ( 14.79) describes the
14.4 Examples of Applications
309
simplest wave process in a continuous medium. It is called the homogeneous wave equation. The quantity a has a simple physical meaning: it is the speed of propagation of an acoustic disturbance in the medium, that is, the speed of sound in it (see Problem 4). In the case of forced oscillations, when certain forces are acting on each element of volume of the medium, the three-dimensional density of whose distribution is given, Eq. (14.79) is replaced by the relation
{)2p
at 2
=
a
2 Llp + f
corresponding to Eq. ( 14.77) , which for f ¢. 0 is called the
wave equation.
( 14.80 )
inhomogeneous
14.4.5 Problems and Exercises 1. Suppose the velocity field v of a moving continuous medium is a potential field. Show that if the medium is incompressible, the potential
a ) Show that Euler's equation ( 14.76) can b e rewritten as
�: + grad (� v2 ) - v x curl v = F - � grad p
(see Problem 1 of Sect. 14. 1 ) . b ) Verify o n the basis of the equation of a ) that a n irrotational flow (curl v = 0 ) of a homogeneous incompressible liquid can occur only i n a potential field F . c ) I t turns out (Lagrange's theorem) that i f at some instant the flow i n a po tential field F = grad U is irrotational, then it always has been and always will be irrotational. Such a flow consequently is at least locally a potential flow, that is, v = grad
)
(
-
-
310
14 Elements of Vector Analysis and Field Theory
3. A flow whose velocity field has the form v = ( Vx , Vy , 0) is naturally called plane parallel or simply a planar flow. a) Show that the conditions div v = 0, curl v = 0 for a flow to be incompressible and irrotational have the following forms: avx _ avy = 0 .
ay
ax
b) Show that these equations at least locally guarantee the existence of functions
'1/J(x, y) and r.p(x, y) such that ( -vy , vx ) = grad 'I/J and (vx , Vy ) = grad r.p. c) Verify that the level curves r.p = c 1 and '1/J = c2 of these functions are or thogonal and show that in the steady-state flow the curves '1/J = coincide with the trajectories of the moving particles of the medium. It is for that reason that the function '1/J is called the current function, in contrast to the function r.p, which is the c
velocity potential. d) Show, assuming that the functions r.p and '1/J are sufficiently smooth, that they are both harmonic functions and satisfy the Cauchy-Riemann equations:
ar.p = a'!jJ ar.p ax ay ' ay Harmonic functions satisfying the Cauchy-Riemann equations are called conjugate harmonic functions. e) Verify that the function f(z) = (r.p + i'!jJ) (x, y) , where z = x + iy, is a dif ferentiable function of the complex variable z. This determines the connection of the planar problems of hydrodynamics with the theory of functions of a complex variable.
4. Consider the elementary version � = a 2 � of the wave equation (14.79) . This is the case of a plane wave in which the pressure depends only on the x-coordinate of the point (x, y, z) of space. a) By making the change of variable u = x - at, v = x + at, reduce this equation to the form aO:Cv = 0 and show that the general form of the solution of the original equation is p = f(x + at) + g(x - at) , where f and g are arbitrary functions of class
c< 2 l .
b) Interpret the solution just obtained as two waves J(x) and g(x) propagating left and right along the x-axis with velocity a. c) Assuming that the quantity a is the velocity of propagation of a dis turbance even in the general case (14. 79) , and taking account of the relation -1 12 , find, following Newton, the velocity CN of sound in air, assum a = ( '1/J'(po) ) ing that the temperature in an acoustic wave is constant, that is, assuming that the process of acoustic oscillation is isothermic. (The equation of state is p = ]ff , R = 8.31 d J 1 is the universal gas constant, and 1-£ = 28.8 __£_ 1 is the molecular eg·mo e mo e weight of air. Carry out the computation for air at a temperature of 0° C, that is, T = 273 K. Newton found that CN = 280 m/s.) d) Assuming that the process of acoustic vibrations is adiabatic, find, following Laplace, the velocity CL of sound in air, and thereby sharpen Newton's result CN . (In an adiabatic process p = cp"� . This is Poisson's formula from Problem 6 of
311
14.4 Examples of Applications
Sect. 13. 1. Show that i f CN = � ' then C L = 'Y � · For air "( � 1 . 4 . Laplace found CL = 330 m/s, which is in excellent agreement with experiment.) Using the scalar and vector potentials one can reduce the Maxwell equations (( 14.12) of Sect. 14. 1) to the wave equation (more precisely, to several wave equations of the same type) . By solving this problem, you will verify this statement. a) It follows from the equation V' · B = 0 that at least locally B = V' x A, where A is the vector potential of the field B . b ) Knowing that B = V' x A, show that the equation V' x E = �� implies that at least locally there exists a scalar function <.p such that E = - V' <.p �� c) Verify that the fields E = -V' <.p - �� and B = V' x A do not change if instead of <.p and A we take another pair of potentials cp and A such that cp = <.p � and A = A + V'"¢, where "1/J is an arbitrary function of class c <2 l . d) The equation V' · E = 1'o implies the first relation -V' 2 c.p - -ft Y' · A = /eo between the potentials <.p and A. e) The equation c2 V' x B - ft = a!o implies the second relation 5.
-
-
•
-
a j o2 A = 2 V' 2 A + c2 V'(V' · A) + EJt V'c.p + EJt 2 co between the potentials <.p and A. f) Using c) , show that by solving the auxiliary wave equation .d"¢ + f = � � ' without changing the fields E and B one can choose the potentials <.p and A so that they satisfy the additional (so-called gauge) condition V' A = � � . g) Show that if the potentials <.p and A are chosen as stated in f) , then the required inhomogeneous wave equations -c
·
-
for the potentials c.p and A follow from d) and e) . By finding <.p and A, we also find the fields E = V'c.p, B = V' x A.
1 5 *Integrat ion of D ifferent ial Forms
on Manifolds
1 5 . 1 A Brief Review of Linear Algebra 15. 1 . 1 The Algebra of Forms Let X be a vector space and p k : X k ---+ � a real-valued k-form on X . If e1 , . . . , en is a basis in X and x 1 = x i ' e i, , . . . , X k = x i k e i k is the expansion of the vectors x 1 , . . . , X k E X with respect to this basis, then by the linearity of p k , with respect to each argument (15.1) Thus, after a basis i s given i n X , one can identify the k-form p k : X k ---+ � with the set of numbers a i , ... i k = F k (e i , , . . . , e i k ). lf e1 , . . . , en is another basis in X and a1, ... Jk = F k (eh , . . . , eJk ) , then, setting e1 = cj e i , j = 1 , . . . , n, we find the ( tensor ) law ( 15.2) for transformation of the number sets a i, ... i k ' aj, ... j k corresponding to the same form p k . The set Fk := {F k : X k ---+ �} of k-forms on a vector space X is itself a vector space relative to the standard operations
(F1k + F�)(x) := Ff(x) + F�(x) , ( >-. Fk )(x) := >-. F k (x) of addition of k-forms and multiplication of a k-form by a scalar. For forms p k and F 1 of arbitrary degrees k and l the following product operation ® is defined:
( 15.3) ( 15.4)
tensor
(F k ® F1 )(x 1 , . . . , x k , X k+l , · · · , X k+ l ) := ( 15.5) = Fk (x 1 , . . . , x k )F1 (x k+l , · · · , x k+ l ) . Thus p k ® F 1 is a form p k+ l of degree k + l. The following relations are
obvious:
15 *Integration of Differential Forms on Manifolds
314
( >-. F k ) F1 = >-. (F k F1 ) , (15.6) (15.7) (Ff + F�) 0 F1 = Ff 0 F1 + F� 0 F1 , k k k (15.8) F ( F{ + F�) F F{ + F F� , k m k m 1 1 F ) F = F ( 15.9) (F (F F ) . Thus the set F = {Fk } of forms on the vector space X is a graded algebra F Q9 Fk with respect to these operations, in which the vectork space operations are carried out inside each space Fk occurring in the direct sum, and if F k E Fk , F 1 E F1 , then F k F 1 E Fk +l . Example 1 . Let X* be the dual space to X ( consisting of the linear function als on X) and e 1 , . . . , e n the basis of X* dual to the basis e 1 , . . . , e n in X, that is, e i (e1 ) = Jj . Since e i (x) e i ( xi e 1 ) xi i ( e 1 ) = xi Jj = x i , taking account of (15. 1) and ( 1 5.9) , we can write any k-form F k : X k JR. as 181
181
181
=
181
181
181
181
181
181
=
181
=
=
e
----t
(15. 10)
1 5 . 1 . 2 The Algebra of Skew-symmetric Forms Let us now consider the space w E f? k if the equality
[l k of skew-symmetric forms in Fk , that is,
w(x 1 , . . . , xi , · · · , xj , · · · , x k ) = -w(x 1 , , x1 , . . . , xi , · · · , xk ) holds for any distinct indices i , j E { 1 , . . . , n}. From any form F k E Fk one can obtain a skew-symmetric form using the operation A : F [l k of alternation, defined by the relation (15. 1 1 ) AFk ( XI ' . . . ' X k ) . - -k.1, F k ( Xi, ' . . . ' Xi k ) u l . ... .k. i k ' .
----t
•
.
s: i ,
. _
where 1,
i, ...ki k Jl...
-1, 0,
( i11 ·. ·. ·. ikk ) is even , if the permutation ( i11 ·. .· .· ikk ) is odd , if ( i � : : : �k ) is not a permutation .
i f the permutation
If F k is a skew-symmetric form, then, as one can see from (15. 1 1 ) , AF k = k F . Thus A(AF k ) AFk and Aw w if w E il k . Hence A : Fk f?k is a mapping of Fk onto [l k . =
=
Comparing Definitions ( 1 5.3) , ( 15.4) , and ( 1 5. 1 1 ) , we obtain
A(Ff + F�) AFf + AF� , A(;>..F k ) >-.AF k . =
=
----t
( 15. 12) (15. 13)
315
1 5 . 1 A Brief Review of Linear Algebra
Example 2 .
(15. 10) that
Taking account of relations (15. 12) and ( 1 5 . 1 3) , we find by
AF k =
eik ) A(ei ' so that it is of interest to find A( e i ' ® . . . ® e i k ) .
a
·
r0-. \CY
·
� l · · · �k
•
•
•
r0-. iCY
'
From Definition ( 1 5. 1 1 ) , taking account of the relation e i (x) = x i , we find A( eJ l '
·
- k!
·
x11 '
·
.
.
.
·
xj1 k
-
(15. 14)
x1k'
x1kk
The tensor product of skew-symmetric forms is in general not skew symmetric, so that we introduce the following exterior product in the class of skew-symmetric forms:
+ l)! A(w k ® w l . w k w l := (kk!l! (15. 15) Thus w k w 1 is a skew-symmetric form w k+ l of degree k + l. Example 3 . Based on the result (15. 14) of Example 2, we find by Definition 1\
1\
)
(15. 15) that
(15. 16)
Example 4. Using the equality obtained in Example 3, relation ( 1 5 . 1 4) , and the definitions ( 1 5. 1 1 ) and ( 1 5 . 1 5 ) , we can write e i i ( e i2 e i3 )(x 1 , x 2 , x3 ) = = ( 1 1+. 2 2). 1· A ( e '' ® (e '2 ® e ' 3 ) ) (x 1 , x 2 , x 3 ) = 1\
1\
1 1
. . .
A similar computation shows that
e i ' ( e i 2 e i3 ) (e i ' e i 2 ) e i 3 1\
1\
=
1\
1\
.
( 1 5 . 1 7)
316
15 *Integration of Differential Forms on Manifolds
Using the expansion of the determinant along a column, we conclude by induction that ei l
1\ · · · 1\
e ik (
x 1 , . . . , X k ) --
(15. 18)
and, as one can see from the computations just carried out, formula (15. 18) holds for any 1-forms e i 1 , . . . , e ik (not just the basis forms of the space X * ) . Taking the properties of the tensor product and the alternation operation listed above into account , we obtain the following properties of the exterior product of skew-symmetric forms: (15. 19) (wt + w�) 1\ w 1 = wt 1\ w 1 + w� 1\ w 1 , k k 1\ 1\ 1 1 (15.20) (.Xw ) w = .X(w w ) , w k 1\ w 1 = ( -1) k 1 w 1 1\ w k , (15.21) (w k 1\ w 1 ) 1\ w m = w k 1\ (w 1 1\ w m ) . (15.22) Proof. Equalities (15. 19) and (15.20) follow obviously from relations ( 15.6) ( 1 5 .8) and ( 1 5 . 12) and ( 1 5 . 1 3 ) . From relations (15. 10)-( 15. 14) and (15. 17) , for every skew-symmetric form w = a; 1 . . . ik e i 1 ® · · · ® e ik we obtain w = Aw = a'· l · · · '· k A ( e '. 1 ® · · · ® e '. k ) = -1 a'· l · · · ' k e '. 1 1\ · · · 1\ e'. k .
k!
·
Using the equalities (15. 19) and (15.20) we see that it now suffices to verify ( 1 5 . 2 1 ) and ( 15.22) for the forms e i 1 1\ · · . 1\ e ik . Associativity (15.22) for such forms was already established by (15.17). We now obtain (15.21) immediately from (15. 18) and the properties of determinants for these particular forms. 0 Along the way we have shown that every form w E fl k can be represented as W= (15.23) l "' i 1 < i 2 < · · - < ik "'n of skew-symmetric forms on the vector space =
Thus, the set fl { fl k } X relative to the linear vector-space operations ( 15.3) and ( 15.4) and the dim X
exterior multiplication ( 1 5 . 1 5 ) is a graded algebra fl = E9
fl k . The vectorspace operations on fl are carried out inside each vector space fl k , and if w k E fl k , w 1 E fl1 , then w k w 1 E fl k + l . In the direct sum E9 fl k the summation runs from zero the dimension of the space X , since the skew-symmetric forms w k : X k JR. of degree larger 1\
k =O
�
than the dimension of X are necessarily identically zero, as one can see by ( 15.21) (or from relations (15.23) and ( 15.8) ) .
15.1 A Brief Review of Linear Algebra
317
15.1.3 Linear Mappings o f Vector Spaces and the Adjoint Mappings of the Conjugate Spaces Let X and Y be vector spaces over the field JR. of real numbers (or any other field, so long as it is the same field for both X and Y ) , and let l : X -+ Y be a linear mapping of X into Y , that is, for every x, x 1 , x 2 E X and every
,\ E JR.,
+
(15 .24) l(x 1 x 2 ) = l(x i ) l(x 2 ) and l(-\x) = ,\l(x) . A linear mapping l : X -+ Y naturally generates its adjoint mapping l* : Fy -+ Fx from the set of linear functionals on Y (Fy) into the analogous set Fx. If F� is a k-form on Y, then by definition
+
( 1 5.25) It can be seen by (15.24) and (15.25) that l* F� is a k-form F� on X, that is, l* (FM C F� . Moreover, if the form F� was skew-symmetric, then ( l* FM = F� is also skew-symmetric, that is, l* ( !!� ) C !!!\: . Inside each vector space F� and !!� the mapping l* is obviously linear, that is,
+
+
l* (Ff F�) = l* Ff l* F�
and
l*(,\F k ) = -\l* p k .
( 1 5. 26)
Now comparing definition ( 1 5.25) with the definitions ( 1 5.5) , ( 1 5. 1 1 ) , and (15. 15) of the tensor product , alternation, and exterior product of forms, we conclude that ( 1 5.27) l*(FP ® Fq ) = (l *FP ) ® (l*Fq ) , ( 1 5.28) l*(AFP ) = A(l* FP ) ' P ( 1 5 . 29) l*(w w q ) = (l *wP ) (l*wq) . Example 5 . Let e 1 , . . . , e m be a basis in X , e1 , . . . , en a basis in Y , and l(e i ) = d; ej , i E { 1 , . . . , m} , j E { 1 , . . . , n} . If the k-form F� has the coordinate representation Fyk ( Y 1 , . . . , yk ) -- b)l ...)k Y)l 1 · . . . yJkk in the basis e1 , . . . , en , where bj 1 ... )k = F�( ej 1 , , e)k ), then (l*Fyk )( X 1 , . . . , X k ) = ai 1 ... i k Xil1 . . . · Xikk , where ai 1 i k = bj 1 ...j k c1,� . . . · c1: , since ai 1 . i k =: (l* F� )(ei1 , , e i k ) := F� (le i 1 , , le i k ) = = F� ( c;� e1 1 ' . . . , c1:e1 k ) = F� ( ej p . . . , ej k )d;� . . . · c;: . Example 6. Let e 1 , . . . , e m and e1 , . . . , e'' be the bases of the conjugate spaces X* and Y* dual to the bases in Example 5. Under the hypotheses of Example 5 we obtain (l* el )(x) = (l*eJ )(xi e i ) = el (x i le i ) = xi cJ (c� ek ) = = xi c� eJ (ek ) = xi c� J� = q xi = q e i (x) . 1\
1\
·
•
•
•
·
·
• • •
•
•
•
•
•
•
·
1 5 *Integration of Differential Forms on Manifolds
318
Example 7.
Retaining the notation of Example 6 and taking account of rela tions ( 15.22) and ( 15.29) , we now obtain
l* (eh
l* eJ k e:i k ) l* e:il ( d'l. kk e i k ) 2'1. 11 . . . . . d'l. kk e i 1 ( dl1-1 e i l ) dl'l. l
1\ . . . 1\
=
1\ . . . 1\
1\ . . . 1\
=
=
1\ . . . 1\
d'l. lk
eik
=
Keeping Eq. ( 1 5. 26) in mind, we can conclude from this that
l�l�ij11 <···
bJ· l ··· J. k
1 5 . 1 .4 Problems and Exercises 1.
Show by examples that in general a) p k 0 p l "I= p l 0 F k ; b) A(F k ® F 1 ) "/= AF k ® AF 1 ; c) if F k , F 1 E fl, then it is not always true that F k ® F 1
E
fl.
2. a) Show that if e 1 , . . . , e n is a basis of the vector space X and the linear func tionals e 1 , , en on X (that is elements of the conjugate space X*) are such that ei (ei ) = 8f , then e1, . . . , en is a basis in X* . b) Verify that one can always form a basis of the space Fk = Fk ( X ) from k-forms of the form ei 1 ® · · · ® ei k , and find the dimension (dim Fk ) of this space, knowing that dim X = n . c) Verify that one can always form a basis of the space fl k from forms of the form ei 1 1\ · · · 1\ ei k , and find dim fl k knowing that dim X = n . • • •
d) Show that if f1 = E9 fl k , then dim f1 = 2n.
k=n k =O=
15.1 A Brief Review of Linear Algebra
319
The exterior {Grassmann/ algebra G over a vector space X and a field P (usually denoted 1\ (X) in agreement with the symbol 1\ for the multiplication operation in G) is defined as the associative algebra with identity 1 having the following properties: 1 ° G is generated by the identity and X, that is, any subalgebra of G containing 1 and X is equal to G; 2 ° x 1\ x = 0 for every vector x E X; 3 ° dim G = 2 d i m X . a) Show that if e 1 , . . . , e n is a basis in X , then the set 1 , e 1 , . . . , e n , e 1 1\ e 2 , . . . , e n - 1 1\ e n , . . . , e 1 1\ · · · 1\ e n of elements of G of the form e;1 1\ · · · 1\ e; k = e1 , where I = { i 1 < · · · < i k } C { 1 , 2, . . . , n} , forms a basis in G. b) Starting from the result in a) one can carry out the following formal con struction of the algebra G = 1\(X). For the subsets I = {i 1 , . . . , i k } of {1, 2, . . . , n} shown in a) we form the formal elements e1 , (by identifying € { i } with e ; , and ee; with 1 ) , which we take as a basis of the vector space G over the field P. We define multiplication in G by the formula 3.
(� ) (� ) a1 e 1
bJeJ
=
f.;
ai bJ E (I, J ) e w J
,
where E(I, J) = sgn IT (j i) . Verify that the Grassmann algebra /\ (X) is iEl, jEJ obtained in this way. c) Prove the uniqueness (up to isomorphism) of the algebra /\(X) . k=n d) Show that the algebra /\ (X) is graded: /\(X) = ffi 1\ k (X) , where 1\ k (X) k =O is the linear span of the elements of the form e;1 1\ · · · I\ e; k ; here if a E /\P(X) and p b E N (X) , then a 1\ b E 1\ +q (X) . Verify that a 1\ b = ( - l ) Pq b 1\ a. -
a) Let A : X -+ Y be a linear mapping of X into Y. Show that there exists a unique homomorphism 1\ (A) : /\(X) -+ 1\(Y) from /\( X) into 1\(Y) that agrees with A on the subspace /\' (X) C 1\ (X) identified with X. b ) Show that the homomorphism A (A) maps A k (X) into A k (Y). The restriction of 1\ (A) to 1\ k (X) is denoted by 1\ k (A) . c) Let { e; : i = 1 , . . , m } be a basis in X and { e i : j = 1 , . . . , n} a basis in Y, and let the matrix ( aj ) correspond to the operator A in these bases. Show that if { e1 : I C { 1 , . . . , m } } , { e J : J C { 1 , . . . , n}} are the corresponding bases of the spaces /\ (X) and 1\ (Y) , then the matrix of the operator 1\ k (A) has the form a � = det(a; ) , i E I, j E J, where card ! = card J = k . d) Verify that if A : X -+ Y, B : Y -+ Z are linear operators, then the equality 1\ ( B o A) = 1\ ( B ) o 1\ (A) holds. 4.
.
1 H. Grassmann (1809-1877) - German mathematician, physicist and philologist; in particular, he created the first systematic theory of multidimensional and Euclidean vector spaces and gave the definition of the inner product of vectors.
320
15 *Integration of Differential Forms on Manifolds
1 5 . 2 Manifolds 1 5 . 2 . 1 Definition of a Manifold Definition 1. A Hausdorff topological space whose topology has a count able base 2 is called an n-dimensional manifold if each of its points has a neighborhood U homeomorphic either to all of JR. n or to the half-space
Hn = {x E 1Rn l x 1 :::; o}.
cp : JR.n U M (or Hn U M ) that realizes the homeomorphism of Definition 1 is a local chart of the manifold M, JR.n (or H n ) is called the parameter domain, and U the range of the chart Definition 2 . A mapping
-+
C
cp
:
-+
C
on the manifold M.
A local chart endows each point x E U with the coordinates of the point cp - 1 ( x) E JR.n corresponding to it. Thus, a local coordinate system is introduced in the region U; for that reason the mapping cp, or, in more ex panded notation, the pair (U, cp) is a map of the region U in the ordinary meaning of the term.
t
=
Definition 3. A set of charts whose ranges taken together cover the entire manifold is called an atlas of the manifold. Example 1 . The sphere 82 = { x E JR3 I I x l = 1 } is a two-dimensional mani fold. If we interpret 8 2 as the surface of the Earth, then an atlas of geograph ical maps will be an atlas of the manifold 8 2 . The one-dimensional sphere 8 1 = {x E JR 2 I Ix l = 1 } - a circle in JR 2 is obviously a one-dimensional manifold. In general, the sphere 8n = {X E JR.n + 1 l l x l = 1 } is an n-dimensional manifold. (See Sect. 1 2 . 1 . )
Remark 1 . The object (the manifold M) introduced by Definition 1 obviously does not change if we replace JR. n and H n by any parameter domains in JR. n homeomorphic to them. For example, such a domain might be the open cube In = {x E JR.n l 0 < xi < 1 , i = 1 , . . . , n} and the cube with a face attached Jn = {x E JR.n l o < x 1 :::; 1 , 0 < xi < 1 , i = 2, . . . , n} . Such standard parameter domains are used quite often. It is also not difficult to verify that the object introduced by Definition 1 does not change if we require only that each point x E M have a neighborhood U in M homeomorphic to some open subset of the half-space Hn .
Example 2. If X is an m-dimensional manifold with an atlas of charts {(U0, cp0)} and Y is an n-dimensional manifold with atlas {(V,B , �,B)}, then X x Y can be regarded as an ( m + n )-dimensional manifold with the atlas {(Wa,B, Xa,B) }, where Wa,B = Ua X V,13 and the mapping Xa,B = (cpa, �,B) maps the direct product of the domains of definition of 'Pa and �/3 into Wa ,B · 2 See Sect. 9.2 and also Remarks 2 and 3 in the present section.
15.2 Manifolds
321
In particular, the two-dimensional torus = S 1 x S 1 ( Fig. 12. 1 ) or the 1 1 n-dimensional torus Tn = 8 X · · · X 8 is a manifold of the corresponding '--v----" n factors dimension.
T2
If the ranges Ui and Uj of two charts ( Ui , 'P i ) and ( Uj , r.pj ) of a manifold M intersect, that is, Ui n Uj =f. 0, mutually inverse homeomorphisms 'Pij : Iij -+ Ij i and1 '{!j i : Ij i -+ Iij naturally arise between the sets Iij = 1'Pi 1 (Uj) and Ij i = r.pj (Ui ) · These homeomorphisms are given by 'f!i j = r.pj o 'P i l rij and '{!j i = r.pi 1 o 'Pj l rJ • . These homeomorphisms are often called changes of coordinates, since they effect a transition from one local coordinate system to another system of the same kind in their common range Ui n Uj ( Fig. 15. 1 ) .
Fig. 1 5 . 1 .
Definition 4. The number n i n Definition 1 is the
M and is usually denoted dim M.
dimension of the manifold
Definition 5 . If a point r.p - 1 (x) on the boundary 8Hn of the half-space Hn corresponds to a point x E U under the homeomorphism r.p : Hn -+ U, then x is called a boundary point of the manifold M ( and of the neighborhood U). The set of all boundary points of a manifold M is called the boundary of this manifold and is usually denoted 8M.
By the topological invariance of interior points ( Brouwer ' s theorem3 ) the concepts of dimension and boundary point of a manifold are unambiguously 3 This theorem asserts that under a homeomorphism r.p : E -t r.p(E) of a set E C IR. n onto a set r.p(E) C IR. n the interior points of E map to interior points of r.p(E) .
322
15 *Integration of Differential Forms on Manifolds
defined, that is, independent of the particular local charts used in Defini tions 4 and 5. We have not proved Brouwer ' s theorem, but the invariance of interior points under diffeomorphisms is well-known to us (a consequence of the inverse function theorem) . Since it is diffeomorphisms that we shall be dealing with, we shall not digress here to discuss Brouwer ' s theorem.
Example 3. The closed ball If' = {x E lRn [ l x l :S: I } or, as we say, the n-dimensional disk, is an n-dimensional manifold whose boundary is the (n - I )-dimensional sphere sn - l = {x E lRn [ l x l = I } . Remark 2. A manifold M having a non-empty set of boundary points is usu ally called a manifold with boundary, the term manifold (in the proper sense
of the term) being reserved for manifolds without boundary. In Definition I these cases are not distinguished.
The boundary oM of an n-dimensional manifold with boundary M is an (n - I)-dimensional manifold without boundary. Proof. Indeed, oHn = JRn - l , and the restriction to oH n of a chart of the form 'P i H n -+ Ui belonging to an atlas of M generates an atlas of oM. D Proposition 1 .
:
Example 4- Consider the planar double pendulum (Fig. I5.2) with arm a shorter than arm b, both being free to oscillate, except that the oscillations of b are limited in range by barriers. The configuration of such a system is characterized at each instant of time by the two angles o: and (3. If there were no constraints, the configuration space of the double pendulum could be identified with the two-dimensional torus = s; X s� . Under these constraints, the configuration space of the double pendulum is parametrized by the points of the cylinder s; X I� , where s; is the circle, corresponding to all possible positions of the arm a , and I� = {(3 E 1R [ I f3 1 ::::;
T2
Fig. 1 5 . 2 .
15.2 Manifolds
323
,1} is the interval within which the angle (3 may vary, characterizing the position of the arm b. In this case we obtain a manifold with boundary. The boundary of this manifold consists of the two circles S}, x { - ,1} and S}, x { ,1 } , which are the products of the circle S}, and the endpoints { -Ll} and {Ll} of the interval
IJ . Remark 3.
It can be seen from Example 4 just considered that coordinates sometimes arise naturally on M ( a and (3 in this example) , and they them selves induce a topology on M. Hence, in Definition 1 of a manifold, it is not always necessary to require in advance that M have a topology. The essence of the concept of a manifold is that the points of some set M can be parametrized by the points of a set of subdomains of JR n . A natural connec tion then arises between the coordinate systems that thereby arise on parts of M, expressed in the mappings of the corresponding domains of JR n . Hence we can assume that M is obtained from a collection of domains of JR n by exhibiting some rule for identifying their points or, figuratively speaking, ex hibiting a rule for gluing them together. Thus defining a manifold essentially means giving a set of subdomains of JR n and a rule of correspondence for the points of these subdomains. We shall not take the time to make this any more precise by formalizing the concept of gluing or identifying points, introducing a topology on M, and the like. Definition 6. A manifold is compact (resp. connected) as a topological space.
connected) if it is compact (resp.
The manifolds considered in Examples 1-4 are compact and connected. The boundary of the cylinder s; X IJ in Example 4 consists of two indepen dent circles and is a one-dimensional compact , but not connected, manifold. The boundary sn - l = 8B n of the n-dimensional disk of Example 3 is a com pact manifold, which is connected for n > 1 and disconnected (it consists of two points) if n = 1 . The space JR n itself i s obviously a connected noncompact mani fold without boundary, and the half-space H n provides the simplest example of a connected noncompact manifold with boundary. (In both cases the at las can be taken to consist of the single chart corresponding to the identity mapping. )
Example 5 .
Proposition 2.
If a manifold M is connected, it is path connected.
x0
Proof. After fixing a point E M, consider the set Exo of points of M that can be joined to by a path in M. The set Ex o , as one can easily verify from the definition of a manifold, is both open and closed in M. But that means that Ex 0 = M. D
x0
324
15 *Integration of Differential Forms on Manifolds
If to each real n x n matrix we assign the point of � n 2 whose coordinates are obtained by writing out the elements of the matrix in some fixed order, then the group GL(n, �) of nonsingular n x n matrices becomes a manifold of dimension n 2 . This manifold is noncompact (the elements of the matrices are not bounded) and nonconnected. This last fact follows from the fact that GL(n, �) contains matrices with both positive and negative determinants. The points of GL(n , �) corresponding to two such matrices cannot be joined by a path. (On such a path there would have to be a point corresponding to a matrix whose determinant is zero. )
Example 6.
The group 80 ( 2 , �) of orthogonal mappings of the plane � 2 hav. determmant · · · mg equa 1 to 1 consists of matnces of t he c10rm cos. a sin a - sm a cos a and hence can be regarded as a manifold that is identified with the circle the domain of variation of the angular parameter a . Thus 80 (2, �) is a one dimensional compact connected manifold. If we also allow reflections about lines in the plane � 2 , we obtain the group 0 ( 2 , �) of all real orthogonal 2 x 2 matrices. It can be naturally identified with two different circles, correspond ing to matrices with determinants + 1 and - 1 respectively. That is, 0(2, �) is a one-dimensional compact, but not connected manifold.
(
Example 7.
)
Let a be a vector in � 2 and Ta the group of rigid motions of the plane generated by a. The elements of Ta are translations by vectors of the form na, where n E Z. Under the action of the elements of the group Ta each point of the plane is displaced to a point ) of the form + na. The set of all points to which a given point E � 2 passes under the action of the elements of this group of transformations is called its orbit. The property of points of � 2 of belonging to the same orbit is obviously an equivalence relation on � 2 , and the orbits are the equivalence classes of this relation. A domain in � 2 containing one point from each equivalence class is called a fundamental domain of this group of automorphisms (for a more precise statement see Problem 5d) ) . In the present case we can take as a fundamental domain a strip of width lal bounded by two parallel lines orthogonal to a. We need only take into account that these lines themselves are obtained from each other through translations by a and -a respectively. Inside a strip of width less than lal and orthogonal to a there are no equivalent points, so that all orbits having representatives in that strip are endowed uniquely with the coordinates of their representatives. Thus the quotient set � 2 /Ta consisting of orbits of the group Ta becomes a manifold. From what was said above about a fundamental domain, one can easily see that this manifold is homeomorphic to the cylinder obtained by gluing the boundary lines of a strip of width lal together at equivalent points.
Example 8.
x
Ta.h
g x
Now let a and b be a pair of orthogonal vectors of the plane � 2 the group of translations generated by these vectors. In this case a
Example 9.
and
x g(x
15.2 Manifolds
325
fundamental domain is the rectangle with sides a and b. Inside this rectangle the only equivalent points are those that lie on opposite sides. After gluing the sides of this fundamental rectangle together, we verify that the resulting manifold IR 2 /Ta,b is homeomorphic to the two-dimensional torus.
Example 10. Now consider the group Ga , b of rigid motions of the plane IR 2 generated by the transformations a(x, y) = (x+ 1 , 1 -y) and b(x, y) = (x, y+ 1 ) . A fundamental domain for the group Ga , b i s the unit square whose hori zontal sides are identified at points lying on the same vertical line, but whose vertical sides are identified at points symmetric about the center. Thus the resulting manifold IR 2 /G a , b turns out to be homeomorphic to the Klein bottle (see Sect. 1 2 . 1 ) .
We shall not take time t o discuss here the useful and important examples studied in Sect. 12. 1 . 1 5 . 2 . 2 Smooth Manifolds and Smooth Mappings Definition 7. An atlas of a manifold is
smooth (of class C ( k ) or analytic)
if all the coordinate-changing functions for the atlas are smooth mappings ( diffeomorphisms) of the corresponding smoothness class. Two atlases of a given smoothness (the same smoothness for both) are
equivalent if their union is an atlas of this smoothness. Example 11. An atlas consisting of a single chart can be regarded as having1 any desired smoothness. Consider in this connection the atlas on the line IR generated by the identity mapping IR 1 x tp(x) = x E IR 1 , and a second atlas - generated by any strictly monotonic function IR 1 x ip( x) E IR 1 , mapping IR 1 onto IR 1 . The union of these atlases is an atlas having smoothness equal to the smaller of the smoothnesses of ;:p and ;p- 1 . In particular, if ip( x) = x 3 , then the atlas consisting of the two charts what has just been said, {x, x3 } is not smooth, since ;p- 1 (x) = x 1 1 3 . Using we can construct infinitely smooth atlases in IR 1 whose union is an atlas of a preassigned smoothness class C ( k ) . Definition 8. A smooth manifold (of class C ( k ) or analytic) is a manifold M with an equivalence class of atlases of the given smoothness. After this definition the following terminology is comprehensible: topolog ical manifold (of class c< 0 l ) , C ( k ) _manifold, analytic manifold. To give the entire equivalence class of atlases of a given smoothness on a manifold M it suffices to give any atlas A of this equivalence class. Thus we can assume that a smooth manifold is a pair (M, A), where M is a manifold and A an atlas of the given smoothness on M. 3
H
3
H
326
15 *Integration of Differential Forms on Manifolds
The set of equivalent atlases of a given smoothness on a manifold is often called a structure of this smoothness on the manifold. There may be differ ent smooth structures of even the same smoothness on a given topological manifold ( see Example 11 and Problem 3) . Let us consider some more examples in which our main attention is di rected to the smoothness of the coordinate changes.
Example 12. The one-dimensional manifold !RlP' 1 called the real projective line, is the pencil of lines in IR 2 passing through the origin, with the natural
notion of distance between two lines ( measured, for example, by the magni tude of the smaller angle between them ) . Each line of the pencil is uniquely determined by a nonzero direction vector ( x 1 , x 2 ), and two such vectors give the same line if and only if they are collinear. Hence !RlP' 1 can be regarded as a set of equivalence classes of ordered pairs (x 1 , x 2 ) of real numbers. Here at least one of the numbers in the pair must be nonzero, and two pairs are considered equivalent ( identified ) if they are proportional. The pairs (x 1 , x 2 ) are usually called homogeneous coordinates on !RlP' 1 . Using the interpretation of !RlP' 1 in homogeneous coordinates, it is easy to construct an atlas of two charts on !RlP' 1 . Let Ui , i = 1, 2 , be the lines ( classes of pairs (x 1 , x 2 )) in !RlP' 1 for which x i # 0. To each point ( line ) p E U1 there corresponds a unique 2 2 pair ( 1, � 1 ) determined by the number t 21 = � 1 • Similarly the points of the 1 region U2 are in one-to-one correspondence with pairs of the form ( � , 1 ) I and are determined by the number t§ = � . Thus local coordinates arise in u1 and u2 , which obviously correspond to the topology introduced above on !RlP' 1 . In the common range U1 n U2 of these local charts the coordinates they introduce are connected by the relations t§ = (t?) - 1 and t i = (t§) - 1 , which shows that the atlas is not only C ( oo ) but even analytic. It is useful to keep in mind the following interpretation of the manifold !RlP' 1 . Each line of the original pencil of lines is completely determined by its intersection with the unit circle. But there are exactly two such points, diametrically opposite to each other. Lines are near if and only if the cor responding points of the circle are near. Hence !RlP' 1 can be interpreted as a circle with diametrically opposite points identified ( glued together ) . If we take only a semicircle, there is only one pair of identified points on it , the end-points. Gluing them together, we again obtain a topological circle. Thus !RlP' 1 is homeomorphic to the circle as a topological space.
Example 13. If we now consider the pencil of lines passing through the ori gin in JR 3 , or, what is the same, the set of equivalence classes of ordered triples of points (x 1 , x 2 , x 3 ) of real numbers that are not all three zero, we obtain the real projective plane !RlP' 2 . In the regions U1 , U2 , and U3 where x 1 # 0, x 2 # 0, x 3 # 0 respectively, we introduce local coordinate systems ( 1 ' �� ' �� ) = ( 1 ' ti ' t y ) ( t r' t i ) ' ( � ' 1 ' �� ) = ( t§ ' 1 ' t � ) ( t§ ' t � ) ' and ( �31 , �21 , 1 ) = (t 1 , t�, 1) (t§, t�), which are obviously connected by the relarv
rv
rv
327
15.2 Manifolds
tions ti = (tj) - I , ti = t1 (tk) - I , which apply in the common portions of the ranges of these charts. For example, the transition from ( ti , tr) to ( t� , t�) in the domain UI n U2 is given by the formulas
3
The Jacobian of this transformation is - ( ti) and since ti �� , it is defined and nonzero at points of the set UI n u2 under consideration. Thus 1RlP'2 is a two-dimensional manifold having an analytic atlas consist ing of three charts. By the same considerations as in Example 1 2 , where we studied the projective line lRlP' I , we can interpret the projective plane 1RlP' 2 as the two- dimensional sphere S2 c JR 2 with antipodal points identified, or as a hemi sphere, with diametrically opposite points of its boundary circle identified. Projecting the hemisphere into the plane, we obtain the possibility of inter preting 1RlP'2 as a (two-dimensional) disk with diametrically opposite points of its boundary circle identified. -
,
Example 14. The set of lines in the plane JR 2 can be parititioned into two sets: U, the nonvertical lines, and V, the nonhorizontal lines. Each line in U has an equation of the form y u x + u 2 , and hence is characterized by the coordinates ( U I , u 2 ) , while each line in V has an equation x V I Y + v 2 and is determined by coordinates (V I , v2 ) . For lines in the intersection U V we I I =
i
=
n
have the coordinate transformation V I = u1I , v 2 = -u 2 u1 and U I = v1 , u 2 = -v2 v1 I · Thus this set is endowed with an analytic atlas consisting of two charts. Every line in the plane has an equation ax+by+c = and is characterized by a triple of numbers (a, b, c), proportional triples defining the same line. For that reason, it might appear that we are again dealing with the projective plane 1RlP'2 considered in Example 13. However, whereas in 1RlP' 2 we admitted any triples of numbers not all zero, now we do not admit triples of the form c) where c -=f. A single point in 1RlP'2 corresponds to the set of all such triples. Hence the manifold obtained in our present example is homeomorphic to the one obtained from 1RlP' 2 by removing one point . If we interpret 1RlP' 2 as a disk with diametrically opposite points of the boundary circle identified, then, deleting the center of the circle, we obtain, up to homeomorphism, an annulus whose outer circle is glued together at diametrically opposite points. By a simple incision one can easily show that the result is none other than the familiar Mobius band. Definition 9. Let M and N be C (k Lmanifolds. A mapping f : M -+ N is l-smooth (a c U l_mapping) if the local coordinates of the point f(x) E N are c ( lLfunctions of the local coordinates of X E M.
0
(0, 0,
0.
This definition has an unambiguous meaning (one that is independent of the choice of local coordinates) if l :::; k.
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15 *Integration of Differential Forms on Manifolds
In particular, the smooth mappings of M into JR. 1 are smooth functions on M, and the smooth mappings of JR. 1 (or an interval of JR. 1 ) into M are smooth paths on M. Thus the degree of smoothness of a function f : M --+ N on a manifold M cannot exceed the degree of smoothness of the manifold itself. 1 5 . 2 . 3 Orientation of a Manifold and its Boundary Definition 10. Two charts of a smooth manifold are consistent if the tran sition from the local coordinates in one to the other in their common range is a diffeomorphism whose Jacobian is everywhere positive.
In particular, if the ranges of two local charts have empty intersection, they are considered consistent.
A of a smooth manifold (M, A) is an orienting atlas of M if it consists of pairwise consistent charts. Definition 1 2 . A manifold is orientable if it has an orienting atlas. Other wise it is nonorientable. Two orienting atlases of a manifold will be regarded as equivalent (in the Definition 1 1 . An atlas
sense of the question of orientation of the manifold considered just now) if their union is also an orienting atlas of the manifold. It is easy to see that this relation really is an equivalence relation. Definition 1 3 . An equivalence class of orienting atlases of a manifold in the relation just defined is called an orientation class of atlases of the manifold or an orientation of the manifold.
oriented manifold is a manifold with this class of orien tations of its atlases, that is, with a fixed orientation on the manifold.
Definition 14. An
Thus orienting the manifold means exhibiting (by some means or other) a certain orientation class of atlases on it. To do this, for example, it suffices to exhibit any specific orienting atlas from the orientation class. Various methods used in practice to define an orientation . of manifolds embedded in JR.n are described in Sects. 12.2 and 12.3.
A connected manfiold is either nonorientable or admits ex actly two orientations. Proof. Let A and A be two orienting atlases of the manifold M with diffeo morphic transitions from the local coordinates of charts of one to charts of the other. Assume that;_there is a point p0 E M and two charts of these atlases whose ranges Ui o and Ui o contain po ; and suppose the Jacobian of the change Proposition 3 .
of coordinates of the charts at points of the parameter space corresponding
329
15.2 Manifolds
Po
p
to the point is positive. We shall show that then for every point E M and any charts of the atlases A and A whose ranges contain the Jacobian of the coordinate transformation at corresponding coordinate points is also positive. We begin by making the obvious observation that if the Jacobian of the transformation is positive (resp. negative) at the point for any pair of charts containing in the atlases A and A, then it is positive (resp. negative) at for any such pair of charts, since inside each given atlas the coordinate trans formations occur with positive Jacobian, and the Jacobian of a composition of two mappings is the product of the Jacobians of the individual mappings. Now let E be the subset of M consisting of the points E M at which the coordinate transformations from the charts of one atlas to those of the other have positive Jacobian. The set E is nonempty, since E E . The set E_is open in M . Indeed, for every point E E there exist ranges Ui and Uj of certain charts of the atlases A and A containing The sets Ui and Uj are open in M, so that the set Ui n Uj is open in M. On the connected component of the set Ui n Ui containing which is open in Ui n Ui and in M, the Jacobian of the transformation cannot change sign without vanishing at some point. That is, in some neighborhood of the Jacobian remains positive, which proves that E is open. But E is also closed in M. This follows from the continuity of the Jacobian of a diffeomorphism and the fact that the Jacobian of a diffeomorphism never vanishes. Thus E is a non-empty open�losed subset of the connected set M. Hence E = M, and the atlases A and A define the same orientation on M. Replacing one coordinate, say t 1 by -t 1 in every chart of the atlas A, we obtain the orienting atlas - A belonging to a different orientation class. Since the Jacobians of the coordinate transformations from an arbitrary chart to the charts of A and - A have opposite signs, every atlas that orients M is equivalent either to A or to - A. D Definition 15. A finite sequence of charts of a given atlas will be called a chain of charts if the ranges of any pair of charts having adjacent indices have a non-empty intersection (Ui n Ui + 1 f. 0). Definition 16. A chain of charts is contradictory or disorienting if the Ja cobian of the coordinate transformation from each chart in the chain to the next is positive and the ranges of the first and last charts of the chain inter sect , but the coordinate transformation from the last to the first has negative Jacobian.
p
p
p
p
p
p0 p.
p
p,
p
manifold is orientable if and only if there does not exist a contradictory chain of charts on it. Proof Since every manifold decomposes into connected components whose
Proposition 4. A
orientations can be defined independently, it suffices to prove Proposition 4 for a connected manifold M.
330
15 *Integration of Differential Forms on Manifolds
N e c e s s i t y. Suppose the connected manifold M is orientable and A is an atlas defining an orientation. From what has been said and Proposition 3, every smooth local chart of the manifold M connected with the charts of the atlas A is either consistent with all the charts of A or consistent with all the charts of -A. This can easily be seen from Proposition 3 itself, if we resrict charts of A to the range of the chart we have taken, which can be regarded as a connected manifold oriented by one chart . It follows from this that there is no contradictory chain of charts on M. S u f f i c i en c y. It follows from Definition 1 that there exists an atlas on the manifold consisting of a finite or countable number of charts. We take such an atlas A and number its charts. Consider the chart (U1 , cp1) and any chart (Ui , 'P i ) such that U1 n Ui # 0 . Then the Jacobians of the coordinate transformations 'P l i and 'Pi l are either everywhere negative or everywhere positive in their domains of definition. The Jacobians cannot have values of different signs, since otherwise one could exhibit connected subsets U_ and u+ in ul uui where the Jacobian is negative and positive respectively, and the chain of charts (U1 , cp l ), (U+ , cp1), (Ui , 'P i ), (U_ , 'Pi ) would be contradictory. Thus, changing the sign of one coordinate if necessary in the chart ( Ui , 'Pi ), we could obtain a chart with the same range Ui and consistent with (U1 , cp1). After that procedure, two charts (Ui , 'P i ) and (Uj , 'Pj ) such that U1 n Ui # 0 , ul n uj "I 0 , ui n uj "I 0 are themselves consistent: otherwise we would have constructed a contradictory chain of three charts. Thus, all the charts of an atlas whose ranges intersect U1 can now be considered consistent with one another. Taking each of those charts now as the standard, one can adjust the charts of the atlas not covered in the first stage so that they are consistent . No contradictions arise when we do this, since by hypothesis, there are no contradictory chains on the manifold. Con tinuing this process and taking account of the connectedness of the manifold, we construct on it an atlas consisting of pairwise consistent charts, which proves the orientability of the manifold. D
This criterion for orientability of the manifold, like the considerations used in its proof, can be applied to the study of specific manifolds. Thus, the manifold IRIP' 1 studied in Example 1 2 is orientable. From the atlas shown there it is easy to obtain an orienting atlas of IRIP' 1 . To do this, it suffices to reverse the sign of the local coordinates of one of the two charts constructed there. However, the orientability of the projective line IRIP' 1 obviously also follows from the fact that the manifold IRIP' 1 is homoeomorphic to a circle. The projective plane IRIP'2 is nonorientable: every pair of charts in the atlas constructed in Example 13 is such that the coordinate transformations have domains of positivity and domains of negativity of the Jacobian. As we saw in the proof of Proposition 4, it follows from this that a contradictory chain of charts on IRIP'2 exists. For the same reason the manifold considered in Example 14 is nonori entable, which, as was noted, is homeomorphic to a Mobius band.
15.2 Manifolds
331
The boundary of an orientable smooth n-dimensional man ifold is an orientable (n - I)-dimensional manifold admitting a structure of the same smoothness as the original manifold. Proof. The proof of Proposition 5 is a verbatim repetition of the proof of the
Proposition 5 .
analogous Proposition 2 of Subsect. 12.3.2 for surfaces embedded in
JRn .
0
Definition 17. If A(M) {(Hn, 'Pi , Ui )} U { (JRn , tpj , Uj) } is an atlas that orients the manifold M, then the charts A (aM) {(JRn -1 , 'Pi itmn=�n-l , 8U; )} provide an orienting atlas for the boundary 8M of M. The orientation of the boundary defined by this atlas is called the
orientation of the boundary induced by the orientation of the manifold.
Important techniques for defining the orientation of a surface embedded in JRn and the induced orientation of its boundary, which are frequently used in practice, were described in detail in Sects. 12.2 and 12.3. 15.2.4 Partitions of Unity and the Realization of Manifolds as Surfaces in JRn
In this subsection we shall describe a special construction called a partition of unity. This construction is often the basic device for reducing global problems to local ones. Later on we shall demonstrate it in deriving Stokes ' formula on a manifold, but here we shall use the partition of unity to clarify the possibility of realizing any manifold as a surface in JRn of sufficiently high dimension.
One can construct a function f E c<= l (JR, JR) on lR such that f(x) 0 for l x l 2: 3, f(x) 1 for l x l :::; 1, and 0 < f(x) < 1 for 1 < l x l < 3 . Proof. We shall construct one such function using the familiar function -1 f x2 ) for x =f. 0 e< g(x) = 0 X = 0 '. Previously ( see Exercise 2 of Sect. 5.2) we verified that g E c<= l (JR, JR) by showing that g
{
=
c 10f
{ -(x-1 ) -2
In such a case the nonnegative function
G(x) = also belongs to
e
· e -(x+1 ) -2 for for 0
c<= l (JR, JR) , and along with it
F(x) belongs to this class, since
�
lxl < 1 , l x l 2: 1
the function
JCXJ G(t) dt ITG(t) t
-
F'(x) = G(x)
-
oo
d
j_[ G(t) dt .
15 *Integration of Differential Forms on Manifolds
332
The function F is strictly increasing on [ - 1 , 1] , = 1 for x � 1 . We can now take the required function t o be
F ( x)
F( x)
f(x) = F(x + 2) + F( -x - 2) - 1 .
=
0 for x :::; - 1 , and
D
Remark. If f : JR. -t JR. is the function constructed in the proof of the lemma, then the function defined in JR_n is such that () E C ( oo ) (JR.n , JR.), 0 :::; () :::; 1, at every point x E lR.n , ()(x) = 1 on the interval I(a) = {x E JR.n j l x i - ai l :::; 1 , i = 1 , . . . , n } , and the support supp () of the function () is contained in the interval l(a) = {x E JR. n I l x i - ai l :::; 3, i = 1 , . . . , n } . Definition 18. Let M be a cC k Lmanifold and X a subset of M. The system E = a E A} of functions E cC k l ( M, lR. is a
{ea,
ea
)
C ( k ) partition of unity
on X if 1 ° 0 :::; ea(x) :::; 1 for every function ea E E and every x E M; 2 ° each point x E X has a neighborhood U(x) in M such that all but a finite number of functions of E are identically zero on U(x); 3 ° L ea ( x) = 1 on X . ea E E
We remark that by condition 2 ° only a finite number of terms in this last sum are nonzero at each point x E X.
{o,13, j3 E B} be an open covering of X C M. We the partition of unity E = { ea, E A} on X is subordinate to the covering 0 if the support of each function in the system E is contained in at
Definition 19. Let 0 =
a
say that
least one of the sets of the system
0.
Let {(Ui , 'Pi ), i = 1, . . . , m } be a finite set of charts of some C ( k ) atlas of the manifold M, whose ranges Ui , i = 1, . . . , m, form a covering of a compact set K M . Then there exists a C ( k ) partition of unity on K subordinate to the covering {Ui , i = 1 , . . . , m } . Proof. For any point x0 E K we first carry out the following construction. We choose successively a domain Ui containing x0 corresponding to a chart 'P i : JR.n Ui (or 'Pi : H n Ui ), the point to = cpi 1 (xo) E JR.n (or Hn ), the function ()(t - t 0 ) (where ()(t) i s the function shown i n the remark t o the lemma ) , and the restriction ()t o of ()(t - t 0 ) to the parameter domain of 'P i · Let It o be the intersection of the unit cube centered at to E JR. n with the parameter domain of 'P i . Actually ()t o differs from () ( t - t0 ) and It o differs Proposition 6.
c
-t
-t
from the corresponding unit cube only when the parameter domain of the
15.2 Manifolds
333
chart 'Pi is the half-space H n . The open sets 'P i (It ) constructed at each point x E K and the point t = 'Pi 1 (x), taken for all admissible values of i = 1, 2, . . . , m, form an open covering of the compact set K. Let { 'PiJ ( lti ) , j = 1, 2, . . . , l} be a finite covering of K extracted from it. It is obvious that 'Pi)Itj ) c uij " We define on uij the function Oi (x) = Btj 0
0
·
e 2 (x) = 02 (x)(1 - 01 (x)), . . . , e 1 (x) = B; (x) (1 - B;_ l (x))
· . . . ·
(1 (x)) l - 01 form the required partition of unity. We shall verify only that I: e 1 (x) 1 j= l on K, since the system of functions { e 1 , . . . , e l } obviously satisfies the other conditions required of a partition of unity on K subordinate to the covering {Ui, , . . . , Ui } {Ui , i = 1, . . . , m } . But l 1 - I >J (x) = (1 - 01 (x)) (1 - 01 (x)) on K , j= l since each point x E K is covered by some set 'P ij Uti ) on which the corre sponding function oj is identically equal to 1. Corollary 1 . If M is a compact manifold and A a C ( k ) atlas on M, then there exists a finite partition of unity { e 1 , . . . , e 1 } on M subordinate to a covering of the manifold by the ranges of the charts of A. Proof. Since M i s compact , the atlas A can b e regarded as finite. We now have the hypotheses of Proposition 6, if we set K = M in it . 0 Corollary 2 . For every compact set K contained in a manifold M and every open set G M containing K, there exists a function f M lR with smoothness equal to that of the manifold and such that f ( x ) 1 on K and supp f G. Proof. Cover each point x E K by a neighborhood U(x) contained in G and inside the range of some chart of the manifold M. From the open covering {U(x), x E K} of the compact set K extract a finite covering, and construct a l partition of unity { e 1 , . . . , e 1 } on K subordinate to it . The function f = I: e i i= l =
c
1
· . . . ·
=
0
0
C
C
:
=
--+
is the one required. 0
Every (abstractly defined) compact smooth n-dimensional manifold M is diffeomorphic to some compact smooth surface contained in JRN of sufficiently large dimension N . Corollary 3.
334
15 *Integration of Differential Forms on Manifolds
Proof. So as not to complicate the idea of the proof with inessential details, we carry it out for the case of a compact manifold M without boundary. In that case there is a finite smooth atlas A = {'P i : I ---+ i , i = . . . , m} on M, where I is an open n-dimensional cube in JRn . We take a slightly smaller cube I' such that I' C I and the set = 'P i (I'), i = . . . , m} still forms a covering of M. Setting K = I' , G = I, and M = JRn in Corollary 2, we construct a function f E C ( 00) (lR n , JR.) such that f ( t) = for t E I' and supp f C I. We now consider the coordinate functions t} ( x), . . . , ti ( x) of the mappings rpi 1 : ---+ I, i = . . . , m, and use them to introduce the following function on M: for X E i ,
U 1,
{Uf
U1
1,
1
1,
i
=
1, . . . , m ;
U for X � Ui ,
k = 1, . . . , n .
3
f-t
At every point x E M the rank of the mapping M x y(x) = and equal to n. Indeed, (yf , . . . , y]', . . . y;,, . . 1. , y;::, ) (x) E JRm · n is maximal if x E Uf, then
o
o
f-t
o
3
o
f-t
=
o
=
o
0
For information on the general Whitney embedding theorem for an ar bitrary manifold as a surface in JRn the reader may consult the specialized geometric literature. 1 5 . 2 . 5 Problems and Exercises 1 . Verify that the object (a manifold) introduced by Definition 1 does not change if we require only that each point x E M have a neighborhood U(x) C M homeo morphic to an open subset of the half-space H n .
2. Show that a) the manifold GL(n, IR) of Example 6 is noncompact and has exactly two connected components; b) the manifold SO (n, IR) (see Example 7) is connected;
15.2 Manifolds
335
c) the manifold O ( n, IR. ) is compact and has exactly two connected components. 3. Let (M, A) and (M, A) be manifolds with smooth structures of the same degree of smoothness C ( k ) on them. The smooth manifolds (M, A) and (M, A) (smooth structures) are considered isomorphic if there exists a C ( k ) mapping f : M -+ M having a c < k l inverse f - 1 : M -+ M in the atlases A, A. a) Show that all structures of the same smoothness on IR. 1 are isomorphic. b) Verify the assertions made in Example 1 1 , and determine whether they con tradict a) . c) Show that on the circle 8 1 (the one-dimensional sphere) any two c < = l struc tures are isomorphic. We note that this assertion remains valid for spheres of dimen sion not larger than 6, but on 8 7 , as Milnor 4 has shown, there exist nonisomorphic c < = l structures.
4. Let S be a subset of an n-dimensional manifold M such that for every point xo E S there exists a chart x = rp(t) of the manifold M whose range U contains xo , and the k-dimensional surface defined by the relations t k + l = 0, . . . , t n = 0 corresponds to the set S n U in the parameter domain t = (t 1 , , t n ) of rp. In this case S is called a k-dimensional submanifold of M. a) Show that a k-dimensional manifold structure naturally arises on S, induced by the structure of M and having the same smoothness as the manifold M. b) Verify that the k-dimensional surfaces S in IR. n are precisely the k-dimensional submanifolds of IR. n . c) Show that under a smooth homeomorphic mapping f IR. 1 -+ T2 of the line 1 IR. into the torus T2 the image f(IR. 1 ) may be an everywhere dense subset of T 2 and in that case will not be a one-dimensional submanifold of the torus, although it will be an abstract one-dimensional manifold. d) Verify that the extent of the concept "submanifold" does not change if we consider S C M a k-dimensional submanifold of the n-dimensional manifold M when there exists a local chart of the manifold M whose range contains xo for every point Xo E S and some k-dimensional surface of the space IR. n corresponds to the set S n U in the parameter domain of the chart. • • .
:
5. Let X be a Hausdorff topological space (manifold) and G the group of homeo morphic transformations of X. The group G is a discrete group of transformations of X if for every two (possibly equal) points x 1 , x 2 E X there exist neighborhoods U1 and U2 of them respectively, such that the set {g E G l g(U1 ) n U2 =/= 0} is finite. a) It follows from this that the orbit {g(x) E X I g E G } of every point x E X is discrete, and the stabilizer Gx = {g E G l g(x) = x} of every point x E X is finite. b) Verify that if G is a group of isometries of a metric space, having the two properties in a) , then G is a discrete group of transformations of X. c) Introduce the natural topological space (manifold) structure on the set X/G of orbits of the discrete group G.
4 J. Milnor (b. 1931) - one of the most outstanding modern American mathemati cians; his main works are in algebraic topology and the topology of manifolds.
336
15 *Integration of Differential Forms on Manifolds
d) A closed subset F of the topological space (manifold) X with a discrete group G of transformations is a fundamental domain of the group G if it is the closure of an open subset of X and the sets g(F) , where g E G, have no interior points in common and form a locally finite covering of X. Show using Examples 8-10 how the quotient space X/G (of orbits) of the group G can be obtained from F by "gluing" certain boundary points. 6. a) Using the construction of Examples 12 and 13, construct n-dimensional pro jective space JR]pm . b) Show that IRIP'n is orientable if n is odd and nonorientable if n is even. c) Verify that the manifolds 80 (3, JR) and IRIP'3 are homeomorphic. 7. Verify that the manifold constructed in Example 14 is indeed homeomorphic to the Mobius band. 8 . a) A Lie group 5 is a group G endowed with the structure of an analytic manifold such that the mappings (9 1 , 92 ) >-+ 91 · 92 and 9 >-+ 9 - 1 are analytic mappings of G x G and G into G. Show that the manifolds in Examples 6 and 7 are Lie groups. b) A topological group (or continuous group) is a group G endowed with the structure of a topological space such that the group operations of multiplication and inversion are continuous as mappings G x G -+ G, and G -+ G in the topology of G. Using the example of the group Q of rational numbers show that not every topological group is a Lie group. c) Show that every Lie group is a topological group in the sense of the definition given in b) . d) It has been proved 6 that every topological group G that is a manifold is a Lie group (that is, as a manifold G admits an analytic structure in which the group becomes a Lie group) . Show that every group manifold (that is, every Lie group) is an orientable manifold. 9. A system of subsets of a topological space is locally finite if each point of the space has a neighborhood intersecting only a finite number of sets in the system. In particular, one may speak of a locally finite covering of a space. A system of sets is said to be a refinement of a second system if every set of the first system is contained in at least one of the sets of the second system. In particular it makes sense to speak of one covering of a set being a refinement of another. a) Show that every open covering of !R n has a locally finite refinement. b) Solve problem a) with !R n replaced by an arbitrary manifold M. c) Show that there exists a partition of unity on !R n subordinate to any preas signed open covering of !R n . d) Verify that assertion c) remains valid for an arbitrary manifold.
5 S. Lie ( 1842-1899) - outstanding Norwegian mathematician, creator of the theory of continuous groups (Lie groups) , which is now of fundamental importance in geometry, topology, and the mathematical methods of physics; one of the winners of the International Lobachevskii Prize (awarded in 1897 for his work in applying group theory to the foundations of geometry) . 6 This i s the solution t o Hilbert's fifth problem.
15.3 Differential Forms and Integration on Manifolds
337
15.3 Differential Forms and Integration on Manifolds 15.3. 1 The Tangent Space to a Manifold at a Point We recall that to each smooth path JR. 3 t 8 x(t) E JR.n (a motion in JR.n ) passing through the point x0 = x(t0) E JR.n at time t 0 we have assigned the instantaneous velocity vector � = (e ' . . . ' � n ) : �( t ) = x( t ) = (x 1 ' . . . ' x n ) ( to ) . The set of all such vectors � attached to the point x0 E JR.n is naturally identified with the arithmetic space JR.n and is denoted TlR.�0 (or Tx 0 ( JR. n ) ) . In TlR.�0 one introduces the same vector operations on elements � E TlR.�0 as on the corresponding elements of the vector space JR.n . In this way a vector space TlR.�0 arises, called the tangent space to JR.n at the point x0 E JR.n . Forgetting about motivation and introductory considerations, we can now say that formally TlR.�0 is a pair (x0, JR.n ) consisting of a point x 0 E JR.n and a copy of the vector space JR.n attached to it. Now let M be a smooth n-dimensional manifold with an atlas A of at least C ( l ) smoothness. We wish to define a tangent vector � and a tangent space TMp0 to the manifold M at a point Po E M. To do this we use the interpretation of the tangent vector as the instan taneous velocity of a motion. We take a smooth path JR.n 3 t 8 p ( t ) E M on the manifold M passing through the point p0 = p( t0) E M at time to. The parameters of charts (that is, local coordinates) of the manifold M will be denoted by the letter x here, with the subscript of the corresponding chart and a superscript giving the number of the coordinate. Thus, in the pa rameter domain of each chart ( Ui , Cf' i ) whose range Ui contains p0 , the path t ..24 cpi 1 o p(t) = x i (t) E JR.n (or H n ) corresponds to the path 'Y · This path is smooth by definition of the smooth mapping JR. 3 t 8 p(t) E M . Thus, in the parameter domain of the chart (Ui , Cf' i ), where Cf' i is a mapping p = Cf'i (xi ), there arises a point x i (to) = cpi 1 (po ) and a vector �i = x i (to) E TJR.�i ( t o ) " In another such chart (Uj, Cf'J ) these objects will be respectively the point Xj ( to ) = cpj 1 (po ) and the vector �j = Xj ( t o ) E TJR.�i ( t o ) " It is natural to regard these as the coordinate expressions in different charts of what we would like to call a tangent vector � to the manifold M at the point p0 E M . Between the coordinates Xi and Xj there are smooth mutually inverse transition mappings (15.30) as a result of which the pairs by the relations
(x i (t0), �i) , (xJ (t0), �j) turn out to be connected
Xi (to) = Cf/Ji ( Xj (to)) , �i = cpj i (xJ (to))�J ,
Xj (to) = Cf'iJ ( xi (to) ) , �j = cp�j (x i (to))�i .
( 15.31) ( 1 5.32)
338
15 *Integration of Differential Forms on Manifolds
Equality ( 1 5.32) obviously follows from the formulas
obtained from (15.30) by differentiation.
tangent vector � to the manifold M at the point p E M is defined if a vector �i is fixed in each space TJR�, tangent to JR n at the point x i corresponding to p in the parameter domain of a chart (Ui , VJi ), where Ui p, in such a way that ( 1 5.32) holds. If the elements of the Jacobian matrix IPJ i of the mapping IPi i are written
Definition 1. We shall say that a 3
out explicitly as ::� , we find the following explicit form for the connection J between the two coordinate representations of a given vector �:
k = 1 , 2, . . . , n ,
(15.33)
where the partial derivatives are computed at the point Xj =
It can be seen from formulas (15.32) and ( 1 5.33) that the vector-space structure introduced in TMp is independent of the choice of individual chart, that is, Definition 2 is unambiguous in that sense. Thus we have now defined the tangent space to a manifold. There are various interpretations of a tangent vector and the tangent space (see Problem 1 ) . For example, one such interpretation is to identify a tangent vector with a linear functional. This identification is based on the following observation, which we make in lR n . Each vector � E TlR�0 is the velocity vector corresponding to some smooth path x = x ( t ) , that is, � = x ( t ) l t = t o with x0 = x ( t0 ) . This makes it possible to define the derivative D� f ( ) of a smooth function f defined on JR n (or in a neighborhood of x0 ) with respect to the vector � E TlR�0 . To be specific,
x0
d Dd(x0 ) : = - ( ! x ) ( t ) l t -- t0 , dt o
(15.34)
15.3 Differential Forms and Integration on Manifolds
339
that is,
( 1 5 .35) Def(xo) = f'(xo)� , where f'(x0) is the tangent mapping to f (the differential of f) at a point xo. The functional De CC l ) ( JR. n , JR. ) JR. assigned to the vector � E TlR.�0 by the formulas ( 15.34) and ( 15.35) is obviously linear with respect to f. It is also clear from ( 15.35) that for a fixed function f the quantity Def(xo) is a linear function of �, that is, the sum of the corresponding linear functionals corresponds to a sum of vectors, and multiplication of a functional De by a number corresponds to multiplying the vector � by the same number. Thus there is an isomorphism between the vector space TlR.�0 and the vector space of corresponding linear functionals De. It remains only to define the linear functional De by exhibiting a set of characteristic properties of it , in order :
-+
to obtain a new interpretation of the tangent space TlR.�0 , which is of course isomorphic to the previous one. We remark that , in addition to the linearity indicated above, the func tional De possesses the following property:
De( ! · g)(xo) = Def(xo) · g(xo) + f(xo) · Deg(xo) .
( 1 5.36)
This is the law for differentiating a product . In differential algebra an additive mapping a f-t a' of a ring A satisfying the relation (a· b)' = a' · b+a·b' is called derivation (more precisely derivation of lthe ring A) Thus the functional De : C ( l ) ( JR.n , JR. ) is a derivation of the ring C ( ) (JR.n , JR. ) . But De is also linear relative to the vector-space structure of
cC l l (JR.n , JR. ) .
One can verify that a linear functional the properties
l C ( 00 ) ( JR.n , JR. ) :
-+
JR. possessing
( 15.37) l( cx f + {3g) = cx l(f) + {3l(g) , cx , {3 E JR. , ( 1 5.38) l(f · g) = l(f)g(xo) + f(xo)l(g) , has the form De, where � E TlR.�0 • Thus the tangent space TlR.�0 to JR.n at xo can be interpreted as a vector space of functionals (derivations) on cC = ) (JR.n , JR. ) satisfying conditions ( 1 5.37) and ( 15.38) . The functions De , f(xo) = �f(x) i x = x o that compute the corresponding partial derivative of the function f at x0 correspond to the basis vectors e 1 , . . . , e n of the space TlR.�0 • Thus, under the functional interpretation of TlR.�0 one can say that the functionals { £r, . . . , a�n } i x = xo form a basis of �o · If � = ( e , . . . , � n ) E TlR.�0 , then the operator De corresponding to the vector � has the form De = � k a�• . In a completely analogous manner the tangent vector � to an n dimensional C ( oo ) manifold M at a point Po E M can be interpreted (or defined) as the element of the space of derivations l on cC =l (M, JR. ) having TJR.
340
15 *Integration of Differential Forms on Manifolds
xo
Po
properties ( 1 5 .37) and (15.38) , of course being replaced by in rela tion (15.38) , so that the functional l is connected with precisely the point E M. Such a definition of the tangent vector and the tangent space TMpo does not formally require the invocation of any local coordinates, and in that sense it is obviously invariant. In coordinates . . . , x n ) of a local chart ( Ui , 'P i ) the operator l has the form + · · · + 8�7! De; . The are naturally called the coo �dinates of the tangent vector numbers (el , . . . l E T Mp0 in coordinates of the chart ( Ui , 'P i ) · By the laws of differentiation, the coordinate representations of the same functional l E T Mp0 in the charts ( Ui , 'Pi ) , ( Uj , rp1 ) are connected by the relations
Po
e
(x 1 ei, =
ef/;r
, er)
n n n a k a a a , e = = ) : f ( m I: I: I: cr I: a cr k=l ax� m=l x k= l m=l a :n axk n
•
which of course duplicate (15.33 ) .
1
1
(15.33' )
•
1 5 . 3 . 2 Differential Forms on a Manifold
Let us now consider the space T * Mp conjugate to the tangent space that is, T * Mp is the space of real-valued linear functionals on TMp.
TMP ,
T * Mp conjugate to the tangent space TMp to the M at the point p E M is called the cotangent space to M at p. If the manifold M is a c < oo l manifold, f E c < oo l ( M, IR ) , and le is the derivation corresponding to the vector E TM , then for a fixed f E C ( 00 ) ( M, lR) the mapping H l£.1 will obviously be an element of the space T * Mp. In the case M !R n we obtain H De f(p) f' (p)e, so that the Definition 3 . The space
manifold
e o e e = = resulting mapping e le f is naturally called the differential of the function f at p, and is denoted by the usual symbol df(p) . If T!R n ( ) (or TH n _ 1 when p E a M) is the space corresponding to (P ) P H
_1
the tangent space TMP in the chart ( U0 , rp0 ) on the manifold M, it is natural to regard the space T * !R n'Pcx_ 1 ( conjugate to TJR1'Pcx_ 1 as the representative of (P) P) the space T * Mp in this local chart. In coordinates of a local chart . . . , dx n } in the conjugate space corresponds to ( Uo , rp0 ) the dual basis { 8 8 } of T!Rn'Pcx. 1 (p) (or TH'Po.n 1 (p ) if p E We recall that the basis { a;:r , . . . , 8 Xa Xa 8j . The expressions for these dual bases in = i , so that dx i ( 8�j ) [)x i [) [) another chart ( Uf3 , 'P f3 may turn out to be not so simple, for {)T at x fl x fl axr"' • j dX i - � d X . 'P cx
'Pcx
dx 1 ,
n
dxi (e) e o
Bx 1
)
{3
Definition 4 . We say that a
=
_
_
(x�, . . . , x�) aM).
=
differential form w m of degree m is defined on
an n-dimensional manifold M if a skew-symmetric form lR is defined on each tangent space TMp to M, p E M.
wm (p) : (TMp ) m
--t
15.3 Differential Forms and Integration on Manifolds
341
In practice this means only that a corresponding m-form wa (x a ) , where , ) ) corresponding X a = cp� 1 (p), is defined on each space TlRn'Pa_ , ( ) (or THn_ 'Pa ( to TMo in the chart (Ua , 'Pa ) of the manifold M. The fact that two such forms wa (x a ) and w13 (xf3) are representatives of the same form w(p ) can be P
P
expressed by the relation
( 15.39)
X a and Xf3 are the representatives of the point p E M, and ( 6 ) a , . . . , ( �m ) a and ( 6 )f3, . . . , ( �m )/3 are the coordinate representations of the vectors 6 , . . . , �m E T Mp in the charts (Ua , 'Pa ), (U13 , 'P!3 ) respectively. in which
In more formal notation this means that =
( 15.31') X a = 'Pf3a (Xf3) , Xf3 'Pa f3(X a ) , = 'P (x/3) /3 = 'P� (15.32') ) (x , ' �a �a �!3 � f3 a �a where, as usual, 'Pf3 a and 'Pa /3 are respectively the functions cp�1 'P/3 and cp� 1 'Pa for the coordinate transitions, and the tangent mappings to them 'P�a =: ('Pf3a ) * , cp�13 =: ('Pa f3 ) * provide an isomorphism of the tangent spaces to !Rn (or Hn ) at the corresponding points X a and X f3 · As stated in Subsect . 1 5 . 1 .3, the adjoint mappings ('P �a ) * =: 'P� a and (cp �13 ) * =: cp� 13 provide the o
o
transfer of the forms, and the relation ( 15.39) means precisely that
( 15.391) where a and f3 are indices (which can be interchanged) . The matrix (d;, ) of the mapping cp �13 (x a ) is known: (c{ ) Thus, if ( 1 5.40) and
bj 1 , ... ,j m dx�1 1\ · · · 1\ dx�m ,
( 15.41)
then according to Example 7 of Sect. 15. 1 we find that
where �, as always, denotes the determinant of the matrix of corresponding partial derivatives.
342
15 *Integration of Differential Forms on Manifolds
Thus different coordinate expressions for the same form w can be ob tained from each other by direct substitution of the variables (expanding the corresponding differentials of the coordinates followed by algebraic transfor mations in accordance with the laws of exterior products) . If we agree to regard the form W0 as the transfer of a form w defined on a manifold to the parameter domain of the chart (U0 ,
a C ( k)
i,. ( l:<:; h <· · < i,. :<:; n
form if the coefficients ai 1
. . .
w on an n-dimensional manifold M is xa
) of its coordinate representation
are C ( k ) functions in every chart ( U0 ,
exterior differential
is the linear operator d : ilk'
DJ:'_i1 possessing the following properties: 1 ° On every function f E D� the differential d : D�
·
--t
--t DL equals the 1 usual differential df of this function. 2 ° d : (w m 1 l\ w m 2 ) = dw m 1 l\ w m 2 + ( - l ) m1 W m1 /\ dw m 2 , where w m 1 E DJ:'1 and w m 2 E D;:' 2 . 3 ° d 2 : = d 0 d = 0.
This last equality means that d ( dw ) is zero for every form w. Requirement 3 ° thus presumes that we are talking about forms whose smoothness is at least c < 2 l . In practice this means that we are considering a c < 00 ) manifold M and the operator d mapping ,a m to ,a m +I .
15.3 Differential Forms and Integration on Manifolds
343
A formula for computing the operator d in local coordinates of a specific chart (and at the same time the uniqueness of the operator d) follows from the relation d
L Ci 1 ... i m ( x ) dx i ' A · · · A dxi m ) = ( 1-<;i, <···
L dci, ... i m ( x ) A dx i ' A · · · A dx i m + 1-<;i, <··-
( 15.43)
0)
The existence of the operator d now follows from the fact that the operator defined by (15.43) in a local coordinate system satisfies conditions 1 ° , 2 ° , and 3° of Definition 6. It follows in particular from what has been said that if W a = cp�w and wp = cp{J w are the coordinate representations of the same form w , that is, Wa = 'P� p Wf3 , then dwa and dwp will also be the coordinate representations of the same form (dw ) , that is, dw a = 'P� p dwp . Thus the relation d(cp� {3 wf3 ) = 'P� p (dwp) holds, which in abstract form asserts the commutativity dcp*
= cp* d
( 1 5.44)
of the operator d and the operation cp* that transfers forms. 15.3.4 The Integral of a Form over a Manifold Definition 7. Let M be an n-dimensional smooth oriented manifold on
which the coordinates x 1 , . . . , x n and the orientation are defined by a sin gle chart 'P x : Dx ---+ M with parameter domain Dx C JR n . Then
j = J a (x) dx 1 A · · · A dxn , Dx w:
( 1 5.45)
M
where the left-hand side is the usual integral of the form w over the oriented manifold M and the right-hand side is the integral of the function f ( x ) over the domain Dx . If 'P t : D t ---+ M i s another atlas of M consisting o f a single chart defining the same orientation on M as 'P x : Dx ---+ M, then the Jacobian det cp' ( t ) of the function x = cp( t ) of the coordinate change is everywhere positive in Dt . The form cp* ( a ( x ) dx 1
A · · · A dx n ) = a ( x ( t ) ) det cp ( t ) dt 1 A · · · A dtn '
in Dt corresponds to the form w . By the theorem on change of variables in a multiple integral we have the equality
15 *Integration of Differential Forms on Manifolds
344
J a(x) dx 1 · · · dxn = J a (x(t) ) det cp' (t) dt 1 . . · dtn , D,
Dx
which shows that the left-hand side of (15.45) is independent of the coordinate system chosen in M. Thus, Definition 7 is unambiguous. Definition 8 . The support of a form w defined on a manifold M closure of the set of points x E M where w (x) =/= 0.
IS
the
The support of a form w is denoted by supp w. In the case of 0-forms, that is, functions, we have already encountered this concept. Outside the support the coordinate representation of the form in any local coordinate system is the zero form of the corresponding degree. Definition 9. A form w defined on a manifold supp w is a compact subset of M.
M is of compact support if
Definition 10. Let w be a form of degree n and compact support on an n dimensional smooth manifold M oriented by the atlas A. Let 'P i : Di ---t Ui , { (Ui , cp) , i = 1 , . . . , m } be a finite set of charts of the atlas A whose ranges U1 , . . . , Um cover supp w, and let e 1 , . . . , e k be a partition of unity subordinate to that covering on supp w. Repeating some charts several times if necessary, we can assume that m = k, and that supp e i C Ui , i = 1 , . . . , m. The integral of a form w of compact support over the oriented manifold M is the quantity
J w : = f J cp; (ei w) ,
M
(15.46)
t= I D;
where cp; ( e i w) is the coordinate representation of the form e i w I U; in the domain D i of variation of the coordinates of the corresponding local chart. Let us prove that this definition is unambiguous. Let A = { 0j : jjj ---t uj } be a second atlas defining the same smooth structure and orientation on M as the atlas A, let ul ' . . . ' Um be the cor responding covering of supp w ' and let el ' . . . ' eii, a partition of unity on supp w subordinate to this covering. We introduce the functions fij = e i ej , i = 1 , . . . , m , j = 1 , . . . , iii , and we set Wij = jijW · We remark that supp Wij c wij = ui n uj . From this and from the fact that Definition 7 of the integral over an oriented manifold given by a single chart is unambiguous it follows that
Proof.
J cp; (wij ) = J
D;
'Pil(Wij )
cpt (w ij ) =
J
0j (wij ) =
J 0j (wij ) .
Dj
15.3 Differential Forms and Integration on Manifolds
345
Summing these equalities on i from 1 to m and on j from 1 to iii , taking iii
m
account of the relation L fiJ interested int.
i= 1 = ej , jL= 1 fiJ = e i , we find the identities we are
D
15.3.5 Stokes' Formula
Let M be an oriented smooth n-dimensional manifold and w a smooth differential form of degree n - 1 and compact support on M. Then (15.47)
Theorem.
where the orientation of the boundary aM of the manifold M is induced by the orientation of the manifold M. If aM = 0 , then f dw = 0. M
Proof.
Without loss of generality we may assume that the domains of vari ation of the coordinates (parameters) of all local charts of the manifold M are either the open cube I = {x E JR n l 0 < x i < 1, i = 1, . . . , n} , or the cube i = {x E JRn l 0 < x 1 ::::; 1 /\ 0 < x i < 1, i = 1, . . . , n} with one (definite! ) face adjoined to the cube I. By the partition of unity the assertion of the theorem reduces to the case when supp w is contained in the range U of a single chart of the form r.p : I --+ U or r.p : i --+ U. In the coordinates of this chart the form w has the form n w = L a i ( x ) dx 1 1\ · · · I\ dx i 1\ · · · 1\ dxn ,
i= 1
where the frown � , as usual, means that the corresponding factor is omitted. By the linearity of the integral, it suffices to prove the assertion for one term of the sum:
wi = ai ( x ) dx 1
1\
· · · I\
dx i
The differential of such a form is the n-form
1\
···
1\
dx n
.
(
15.48 )
aai 1 n ( 15.49 ) = ( - 1)i- 1 � ux' ( X ) dX · · · dX . For a chart of the form : I --+ U both integrals in ( 15.47 ) of the cor responding forms ( 15.48 ) and ( 15.49 ) are zero: the first because supp a i I and the second for the same reason, if we take into account Fubini ' s theorem 1 and the relation J � dx i = a i (1) - a i (O) = 0. This argument also covers the 0 case when aM = 0. dW i
r.p
1\
1\
C
346
15 *Integration of Differential Forms on Manifolds
Thus it remains to verify (15.47) for a chart rp : i ---+ U. If i > 1, both integrals are also zero for such a chart, which follows from the reasoning given above. And if i 1, then
=
I I ( I aa i I ) = 1 . . . 1 1 ox i (x) dx dx 2 . . · dxn = 0I 0I 0 = 1 · · · 1 a i (1, x2 , · · · , xn ) dx2 · · · dxn = 1 WI = 1 WI . o
o
au
aM
Thus formula (15.47) is proved for n > 1. The case n 1 is merely the Newton-Leibniz formula (the fundamental theorem of calculus) , if we assume that the endpoints a and f3 of the oriented interval [a, {3] are denoted a_ and {3+ and the integral of a 0-form g (x) over such an oriented point is equal to -g(a) and + g ( f3 ) respectively. D
=
We now make some remarks on this theorem.
Remark 1 .
Nothing is said in the statement of the theorem about the smooth ness of the manifold and the form w. In such cases one usually assumes that each of them is c < = l . It is clear from the proof of the theorem, however, that formula (15.47) is also true for forms of class c < 2 l on a manifold admitting a form of this smoothness.
M
M
Remark 2. It is also clear from the proof of the theorem, as in fact it was already from the formula (15.47), that if supp w is a compact set contained strictly inside that is, supp w n 0, then f dw 0.
aM = M, = M Remark 3. If M is a compact manifold, then for every form w on M the support supp w, being a closed subset of the compact set M, is compact. Consequently in this case every form w on M is of compact support and Eq. (15.47) holds. In particular, if M is a compact manifold without boundary, then the equality J dw = 0 holds for every smooth form on M. M
Remark 4.
For arbitrary forms w (not of compact support) on a manifold that is not itself compact, formula (15.47) is in general not true. Let us consider, for example, the form w x �� ���x in a circular annulus { ( x, y ) E IR 2 1 1 ::; x2 + y2 ::; 2 } , endowed with standard Cartesian coordinates. In this case is a compact two-dimensional oriented manifold,
M=
=
M
15.3 Differential Forms and Integration on Manifolds
whose boundary aM consists of the two circles Ci = {(x, y) E i = 1 , 2. Since dw = 0 , we find by formula ( 15.47) that
where both circles
347
� 2 l x 2 +y 2 = i},
C1 and C2 are traversed counterclockwise. We know that M = M \ C1 , then aM = C2 and dw = 0 f 27r = w.
Hence, if we consider the manifold
j
M 15.3.6
j
a Ni
Problems and Exercises
a) We call two smooth paths 'Yi : JR. -+ M, i = 1 , 2 on a smooth manifold M tangent at a point p E M if "{ 1 (0) = "{2 (0) = p and the relation ( 15.50) lrp - 1 o "{ 1 (t) - rp - 1 o "{2 (t) l = o (t) as t -+ 0 holds in each local coordinate system rp : JRn -+ U (or rp : H n -+ U) whose range U contains p. Show that if (15.50) holds in one of these coordinate systems, then it holds in any other local coordinate system of the same type on the smooth manifold M. b) The property of being tangent at a point p E M is an equivalence relation on the set of smooth paths on M passing through p. We call an equivalence class a bundle of tangent paths at p E M. Establish the one-to-one correspondence ex hibited in Subsect. 15.3. 1 between vectors of T Mp and bundles of tangent paths at the point p E M. c) Show that if the paths "! 1 and "!2 are tangent at p E M and f E C (ll (M, JR) , then df 0 "{ 1 (0) = df 0 "{2 (0) . dt dt d) Show how to assign a functional [ = l � (= D� ) : cC ool (M, JR) -+ JR. possessing properties ( 15.37) and (15.38) , where Xo = p, to each vector e E TMp . A functional possessing these properties is called a derivation at the point p E M. Verify that differentiation l at the point p is a local operation, that is, if /1 , h E cC oo ) and /1 (x) = h (x) in some neighborhood of p, then l/1 = l/2 . e) Show that if x 1 , . . . , xn are local coordinates in a neighborhood of the point p, then l = E (lx i ) a� ' , where a� ' is the operation of computing the partial derivative i= l with respect to x i at the point x corresponding to p. (H i n t. Write the function 1.
15 *Integration of Differential Forms on Manifolds
348
J
l
U (p )
: M -+ JR. in local coordinates; remember that the expansion f(x)
i it= l x gi(x) holds for the function f E Bi(o) l!r Co), i 1 , ) =
=
c < = l (JR. n , JR) , where
9i
E
=
f(O) +
c < = l (JR. n , JR) and
. . . , n.
f) Verify that if M is a c < = > manifold, then the vector space of derivations at the point p E M is isomorphic to the space T Mp tangent to M at p constructed in Subsect. 15.3. 1 . a ) I f we fix a vector �(p) E TMp at each point p E M of a smooth manifold M, we say that a vector field is defined on the manifold M. Let X be a vector field on M. Since by the preceding problem every vector X (p) = � E TMp can be interpreted as differentiation at the corresponding point p, from any function f E c < = > (M, JR) one can construct a function X f(p) whose value at every point p E M can be computed by applying X (p) to J, that is, to differentiating f with respect to the vector X (p) in the field X. A field X on M is smooth ( of class c < = > ) if for every function f E c < = > (M, JR.) the function X f also belongs to c < = > (M, JR.) . Give a local coordinate expression for a vector field and the coordinate definition of a smooth c c < = > ) vector field on a smooth manifold equivalent to the one just given. b) Let X and Y be two smooth vector fields on the manifold M. For functions f E c < = > (M, JR.) we construct the functional [X, Y]f = X(Y f) - Y(X f). Verify that [X, Y] is also a smooth vector field on M. It is called the Poisson bracket of the vector fields X and Y. c) Give a Lie algebrastructure to the smooth vector fields on a manifold. 2.
3 . a) Let X and w be respectively a smooth vector field and a smooth 1-form on a smooth manifold M. Let wX denote the application of w to the vector of the field X at corresponding points of M. Show that wX is a smooth function on M. b) Taking account of Problem 2, show that the following relation holds:
where X and Y are smooth vector fields, dw 1 is the differential of the form w 1 , and dw 1 (X, Y) is the application of dw 1 to pairs of vectors of the fields X and Y attached at the same point. c) Verify that the relation
i= l + 2:: (-1) i +j w ([xi, xj ], xl , . . . , xi , . . . , xj , . . . , xm+l ) l :'Oi<j:C:: m + l
holds for the general case of a form w of order m. Here the frown � denotes an omitted term, [Xi , Xi] is the Poisson bracket of the fields Xi and Xi , and Xiw represents differentiation of the function w (X 1 , . . . , Xi, . . . , Xm+l ) with respect to the vectors of the field Xi. Since the Poisson bracket is invariantly defined, the
15.3 Differential Forms and Integration on Manifolds
349
resulting relation can be thought of as a rather complicated but invariant definition of the exterior differential operator d : n --+ n. d) Let w be a smooth m-form on a smooth n-dimensional manifold M. Let (6 , . . . , �m + l ); be vectors in !R n corresponding to the vectors 6 , . . . , �m + l E T Mp in the chart <.p; : IR n --+ U C M. We denote by II; the parallelepiped formed by the vectors (6 , . . . , �m + l ) ; in !R n , and let >... II; be the parallelepiped spanned by the vectors (>...6 , . . . , A�m + d i · We denote the images <.p; (II; ) and <.p; ( A II; ) of these parallelepipeds in M by II and >... II respectively. Show that 1 dw (p) (6 , . . . , �m + I ) = l� ).. n l W· +
I
8 ( >- II )
a) Let f : M --+ N be a smooth mapping of a smooth m-dimensional manifold M into a smooth n-dimensional manifold N. Using the interpretation of a tangent vector to a manifold as a bundle of tangent paths (see Problem 1 ) , construct the mapping f. (p) : TMp --+ TNJ ( p ) induced by f. b) Show that the mapping f. is linear and write it in corresponding local coor dinates on the manifolds M and N. Explain why f. (p) is called the differential of f at p or the mapping tangent to f at that point. Let f be a diffeomorphism. Verify that f. [X, Y] = [f. X, f. Y] . Here X and Y are vector fields on M and [· , ·] is their Poisson bracket (see Problem 2). c ) As is known from Sect. 1 5 . 1 , the tangent mapping f. (p) : TMp --+ TN = f p q () of tangent spaces generates the adjoint mapping j* (p) of the conjugate spaces and in general a mapping of k-forms defined on T Nf ( p ) and T Mp . Let w be a k-form on N. The k-form j* w on M is defined by the relation 4.
where 6 , . . . , � k E TMp. In this way a mapping j* : st k ( N ) --+ st k (M ) arises from the space st k (N ) of k-forms defined on N into the space st k (M) of k-forms defined on M. Verify the following properties of the mapping j* , assuming M and N are c < oo) manifolds: 1 ° j* is a linear mapping; 2° j* ( w 1 1\ w2 ) = j* w 1 1\ j* w 2 ; 3° d o j* = j* o d, that is d (j* w ) = j* ( dw ) ; 4° ( h o h ) * = H o f2 . d) Let M and N be smooth n-dimensional oriented manifolds and <.p : M --+ N a diffeomorphism of M onto N. Show that if w is an n-form on N with compact support, then w = e: <.p * w ,
{ 11,
I
I
M
if <.p preserves orientation , . . 1'f <.p reverses onentatwn . e) Suppose A ::::> B. The mapping i : B --+ A that assigns to each point x E B that same point as an element of A is called the canonical embedding of B in A.
where f =
-
,
15 *Integration of Differential Forms on Manifolds
350
If w is a form on a manifold M and M' is a submanifold of M, the canonical embedding i : M' -+ M generates a form i* w on M' called the restriction of w to M'. Show that the proper expression of Stokes' formula (15.47) should be
where i : 8M -+ M is the canonical embedding of 8M in M, and the orientation of 8M is induced from M. a) Let M be a smooth (C (oo ) ) oriented n-dimensional manifold and D:C: (M) the space of smooth (C (oo) ) n-forms with compact support on M. Show that there exists a unique mapping I : n;; (M) -+ lR having the following properties: 5.
M
1 ° the mapping I is linear; M
2 ° if 'P : I n -+ U C M (or 'P : Jn -+ U C M) is a chart of an atlas defining the orientation of M, supp w C U, and w = a(x) dx 1 1\ · · · I\ dx n in the local coordinates x 1 , , x n of this chart, then . . •
J w = J a(x) dx \ . . . , dxn
M
Jn
(or
J w = J a(x) dx 1 ,
M
• • •
, dx n ) ,
jn
where the right-hand side contains the Riemann integral of the function a over the corresponding cube I n (or Jn ) . b ) Can the mapping just exhibited always be extended to a mapping I M n n (M) -+ lR of all smooth n-forms on M, retaining both of these properties? c) Using the fact that every open covering of the manifold M has an at most countable locally finite refinement and the fact that there exists a partition of unity subordinate to any such covering (see Problem 9), define the integral of an n-form over an oriented smooth n-dimensional (not necessarily compact) manifold so that it has properties 1 ° and 2 ° above when applied to the forms for which the integral is finite. Show that for this integral formula ( 1 5.47) does not hold in general, and give conditions on w that are sufficient for (15.47) in the case when M = lR n and in the case when M = H n . 6 . a) Using the theorem on existence and uniqueness of the solution of the differen tial equation x = v(x) and also the smooth dependence of the solution on the initial data, show that a smooth bounded vector field v(x) E lR n can be regarded as the velocity field of a steady-state flow. More precisely, show that there exists a family of diffeomorphisms 'Pt : lR n -+ lR n depending smoothly on the parameter t (time) such that 'Pt (x) is an integral curve of the equation for each fixed value of x E lR n , that is, 8 "'Jix) = v ( 'P t (x) ) and 'Po (x) = x. The mapping 'Pt : lR n -+ lR n obviously characterizes the displacement of the particles of the medium at time t. Verify that the family of mappings 'Pt : lR n -+ lR n is a one-parameter group of diffeomorphisms, that is, ( 'Pt ) - l = 'P - t , and 'Pt 2 o 'Pt 1 = 'Pt , + t 2 .
15.3 Differential Forms and Integration on Manifolds
351
b ) Let v be a vector field on ne and 'P t a one-parameter group of diffeomor phisms of ne generated by v. Veri fy that the relation
lim ! (!( 'P t ( x )) - f( x ) ) = D v ( x) / t holds for every smooth function f E c < = ) (!R n , IR) . If we introduce the notation v(f) := Dv f, i n consistency with the notation of Problem 2, and recall that f o 'P t =: r.p; J , we can write t -+ 0
lim !t ( rp ; f - f ) ( x ) = v(f) (x) .
t-+0
c ) Differentiation of a smooth form w of any degree defined in !R n along the field v is now naturally defined. To be specific, we set
v(w) (x) : = tlim ! ( rp ; w - w) (x) . -+ 0 t
The form v(w) is called the Lie derivative of the form w along the field v and usually denoted Lvw. Define the Lie derivative Lxw of a form w along the field X on an arbitrary smooth manifold M. d ) Show that the Lie derivative on a c < = ) manifold M has the following prop erties. 1 ° Lx is a local operation, that is, if the fields X I and x2 and the forms W j and w2 are equal in a neighborhood U C M of the point x, then (Lx 1 wi ) (x) = (Lx2 W2 ) (x). 2° Lx fl k (M) c fl k (M) . 3° L x : fl k (M) --+ fl k (M) is a linear mapping for every k = 0 , 1 , 2 , . . . . 4° Lx (wi 1\ w2 ) = (Lxwi ) 1\ W2 + WI 1\ Lxw2 . 5° If f E fl 0 (M) , then Lx f = df(X) =: X f. 6° If f E fl 0 (M) , then Lxdf = d(Xf) . e ) Verify that the properties 1 ° �6 ° determine the operation Lx uniquely. Let X be a vector field and w a form of degree k on the smooth manifold M. The inner product of the field X and the form w is the (k - 1 ) -form de noted by ixw or Xj w and defined by the relation (ixw) (X I , . . . , X k - I ) := w(X, X 1 , . . . , Xk- I ) , where X 1 , . . . , Xk - I are vector fields on M. For 0-forms, that is, functions on M, we set X J f = 0. a) Show that if the form w ( more precisely, w l ) has the form 7.
"'""' L.....,
l :S i 1 < · · · < i k :S n
a ' l · · · ' k ( x ) dx i 1 ·
1\ 1\ dx i k = _!_k! a' l · · · ' k dx i 1 1\ 1\ dx i k ·
·
·
·
·
·
·
in the local coordinates x 1 , , x n of the chart r.p : !R n --+ U C M, and X = Xi 8�, , then . 1 i lX W - (k _ 1) ! x i a i ,. 2 . . . i k d X 2 1\ 1\ d X i k . • • •
_
·
·
·
b ) Verify further that if df = j!, dxi , then ix df = Xi j!, = X (f)
=
Dx f.
352
15 *Integration of Differential Forms on Manifolds
c) Let X(M) be the space of vector fields on the manifold M and Jl(M) the ring of skew-symmetric forms on M. Show that there exists only one mapping i : X (M) x Jl(M) --+ Jl(M) having the following properties: 1 ° i is a local operation, that is, if the fields X 1 and X2 and the forms W 1 and w 2 are equal in a neighborhood U of x E M , then (ix 1 w 1 ) (x) = (ix2 w2 ) (x) ; 2 ° ix ( Jl k (M ) ) C n k - 1 (M ) ; 3 ° ix : Jl k ( M) --+ n k - l ( M) is a linear mapping; 4 ° if w 1 E Jl k 1 ( M) and w2 E Jl k 2 (M) , then ix (wu'\w 2 ) = ixwu'\w 2 + ( - 1 ) k 1 wu'\ ixw 2 ; 5 ° if w E Jl 1 ( M ) , then ixw = w (X ) , and if f E Jl 0 ( M ) , then ix f = 0. Prove the following assertions. a) The operators d, ix , and Lx (see Problems 6 and 7) satisfy the so-called homotopy identity (15.51) Lx = ixd + dix , where X is any smooth vector field on the manifold. b) The Lie derivative commutes with d and i x , that is, 8.
Lx o d = d o Lx ,
Lx o ix
=
ix o Lx .
c) (Lx , iy] = i[x,Y] , [Lx , Ly] = L[x,Y) l where, as always, [ A , B] = A o B - B o A for any operators A and B for which the expression A o B - B o A is defined. In this case, all brackets ( , ] are defined. d) Lx fw = f Lxw + df A ixw, where f E Jl 0 ( M ) and w E Jl k (M ) . (H i n t. Part a ) is the main part o f the problem. It can b e verified, for example, by induction on the degree of the form on which the operators act.)
15.4 Closed and Exact Forms on Manifolds 1 5 . 4 . 1 Poincare's Theorem
In this section we shall supplement what was said about closed and exact differential forms in Sect. 14.3 in connection with the theory of vector fields in ffi.n . As before, denotes the space of smooth real-valued forms of degree p on the smooth manifold and =
f!P (M)
M J!(M) U f!P (M). Definition 1 . The form w E f!P (M) is closed if dw 0. Definition 2 . The form w E f!P (M), p > 0, is exact if there exists a form E f!P - 1 (M) such that w d . The set of closed p-forms on the manifold M will be denoted ZP (M), and the set of exact p-forms on M will be denoted B P (M). p
=
a
=
a
353
15.4 Closed and Exact Forms on Manifolds
0
The relation7 d(dw) = holds for every form w E D(M) , which shows that ZP (M) :J BP (M) . We already know from Sect. 14.3 that this inclusion is generally strict. The important question of the solvability (for a ) of the equation d a = w given the necessary condition dw = on the form w turns out to be closely connected with the topological structure of the manifold M. This statement will be deciphered more completely below.
0
Definition 3. We shall call a manifold M contractible (to the point x0 E M) or homotopic to a point if there exists a smooth mapping h : M x I --t M where I = {t E IR I t ::; 1 } such that h ( x, 1 ) = x and h ( x, O ) = x0 .
Example 1 . The h ( x, t) = t x .
0 ::;
space
!Rn can be contracted to a point by the mapping
1. (Poincare) . Every closed (p + 1)-form (p 2': on a manifold that is contractible to a point is exact. Proof. The nontrivial part of the proof consists of the following "cylindrical"
0)
Theorem
construction, which remains valid for every manifold M. Consider the "cylinder" M x I, which is the direct product of M and the closed unit interval I, and the two mappings j; : M --t M x I, where j; ( x ) = ( x, i), i = 1 , which identify M with the bases of the cylinder M x I. Then there naturally arise mappings j;* : f?P (M x I) --t f?P (M) , reducing to the replacement of the variable t in a form of f?P (M x I) by the value i ( = 1 ) , and, of course, di = We construct a linear operator K : f?P + 1 (M x I) --t f?P (M) , which we define on monomials as follows:
0,
0,
0.
K ( a ( x, t) dx i 1 A · · · A dx i v +! ) : = 1 K ( a ( x, t) dt A dx i 1 A · · · A d x i v ) := ( a ( x, t) dt) d x i 1 A · · · A d x i v . 0
0,
J
The main property of the operator K that we need is that the relation K(dw) + d(Kw) = j;w - j�w
( 15.52)
holds for every form w E f?P + 1 (M x I) . It suffices to verify this relation for monomials, since all the operators K, d, Ji , and j0 are linear.
7 Depending on the way in which the operator d is introduced this property is either proved, in which case it is called the Poincare lemma, or taken part of the definition of the operator d. as
15 *Integration of Differential Forms on Manifolds
354
dx i v + 1 , then Kw = 0, d(Kw) 0, and a(x, t) dxi 1 & . dx 'v. + l + [terms not containing dt] , dw = {) dt dx ' 1 t 1 oa dt dx '. 1 dx'v. + 1 K(dw) = {) t 0 (a(x, 1) - a(x, O)) dx i 1 . . I\ dx i v + l Jtw - j�w ,
If w
=
1\
1\ · · · 1\
(/ )
=
1\ · · · 1\
1\ · · · 1\
=
1\
=
·
=
and relation ( 15.52) is valid. If w = a(x, t) dt 1\ dx i 1 1\ · · · 1\ dx i P , then J;w = j0w = 0. Then
K (dw) = K
(
-
io
=
d(Kw)
=
d
( ( j1 a(x, t) dt) dxi 1
=
0
) 1 ( - � J 0��0 dt) dx i o •o 0 dx i v )
.0 . 1 . oa L -ox •o. dt 1\ dx' 1\ dx ' 1\ · · · 1\ dx 'v
1\ . . . 1\
1\
L {)
•o
=
1\ · · · 1\
dx iv ,
=
:io ( J0 a(x, t) dt) dxio dxi 1 1 ( J O{)aXio dt) dx'.0 •o 0 1
=
1\ · · · 1\
"' L.....
(\
dxi v
dx '. 1
=
(\ ' ' ' (\
dx 'P. .
Thus relation ( 15.52) holds in this case also.8 Now let M be a manifold that is contractible to the point x0 E M, let h : M x I ---* M be the mapping in Definition 3, and let w be a (p + 1 )-form on M. Then obviously hoj 1 : M ---* M is the identity mapping and h o j0 : M ---* x0 is the mapping of M to the point x0 , so that (J; o h* )w w and (j0 o h* )w = 0. Hence it follows from ( 15.52 ) that in this case =
K (d(h*w)) + d (K(h*w)) If in addition w is a closed form on we find by ( 15.53) that
=
( 15.53)
w.
M, then, since d(h*w) = h* (dw) = 0,
d (K(h*w)) = w .
Thus the closed form w is the exterior derivative of the form a
QP(M), that is, w is an exact form on M. 8
D
=
K(h*w)
For the justification of the differentiation of the integral with respect to this last equality, see, for example, Sect. 17. 1 .
E
xio in
15.4 Closed and Exact Forms on Manifolds
355
Example 2. Let A, B, and C be smooth real-valued functions of the variables x, y, z E JR3 . We ask how to solve the following system of equations for P, Q,
and R:
( 1 5.54)
An obvious necessary condition for the consistency of the system ( 1 5.54) is that the functions A, B, and C satisfy the relation
aA + aB + ac - 0 ax 8y az
which is equivalent to saying that the form w
_ '
= A dy A dz + B dz A dx + C dx A dy
is closed in IR 3 . The system ( 15.54) will have been solved if we find a form a = P dx + Q dy + R dz
such that d a = w . In accordance with the recipes explained in the proof of Theorem 1 , and taking account of the mapping h constructed in Example 1 , we find, after simple computations,
(/ A(tx, ty, tz)t dt ) (y dz - z dy) + 1
a = K(h*w) =
0
+ (/ B(tx, ty, tz)t dt ) (z dx - x dz) + 1
0
+
1
(/ C(tx, ty, tz)t dt) (x dy - y dx) . 0
One can also verify directly that d a = w .
Remark. The amount o f arbitrariness i n the choice o f a form a satisfying the condition da = w is usually considerable. Thus, along with a , any form a + d1J will obviously also satisfy the same equation. By Theorem 1 any two forms a and (3 on a contractible manifold M satisfying d a = d(3 = w differ by an exact form. Indeed, d( a - (3) = 0, that is, the form ( a - (3) is closed on M and hence exact , by Theorem 1 .
356
15 *Integration of Differential Forms on Manifolds
1 5 . 4 . 2 Homology and Cohomology
By Poincare ' s theorem every closed form on a manifold is locally exact . But it is by no means always possible to glue these local primitives together to obtain a single form. Whether this can be done depends on the topological structure of the manifold. For example, the closed form in the punctured plane IR 2 \ 0 given by w = -y$��-�� d y , studied in Sect . 14.3, is locally the differential of a function cp = cp(x, y) - the polar angle of the point (x, y). However, extending that function to the domain IR 2 \ 0 leads to multivaluedness if the closed path over which the extension is carried out encloses the hole - the point 0. The situation is approximately the same with forms of other degrees. "Holes" in manifolds may be of different kinds, not only missing points, but also holes such as one finds in a torus or a pretzel. The structure of manifolds of higher dimensions can be rather complicated. The connection between the structure of a manifold as a topological space and the relationship between closed and exact forms on it is described by the so-called (co )homology groups of the manifold. The closed and exact real-valued forms on a manifold M form the vector spaces ZP (M) and BP (M) respectively, and ZP (M) :J BP (M) . Definition 4. The quotient space HP (M)
:=
ZP (M)/ BP (M)
(15.55)
is called the p-dimensional cohomology group of the manifold M (with real coefficients) . Thus, two closed forms w 1 , w 2 E ZP (M) lie in the same cohomology class, or are cohomologous, if w 1 - w2 E BP (M) , that is, if they differ by an exact form. The cohomology class of the form w E ZP (M) will be denoted [w] . Since ZP (M) is the kernel of the operator dP : flP (M) ---+ [lP + 1 (M) , and BP (M) is the image of the operator dP - 1 : {}P - 1 (M) ---+ flP(M) , we often write HP (M) = Ker dP /Im dP - 1 . Computing cohomologies, as a rule, is difficult . However, certain trivial general observations can be made. It follows from Definition 4 that if p > dim M, then HP (M) = 0. It follows from Poincare ' s theorem that if M is contractible then HP (M) = 0 for p > 0. On any connected manifold M the group H 0 (M) is isomorphic to IR, since H 0 ( M) = Z 0 ( M), and if d f = 0 holds for the function f : M ---+ lR on a connected manifold M, then f = const. Thus, for example, it results for !R n that HP (JR n ) = 0 for p > 0 and 0 H (1R n ) R This assertion (up to the trivial last relation) is equivalent to Theorem 1 with M = !R n and is also called Poincare ' s theorem. The so-called homology groups have a more visualizable geometrical re lation to the manifold M. ,...,
15.4 Closed and Exact Forms on Manifolds
357
A smooth mapping c : IP --+ M of the p-dimensional cube I C JRP into the manifold M is called a singular p-cube on M.
Definition 5.
This is a direct generalization of the concept of a smooth path to the case of an arbitrary dimension p. In particular, a singular cube may consist of a mapping of the cube I to a single point . Definition 6. A p-chain (of singular cubes) on a manifold M is any finite formal linear combination I: a k c k of singular p-cubes on M with real coeffi.k cients.
Like paths, singular cubes that can be obtained from each other by a diffeomorphic change of the parametrization with positive Jacobian are re garded as equivalent and are identified. If such a change of parameter has negative Jacobian, then the corresponding oppositely oriented singular cubes c and c_ are regarded as negatives of each other, and we set c_ = - c. The p-chains on M obviously form a vector space with respect to the standard operations of addition and multiplication by a real number. We denote this space by Cp (M) .
boundary
Definition 7. The
(p - 1)-chain
8Iof the p-dimensional cube IP in
l
p
ai L L)-l) i +icij i=O j = l :=
JRP is the ( 1 5 . 56)
in JRP , where Cij : IP - l --+ JRP is the mapping of the (p - I )-dimensional cube into JRP induced by the canonical embedding of the corresponding face of IP in JRP . More precisely, if IP - 1 = {x E JRP - 1 1 0 ::::; xm ::::; 1 , m = 1 , . . . , p - 1 } , mp . t hen c,.1· ( X- ) - ( X- 1 , , X-j - 1 , z, X-H 1 , . . . , X-p- i ) E � _
•
•
.
·
It is easy to verify that this formal definition of the boundary of a cube agrees completely with the operation of taking the boundary of the standard oriented cube IP (see Sect . 12.3) .
boundary 8c of the singular p-cube c is the (p - 1 )-chain 1 p 8c L L( - l) i+ic c ii . i=O j = l The boundary of a p-chain I: a k c k on the manifold M is the k
Definition 8. The
:=
Definition 9.
(p - 1 )-chain
o
358
15 *Integration of Differential Forms on Manifolds
Thus on any space of chains Cp (M) we have defined a linear operator Using relation ( 15 .56) , one can verify the relation Consequently a o a = a2 = 0 in general.
a( a!) = 0 for the cube.
p-cycle on a manifold is a p-chain z for which az = 0. Definition 1 1 . A boundary p-cycle on a manifold is a p-chain that is the boundary of some (p + 1 ) -chain. Definition 10. A
Let Zp (M) and Bp (M) be the sets of p-cycles and boundary p-cycles on the manifold M. It is clear that Zp (M) and Bp (M) are vector spaces over the field JR. and that Zp (M) ::) Bp (M) . Definition 1 2 . The quotient space
(15.57) is the p-dimensional cients ) .
homology group of the manifold M ( with real coeffi
Thus, two cycles Z 1 , z2 E Zp ( M ) are in the same homology class, or are homologous, if z 1 - z2 E Bp (M ) , that is, they differ by the boundary of some chain. We shall denote the homology class of a cycle z E Zp (M) by [z]. As in the case of cohomology, relation (15.57) can be rewritten as
Definition 13. If c : I --+ M is a singular p-cube and w is a p-form on the manifold M, then the integral of the form w over this singular cube is
j := j c*w . w
r
(15.58)
I
Definition 14. If 2: o: k ck is a p-chain and k
w
is a p-form on the manifold
M, the integral of the form over such a chain is interpreted as the linear combination 2: O:k J w of the integrals over the corresponding singular cubes. k
Ck
It follows from Definitions 5-8 and 13-14 that Stokes ' formula c
De
(15.59)
holds for the integral over a singular cube, where c and w have dimension p and degree p - 1 respectively. If we take account of Definition 9, we conclude that Stokes ' formula ( 15.59) is valid for integrals over chains.
359
15.4 Closed and Exact Forms on Manifolds
Theorem 2 . a )
The integral of an exact form over a cycle equals zero. b) The integral of a closed form over the boundary of a chain equals zero. c) The integral of a closed form over a cycle depends only on the cohomology class of the form. d) If the closed p-forms w 1 and w2 and the p-cycles z 1 and z2 are such that [w l ] = [w2 ] and [z l ] = [z2 ], then Proof. a ) By Stokes ' formula Jz w dz = J w = 0, since fJ z = 0. 8z ' formula J w J dw = 0, since dw 0. b) By Stokes c 8c c) follows from b). d) follows from a ) . e) follows from ) and d). Corollary. The bilinear mapping DP (M) x Cp (M) ---+ � defined by (w, ) Jc w induces a bilinear mapping ZP (M) x Zv (M) ---+ � and a bilinear mapping HP (M) x Hp (M) ---+ R The latter is given by the formula =
c
=
0
([w], [zl )
c
foo-t
Jw ,
foo-t
(15.60)
z
where w E ZP (M) and z E Zv (M) . Theorem 3. (de Rham9 ) . The bilinear mapping HP (M) x Hp (M) ---+ � given by (15.60) is nondegenerate. 1 0 We shall not take the time to prove this theorem here, but we shall find some reformulations of it that will enable us to present in explicit form some corollaries of it that are used in analysis. We remark first of all that by (15.60) each cohomology class [w] E HP (M) can be interpreted as a linear function [w] ([zl ) = J w. Thus a natural mapz
ping HP (M) ---+ H; (M) arises, where H; (M) is the vector space conjugate to Hp (M) . The theorem of de Rham asserts that this mapping is an isomor phism, and in this sense HP (M) = H; (M) .
9 G. de Rham (1903-1969) - Belgian mathematician who worked mainly in alge braic topology. 10 We recall that a bilinear form L ( x, y) is nondegenerate if for every fixed nonzero value of one of the variables the resulting linear function of the other variable is not identically zero.
360
15 *Integration of Differential Forms on Manifolds
Definition 1 5 . If w is a closed p-form and
then the quantity per (z) : = w
the form over the cycle z.
J z
w
z is a p-cycle on the manifold M, is called the period ( or cyclic constant) of
I n particular, i f the cycle z i s homologous to zero, then, as follows from assertion b ) of Theorem 2, we have per (z) = 0. For that reason the following connection exists between periods:
(15.61) that is, if a linear combination of cycles is a boundary cycle, or, what is the same, is homologous to zero, then the corresponding linear combination of periods is zero. The following two theorems of de Rham hold; taken together, they are equivalent to Theorem 3. Theorem 4. ( de Rham ' s first theorem. ) A
if all its periods are zero.
closed form is exact if and only
Theorem 5. ( de Rham ' s second theorem. ) E Zp(M) M
If a number per (z) is assigned to each p-cycle z on the manifold in such a way that condition (15.61) holds, then there is a closed p-form on M such that J = per (z) for every cycle z E Zp(M) . w
z
w
15.4.3 Problems and Exercises 1. Verify by direct computation that the form a obtained in Example 2 does indeed satisfy the equation da = w.
2 . a ) Prove that every simply-connected domain in ll�? is contractible on itself to a point . b ) Show that the preceding assertion is generally not true in R3 .
Analyze the proof of Poincare's theorem and show that if the smooth mapping h : M x I ---+ M is regarded as a family of mappings ht : M ---+ M depending on the parameter t, then for every closed form w on M all the forms h;w, t E I, will be in the same cohomology class. 4. a ) Let t >--+ ht E c< oo l ( M, N) be a family of mappings of the manifold M into the manifold N depending smoothly on the parameter t E I C R. Verify that for every form w E fl(N) the following homotopy formula holds: 3.
(15.62)
15.4 Closed and Exact Forms on Manifolds
361
Here x E M, X is a vector field on N with X (x, t) E TNh, (x ) > X(x, t) is the velocity vector for the path t' >--+ ht� (x) at t' = t, and the operation ix of taking the inner product of a form and a vector field is defined in Problem 7 of the preceding section. b) Obtain the assertion of Problem 3 from formula ( 15.62) . c) Using formula ( 15.62) , prove Poincare's theorem (Theorem 1) again. d) Show that if K is a manifold that is contractible to a point, then HP ( K x M) = HP (M) for every manifold M and any integer p . e) Obtain relation (15.51) of Sect. 15.3 from formula ( 15.62) . 5. a) Show, using Theorem 4, and also by direct demonstration, that if a closed 2-form on the sphere 8 2 is such that J w = 0, then w is exact.
S2
b) Show that the group H 2 (8 2 ) is isomorphic to JR. c) Show that H 1 (8 2 ) = 0. 6. a) Let t.p : 8 2 --+ 8 2 be the mapping that assigns to each point x E 8 2 the antipodal point -x E 8 2 • Show that there is a one-to-one correspondence between forms on the projective plane IRJP>2 and forms on the sphere 8 2 that are invariant under the mapping c.p , that is, c.p * w = w. b) Let us represent IRJP>2 as the quotient space 8 2 I r, where r is the group of transformations of the sphere consisting of the identity mapping and the antipodal mapping t.p . Let 7r : 8 2 --+ IRJP>2 = 8 2 I r be the natural projection, that is 7r ( x ) = {x, -x } . Show that 1r o t.p = 1r and verify that 2 2 \17] E fJP (8 ) ( c.p * 1J = 17) -¢==} 3 w E fJ P (JRJP> ) (1r * w = 17) . c) Now show, using the result of Problem 5a) , that H 2 ( JRJP>2 ) = 0. d) Prove that if the function f E C(8 2 , IR) is such that f(x) - f ( -x) = const, then f = 0. Taking account of Problem 5c) , deduce from this that H 1 (IRJP>2 ) = 0.
a) Representing IRJP>2 as a standard rectangle II with opposite sides identified as shown by the orienting arrows in Fig. 15.3, show that all = 2c' - 2c, ac = P - Q, and ac' = p - Q. b) Deduce from the observations in the preceding part of the problem that there are no nontrivial 2-cycles on IRJP>2 • Then show by de Rham's theorem that H 2 (IRJP>2 ) = 0. 7.
Q '
c
p
p
II c
Fig. 1 5 . 3 .
Q
362
15 *Integration of Differential Forms on Manifolds
c) Show that the only nontrivial 1-cycle on JRJP'2 (up to a constant factor) is the cycle c' - c , and since c' - c = � 8II, deduce from de Rham's theorem that
H l (JRJP'2 )
=
0.
8. Find the groups H 0 (M) , H 1 (M) , and H 2 (M) if
a) M = 8 1 - the circle; b) M = T 2 - the two-dimensional torus; c) M = K 2 - the Klein bottle.
a) Prove that diffeomorphic manifolds have isomorphic (co)homology groups of the corresponding dimension. b) Using the example of JR 2 and JRJP'2 , show that the converse is generally not true. 9.
10. Let X and Y be vector spaces over the field lR and L(x, y) a nondegenerate bilinear form L : X x Y --+ JR. Consider the mapping X --+ Y* given by the correspondence X 3 x f--7 L(x, · ) E Y* . a) Prove that this mapping is injective. b) Show that for every system Y I , . . . , Y k of linearly independent vectors in Y there exist vectors x 1 , , x k such that x i (yj ) = L(x i , yj ) = 8) , where 8) = 0 if i ¥ j and 8j = 1 if i = j . c) Verify that the mapping X --+ Y* is an isomorphism of the vector spaces X and Y* . d) Show that de Rham's first and second theorems together mean that HP (M) = H; ( M) up to isomorphism. • • •
1 6 Uniform Convergence and the B asic Op erat ions of Analysis on Series and Families of Functions
16. 1 Pointwise and Uniform Convergence
16.1 . 1
Pointwise Convergence
1. We say that the sequence { !n ; n E N} offunctions fn : X ---+ IR converges at the point x E X if the sequence of values at x, Un (x); n E N} ,
Definition
converges.
Definition 2. The set of points E C X at which the sequence Un ; n E N} of functions fn : X ---+ IR converges is called the convergence set of the sequence.
{fn ; n ---+ f(x) := nlim --+ oo fn (x) . This function is called the limit function of the sequence { in ; n E N} or the limit of the sequence Un ; n E N} . Definition 4. If f E ---+ IR is the limit of the sequence { !n ; n E N } , we say that the sequence of functions converges ( or converges pointwise) to f on E. In this case we write f(x) = lim fn (x) on E or fn ---+ f on E as n ---+ oo . n --+ oo Example 1. Let X = { x E IR I x :::>: 0} and let the functions fn X ---+ IR be given by the relation fn ( x) = xn , n E N. The convergence set of this sequence of functions is obviously the closed interval I = [0, 1] and the limit function f I ---+ IR is defined by 0 , if 0 � X < 1 , f(x) =
Definition 3. On the convergence set of the sequence of functions E N} there naturally arises a function : E IR defined by the relation
f
:
:
:
{
1 , if x = 1 .
.
2
The sequence of functions fn ( x) = s m;: x on IR converges on IR to the function f : IR ---+ 0 that is identically 0. Example 3. The sequence fn ( x) = si�2'x also has the identically zero function f : IR ---+ 0 as its limit.
Example 2.
364
16 Uniform Convergence and Basic Operations of Analysis
[0 ,
0
Example 4. Consider the sequence ln (x) = 2(n + 1)x(1 - x 2 ) n on the closed interval I = 1] . Since nqn for l q l < 1 , this sequence tends to zero on -+
/.
the entire closed interval Example 5. Let m, n E N, and let lm (x) := nlim (cos ml7rx) 2 n . If m!x is an -+ao integer, then lm (x) = 1, and if m!x � Z, obviously lm (x) = We shall now consider the sequence Um ; m E N} and show that it con verges on the entire real line to the Dirichlet function V(x)
=
0.
{0'
if x � Q ,
0
1 ' if x E Q . Indeed, if x � Q, then m!x � Z, and lm (x) = for every value of m E N, so that l (x) = But if x = � · where p E Z and q E N, then m!x E Z for m ;::: q, and lm (x) = 1 for all such m, which implies l(x) = 1 . Thus lim lm (x) = V(x) . m-+oo
0.
1 6 . 1 . 2 Statement of the Fundamental Problems
Limiting passages are encountered at every step in analysis, and it is often important to know what kind of functional properties the limit function has. The most important properties for analysis are continuity, differentiability, and integrability. Hence it is important to determine whether the limit is a continuous, differentiable, or integrable function if the prelimit functions all have the corresponding property. Here it is especially important to find conditions that are sufficiently convenient in practice and which guarantee that when the functions converge, their derivatives or integrals also converge to the derivative or integral of the limit function. As the simple examples examined above show, without some additional hypotheses the relation "In -+ I on [a, b] as n -+ oo" does not in general imply either the continuity of the limit function, even when the functions In b b are continuous, or the relation I� -+ f' or J ln (x) dx -+ J l(x) dx, even when a
a
0
all these derivatives and integrals are defined. Indeed, in Example 1 the limit function is discontinuous on [ , 1] although all the prelimit functions are continuous there; in Example 2 the derivatives n cos n 2 x of the prelimit functions in general do not converge, and hence cannot converge to the derivative of the limit function, which in this case is identically zero; 1
J l(x) dx = 0
1
in Example 4 we have J In ( x) dx
0;
0
=
1 for every value of n E N, while
16.1 Pointwise and Uniform Convergence
365
in Example 5 each of the functions fm equals zero except at a finite set of points, so that
b
=
C
J fm (x) dx 0 on every closed interval [a, b] IR, while the (],
limit function V is not integrable on any closed interval of the real line.
At the same time: in Examples 2, 3, and 4 both the prelimit and the limit functions are continuous; in Example 3 the limit of the derivatives co�n x of the functions in the sequence si�;'x does equal the derivative of the limit of that sequence; in Example
1 1 1 we have J fn (x) dx --t J f(x) dx as n --t oo .
0 0 Our main purpose is to determine the cases in which the limiting passage under the integral or derivative sign is legal. In this connection, let us consider some more examples.
Example 6.
We know that for any
1 3!
.
Sill X = X - - X
3
1 +5! X
5
x E IR
- · · ·
m + ( -1)+ X 2m +l + · · · ' (2m l ) !
(16.1)
but after the examples we have just considered, we understand that the re lations
b
sin ' X
=
(-1) m x2m +l ) ' f ( m =O (2m + 1)! 00
(16.2)
b
(- l ) m x 2m + 1 dx / sin x dx """' J ' � -O (2m + 1)! =
a
m
( 16.3)
a
require verification in general. Indeed, if the equality
S(x) = a 1 (x) + a 2 (x) + · · · + am (x) + · · ·
n
is understood in the sense that S(x) = lim Sn (x) , where Sn (x) = 2:::: a m (x), n -+oo m =1 then by the linearity of differentiation and integration, the relations 00
b
S'(x) = L a� (x) , m =1
J S(x) dx a
=
00
b
L J am (x) dx
m= l
a
366
16 Uniform Convergence and Basic Operations of Analysis
are equivalent to
J b
S' (x) = lim S� (x) , n -4oo S(x) dx
a
=
J b
lim n -4oo
a
Sn (x) dx ,
which we must now look upon with caution. In this case both relations ( 16.2) and (16.3) can easily be verified, since it is known that COS X
=1
-
1 2 2!
-X
1 4!
+ -X
4
-
·
·
·
+
( - l) m 2 m + X (2m) !
---
·
·
·
.
However, suppose that Eq. ( 1 6 . 1 ) is the definition of the function sin x. After all, that was exactly the situation with the definition of the functions sin z, cos z, and e z for complex values of the argument . At that time we had to get the properties of the new function (its continuity, differentiability, and integrability) , as well as the legality of the equalities ( 16.2) and ( 16.3) directly from the fact that this function is the limit of the sequence of partial sums of this series. The main concept by means of which sufficient conditions for the legality of the limiting passages will be derived in Sect. 16.3, is the concept of uniform convergence. 1 6 . 1 . 3 Convergence and Uniform Convergence of a Family of Functions Depending on a Parameter
In our discussion of the statement of the problems above we confined our selves to consideration of the limit of a sequence of functions. A sequence of functions is the most important special case of a family of functions f (x) depending on a parameter t. It arises when t E N. Sequences of functions thus occupy the same place occupied by the theory of limit of a sequence in the theory of limits of functions. We shall discuss the limit of a sequence of functions and the connected theory of convergence of series of functions in Sect . 16.2. Here we shall discuss only those concepts involving functions depending on a parameter that are basic for everything that follows.
t
Definition 5. We call a function (x, t) t-t F(x, t) of two variables x and t defined on the set X x T a family of functions depending on the parameter t if one of the variables t E T is distinguished and called the parameter.
The set T is called the parameter set or parameter domain, and the family itself is often written in the form ft (x) or t E T} , distinguishing the parameter explicitly.
Ut i
16. 1 Pointwise and Uniform Convergence
367
As a rule, in this book we shall have to consider families of functions for which the parameter domain T is one of the sets N or JR. or C of natural numbers, real numbers, or complex numbers or subsets of these. In general, however, the set T may be a set of any nature. Thus in Examples 1 - 5 above we had T = N. In Examples 1-4 we could have assumed without loss of content that the parameter n is any positive number and the limit was taken over the base n -+ oo, n E JR. + .
Definition 6 . Let Ut : X -+ JR.; t E T} be a family of functions depending on a parameter and let B be a base in the set T of parameter values. If the limit lim ft (x) exists for a fixed value x E X, we say that the family 8
of functions converges at x. The set of points of convergence is called the convergence set of the family of functions in a given base B. Definition 7. We say that the family of functions converges on the set E X over the base B if it converges over that base at each point x E E. The function f(x) := lim ft(x) on E is called the limit function or the limit of the family of functions ft on the set E over the base B. Example 7. Let ft (x) = e - ( x / t ) 2 , x E X = JR., t E T = JR. \ 0, and let B be the base t -+ 0. This family converges on the entire set JR., and 1 , if x = O , lim ft(x) = t -+ oo O , if x tf: O . C
8
{
We now give two basic definitions.
Ut ; t E T} of functions ft : X -+ JR. converges pointwise (or simply converges) on the set E X over the base B to the function f : E -+ JR. if lim ft (x) = f(x) at every point x E E. In this case we shall often write Ut f on E). Definition 9 . The family {ft; t E T} of functions ft : X -+ JR. converges uniformly on the set E X over the base B to the function f : E -+ JR. if for every c > 0 there exists an element B in the base B such that l f(x)-ft(x) l < c at every value t E B and at every point x E E. In this case we shall frequently write Ut � f on E) . Definition 8. We say that the family 8
C
C
� 8
8
We give also the formal expression of these important definitions: � 8
Ut f on E) := Vc > 0 Vx E E ::JB E B Vt E B (if(x) - ft(x) l < c) , Ut � f on E) := Vc > 0 ::JB E B Vx E E Vt E B (if(x) - ft (x) l < c) . 8
368
16 Uniform Convergence and Basic Operations of Analysis
The relation between convergence and uniform convergence resembles the relation between continuity and uniform continuity on a set. To explain better the relationship between convergence and uniform convergence of a family of functions, we introduce the quantity L1t (x) = l f(x) - ft (x) l , which measures the deviation of the value of the function ft from the value of the function f at the point x E E. Let us consider also the quantity L1t = sup L1t (x) , which characterizes, roughly speaking, the maxi-
mum deviation ( although there may not be a maximum ) of the function ft from the corresponding values of f over all x E E. Thus, at every point x E E we have L1t (x) ::::; L1t . In this notation these definitions obviously can be written as follows: xEE
(ft --,: f on E) .- Vx E E (L1t (x) --+ 0 over B) , ( L1t --+ 0 over B) . (It :::4 f on E) 8
It is now clear that
Ut :::4 f on E) 8
===>
f on E) , Ut --+ 8
that is, if the family ft converges uniformly to f on the set E, it converges pointwise to f on that set . The converse is in general not true.
Example 8. Let us consider the family of functions ft : I --+ lR defined on the closed interval I = { x E lR I 0 ::::; x ::::; 1} and depending on the parameter t E]O, 1]. The graph of the functions y = ft (x) is shown in Fig. 16.1. It is clear that tlim ft (x) = 0 at every point x E I, that is, ft --+ f = 0 as t --+ 0. -tO At the same time L1t = sup l f(x) - ft (x) l = sup l ft (x) l = 1, that is, L1t -/+ 0 xEI xEI as t --+ 0, and hence the family converges, but not uniformly. In such cases we shall say for convenience that the family converges nonuniformly to the limit function. If the parameter t is interpreted as time, then convergence of the family of functions ft on the set E to the function f means that for any preassigned precision c > 0 one can exhibit a time tc: for each point x E E starting from which ( that is, for t > tc: ) the values of all functions ft at x will differ from f(x) by less than c . Uniform convergence means that there is a time tc: , starting from which ( that is, for t > tc: ) the relation l f(x) - ft (x) l < c holds for all x E E. The figure of a traveling bulge of large deviation depicted in Fig. 16.1 is typical for nonuniform convergence.
Example 9. The sequence of functions fn ( x) = x n - x2n defined on the closed interval 0 ::::; x ::::; 1, as one can see, converges to zero at each point as n --+ oo .
369
16. 1 Pointwise and Uniform Convergence y 1
0 t 2t
1
X
Fig. 1 6. 1 .
To determine whether this convergence i s uniform, we find the quantity Ll n = max lfn (x) l . Since f� (x) = nx n - 1 (1 - 2x n ) = 0 for x = 0 and x = 2-1/ n ,
0:5x9
it is clear that Ll n = fn (2- 11 n ) = 1/4. Thus Ll n f+ 0 as n --* oo and our sequence converges to the limit function f(x) = 0 nonuniformly.
Example 1 0.
The sequence of functions fn converges to the function
f(x)
=
{
=
x n on the interval 0 ::; x ::; 1
0 ' if O :S x < 1 , X =
1 , if
1
nonuniformly, since for each n E N
Ll n
=
sup I J (x) - fn (x) l
o::;x::; 1
=
sup l f(x) - fn (x) l
o::;x < 1
=
=
sup l fn (x) l
0 :5x< 1
=
sup l x n l
0:5 x < 1
=
1.
The sequence of functions fn (x) = s in;: x studied in Example 2 converges to zero uniformly on the entire set lR as n --* oo , since in this case 2
Example 1 1 .
lf(x) - fn (x) l
=
lfn (x) l
=
l -nn-2 x l ::; ;1 sin
'
that is, Ll n ::; 1/n, and hence Ll n --* 0 as n --* oo. 1 6 . 1 . 4 The Cauchy Criterion for Uniform Convergence
In Definition 9 we stated what it means for a family of functions ft to converge uniformly on a set to a given function on that set . Usually, when the family of functions is defined the limit function is not yet known, so that it makes sense to adopt the following definition.
370
16 Uniform Convergence and Basic Operations of Analysis
Definition 10. We shall say that the family Ut ; t E T} of functions ft X --+ � converges on the set E X uniformly over the base B if it converges C
on that set and the convergence to the resulting limit function is uniform in the sense of Definition 9. Theorem. ( Cauchy criterion for uniform convergence ) . Let {it ; t E T} be a family of functions it : X --+ � depending on a parameter t E T, and B a
base in T. A necessary and sufficient condition for the family {ft ; t E T} to converge uniformly on the set E X over the base B is that for every E > 0 there exist an element B of the base B such that I it, ( x) - ft2 ( x) I < E for every value of the parameters h , t 2 E B and every point x E E . I n formal language this means that it converges uniformly o n E over the base B {:::::::?- Vc > 0 3B E B 'rft 1 , t z E B Vx E E ( l it, (x) - it2 (x) l < c). Proof. N e c e s s i t y. The necessity of these conditions is obvious, since if f : E --+ � is the limit function and ft f on E over B, there exists an element B in the base B such that l f(x) - ft (x) l < c/2 for every t E B and every x E E. Then for every t 1 , t 2 E B and every x E E we have l it, (x) - ft2 (x) l :::; lf( x) - it, (x) l + lf( x) - it2 (x) l < c/2 + c/2 = E S u f f i c i e n c y. For each fixed value of x E E we can regard ft (x) as a function of the variable t E T. If the hypotheses of the theorem hold, then the C
�
.
hypotheses of the Cauchy convergence criterion for the existence of a limit over the base B are fulfilled. Hence, the family {it ; t E T} converges at least pointwise to some function f : E --+ � on the set E over the base B. If we now pass to the limit in the inequality I ft, ( x) - it2 ( x) I < E, which is valid for any t 1 and t 2 E B and every x E E, one can obtain the inequality l f(x) - it2 (x) l :::; E for every tz E B and every x E E, and this, up to an inessential relabeling and the change of the strict inequality to the nonstrict, coincides exactly with the definition of uniform convergence of the family Ut ; t E T} to the function f : E --+ � on the set E over the base B. D Remark 1 . The definitions of convergence and uniform convergence that we have given for families of real-valued functions ft : X --+ � of course remain valid for families of functions ft : X --+ Y with values in any metric space Y . The natural modification that one must make in the definitions in this case amounts to replacing l f(x) - it (x) l by dy f (x) , it (x) ) , where dy is the metric in Y . For normed vector spaces Y, i n particular for Y
(
y=
=
16.1 Pointwise and Uniform Convergence
371
16. 1 . 5 Problems and Exercises 1. Determine whether the sequences of functions considered in Examples 3-5 con verge uniformly. 2.
Prove Eqs. ( 16.2) and ( 16.3).
3. a) Show that the sequence of functions considered in Example 1 converges uni formly on every closed interval [0, 1 - o] C [0, 1] , but converges nonuniformly on the interval [0, 1 [. b) Show that the same is true for the sequence considered in Example 9. c) Show that family of functions ft considered in Example 8 converges uniformly as t ---+ 0 on every closed interval [o, 1] C [0, 1] but nonuniformly on [0, 1] . d) Investigate the convergence and uniform convergence of the family of func tions ft ( x) = sin( tx) as t ---+ 0 and then as t ---+ oo. 2 e) Characterize the convergence of the family of functions ft (x) = e - t x as t ---+ +oo on an arbitrary fixed set E C JR.
4. a) Verify that if a family of functions converges (resp. converges uniformly) on a set, then it also converges (resp. converges uniformly) on any subset of the set. b) Show that if the family of functions ft : X ---+ JR converges (resp. converges uniformly) on a set E over a base B and g : X ---+ JR is a bounded function, then the family g · ft : X ---+ JR also converges (resp. converges uniformly) on E over the base B. c) Prove that if the families of functions ft : X ---+ JR, 9t : X ---+ JR converge uniformly on E C X over the base B, then the family ht = aft + f3gt , where a, (3 E JR, also converges uniformly on E over B. 5. a) In the proof of the sufficiency of the Cauchy criterion we passed to the limit lim /t1 (x) = f(x) over the base B in T. But h E B, and B is a base in T, not in B . B Can we pass to this limit in such a way that t1 remains in B? b) Explain where the completeness of JR was used in the proof of the Cauchy criterion for uniform convergence of a family of functions ft : X ---+ JR. c) Notice that if all the functions of the family {ft : X ---+ JR; t E T} are constant, then the theorem proved above is precisely the Cauchy criterion for the existence of the limit of the function r.p : T ---+ JR over the base B in T.
6. Prove that if the family of continuous functions ft E C(J, JR) on the closed interval I = { x E JR I a ::; x ::; b} converges uniformly on the open interval ]a, b[, then it converges uniformly on the entire closed interval [a, b] .
16 Uniform Convergence and Basic Operations of Analysis
372
1 6 . 2 Uniform Convergence of Series of Functions 1 6 . 2 . 1 Basic Definitions and a Test for Uniform Convergence of a Series :
{an X ---+ C; n E N} be a sequence of complex-valued (in particular real-valued) functions. The series L an (x) converges or converges n= l uniformly on the set E X if the sequence { s m (x) � an (x); n E N} nl converges or converges uniformly on E. Definition 1. Let
00
C
=
=
m
s m (x) L= an (x), as in the case of numerical nl series, is called the partial sum or, more precisely, the mth partial sum of the series L= an (x) . n l Definition 3 . The sum of the series is the limit of the sequence of its partial Definition 2 . The function 00
sums.
Thus, writing =
00
s(x) L an (x) on E n= l means that s m (x) ---+ s(x) on E as m ---+ oo , and writing the series L a n (x) converges uniformly on E n= l means that s m (x) s(x) on E as m ---+ oo . 00
:::4
Investigating the pointwise convergence of a series amounts t o investigat ing the convergence of a numerical series, and we are already familiar with that .
Example 1 .
Earlier we defined the function exp exp z
:=
00
L 1n1 . z n , n =O
:
C ---+ C by the relation (16.4)
after first verifying that the series on the right converges for every value
z E C.
In the language of Definitions 1-3 one can now say that the series (16.4) n converges on the entire complex plane, and the of functions a n (z) = �z . n function exp z is its sum. By Definitions 1 and 2 just adopted a two-way connection is established between series and their sequences of partial sums: knowing the terms of the
373
16.2 Uniform Convergence of Series of Functions
series, we obtain the sequence of partial sums, and knowing the sequence of partial sums, we can recover all the terms of the series: the nature of the convergence of the series is identified with the nature of the convergence of its sequence of partial sums.
Example 2. In Example 5 of Sect . 16.1 we constructed a sequence {fm ; m E N} of functions that converge to the Dirichlet function 'D( x) on If we set a 1 (x) fi (x) and an (x) fn (x) - fn - l (x) for n > 1, we obtain a series 2::: a (x) that will converge on the entire number line, and 2:::= an (x) 'D(x). n= l n nl Example 3. It was shown in Example 9 of Sect. 16.1 that the sequence of functions fn ( x) x n -x 2n converges, but nonuniformly, to zero on the closed interval [0, 1]. Hence, setting a 1 (x) fi (x) and a n (x) = fn (x) - fn - l (x) for n > 1, we obtain a series 2:::= an (x) that converges to zero on the closed nl interval [0, 1], but converges nonuniformly. =
=
00
=
00
00
1Ft
=
=
The direct connection between series and sequences of functions makes it possible to restate every proposition about sequences of functions as a corresponding proposition about series of functions. Thus, in application to the sequence { s n : X --+
(16 . 5)
1, we obtain the following theorem. Theorem 1. ( Cauchy criterion for uniform convergence of a series ) . The series 2:::= an (x) converges uniformly on a set E if and only if for every > 0 n l N E N such that there exists (16.6) for all natural numbers m, n satisfying m � n > N and every point x E E. Proof. Indeed, setting n 1 m, n 2 n - 1 in (16.5) and assuming that sn (x) is the partial sum of the series, we obtain inequality (16.6), from which relation (16 . 5) in turn follows with the same notation and hypotheses of the From this, taking account of Definition 00
c:
=
theorem.
D
=
Remark 1 . We did not mention the range of values of the functions an (x) in the statement of Theorem 1, taking for granted that it was IR or
374
16 Uniform Convergence and Basic Operations of Analysis
Remark 2. If under the hypotheses of Theorem 1 all the functions an (x) are constant, we obtain the familiar Cauchy criterion for convergence of a 00 numerical series L a n . n= l
Corollary 1 . ( Necessary condition for uniform convergence of a series ) . A 00
necessary condition for the series L= an (x) to converge uniformly on a set n l E is that an 0 on E as n --+ Proof. This follows from the definition of uniform convergence of a sequence to zero and inequality (16.6) if we set m n in it. Example 4. The series (16.4) converges on the complex plane C nonuniformly, since sup I rh z n I for every n E N, while by the necessary condition EIC ::::+
oo .
=
0
= oo
z
for uniform convergence, the quantity sup l an (x) l must tend to zero when xE E
uniform convergence occurs. 00
L z , as we know, converges in the unit disk K = n= l n {z E q l zl < 1}. Since I z: I < � for z E K , we have z: ::::+ 0 on K as n --+ oo.
Example 5.
The series
n
The necessary condition for uniform convergence is satisfied; however, this series converges nonuniformly on K. In fact, for any fixed n E N, by the continuity of the terms of the series, if z is sufficiently close to 1, we can get the inequalities
n
I zn
+
...+
z2n
I
�
>
1.!_
+
...
+
_!_
I
>
�
.
2n 2 n 2n 4 From this we conclude by the Cauchy criterion that the series does not converge uniformly on K. 1 6 . 2 . 2 The Weierstrass M-test for Uniform Convergence of a Series Definition 4. The series
00
L a (x) n= l n
converges absolutely on the set E if the
corresponding numerical series converges absolutely at each point 00
00
x E E.
If the series L= an ( x) and L= bn ( x) are such that I an ( x) I ::; n l n l bn ( x) for every x E E and for all sufficiently large indices n E N, then the uniform convergence of the series L= bn (x) on E implies the absolute and n l uniform convergence of the series L= an (x) on the same set E. Proposition 1 .
00
00
n l
375
16.2 Uniform Convergence of Series of Functions
Proof. Under these assumptions for all sufficiently large indices n and m (let n :S m ) at each point x E E we have l an (x) + · · · + am (x) l :S l an (x) l + · · · + l am ( x ) l :S :S bn ( x ) + · · · + bm (x) = l b n ( x ) + · · · + bm (x) l .
By the Cauchy criterion and the uniform convergence of the series L: bn ( x ) , for each c > 0 we can exhibit an index N E N such that 00
n=l lbn (x) + · · · + bm ( x ) l < for all m 2: n > N and all x E E. But then it follows from the inequalities just written and the Cauchy criterion that the series L: an ( x) and L: I a n ( x) I both converge uniformly. D n=l n=l Corollary 2. (Weierstrass' M-test for uniform convergence of a series) . If for the series n=l L: an (x) one can exhibit a convergent numerical series L: Mn n=l such that sup lan (x) l :S Mn for all sufficiently large indices n E N, then the 00
00
c
00
00
00
xE E
series n=l L: an ( x) converges absolutely and uniformly on the set E. Proof. The convergent numerical series can be regarded as a series of constant functions on the set E, which by the Cauchy criterion converges uniformly on E. Hence the Weierstrass test follows from Proposition 1 if we set bn ( x ) Mn =
in it. D
The Weierstrass M-test is the simplest and at the same time the most frequently used sufficient condition for uniform convergence of a series. As an example of its application, we prove the following useful fact . 00
If a power series n=O L: en (z - z0) n converges at a point ( =I zo , then it converges absolutely and uniformly in any disk Kq {z E C l l z - zo l < ql( - zo l } , where 0 < q < 1. Proof. By the necessary condition for convergence of a numerical series it
Proposition 2.
=
00
follows from the convergence of the series L: en ( ( - zo ) n that Cn ( ( - z) n -+ 0 n=O as n -+ oo. Hence for all sufficiently large values n of n E N we have the estimates l en (z - zo ) n l = len (( - zo) n l · I (=�� I :S l en (( - zo) n l · q n < qn 00 in the disk Kq . Since the series L: qn converges for lql < 1, the estimates n=O len (z - zo) n l < qn and the Weierstrass M-test now imply Proposition 2. D
Comparing this proposition with the Cauchy-Hadamard formula for the radius of convergence of a power series (see Eq. (5.115)), we arrive at the following conclusion.
376
16 Uniform Convergence and Basic Operations of Analysis
( Nature of convergence of a power series ) . A power zo) n converges in the disk K = {z E C l l z - zo l < R}
series whose I: c (z n=O n radius of convergence1 is determined by the Cauchy-Hadamard formula1 R = ( nlim \I[CJ) - . Outside this disk the series diverges. On any closed --+ oo disk contained in the interior of the disk K of convergence of the series, a power series converges absolutely and uniformly. Remark 3. As Examples 1 and 5 show, the power series need not converge Theorem 2. 00
uniformly on the entire disk K. At the same time, it may happen that the power series does converge uniformly even on the closed disk K. 00
Example 6. The radius of convergence of the series I:= � is 1 . But if l z l :::; 1 , n1 then I� I :::; r& , and by the Weierstrass M-test this series converges absolutely and uniformly in the closed disk K = {z E C l l z l :::; 1}. n
1 6 . 2 . 3 The Abel-Dirichlet Test
The following pairs of related sufficient conditions for uniform convergence of a series are somewhat more specialized and are essentially connected with the real-valuedness of certain components of the series under consideration. But these conditions are more delicate than the Weierstrass M-test, since they make it possible to investigate series that converge, but nonabsolutely. Definition 5 . The family F of functions
on a set E for every
C
f
X if there exists a number M
f E F.
X ---+ C is uniformly bounded E lR such that xEE sup l f(x) l :::; M
:
{bn : X ---+ JR; n E N} is called non decreasing (resp. nonincreasing) on the set E X if the numerical sequence {bn (x); n E N} is nondecreasing (resp. nonincreasing) for every x E E. Non decreasing and nonincreasing sequences of functions on set are called mono tonic sequences on the set.
Definition 6 . The sequence of functions
C
We recall ( if necessary, see Subsect. 5.2.3) the following identity, called
Abel 's transformation: m m-1 2.: a k b k A m bm - A n - 1 bn 2.: A k (bk - b k + l ) , k= n k=n where a k = A k - A k - 1 , k = n, . . . , m. =
1 In the exceptional case when lim n -+ oo degenerates to the single point z0 .
+
VfCJ =
oo ,
we take R
=
(16.7)
0 and the disk K
16.2 Uniform Convergence of Series of Functions
377
is a monotonic sequence of real numbers, then, even if , +. ,1a, . . .are, bmcomplex or vectors of a normed space, one can aobtain ,n Ifanb+nthe 1 , b. n. following m estimate,numbers which we need, from the identity (16.7): m (16.8) I k=Ln akbk l :S 4 n-max 1
Proof.
In fact,
I � Ak (bk - bk- 1 ) 1 :S 1 � A I + I b I ( + I b I kl · n m n -1
I Am bm l I An- 1bn l +
=
+
max
=
L...t
n-1:5maxk:S m I Akl · ( l bm l l bn l + I bn - bm l ) :S +
=
In the equality that occurs in this computation we used the monotonicity of the numerical sequence D
bk.
Proposition 3. ( The Abel-Dirichlet test for uniform convergence ) .
oo
A suf-
ficient condition for uniform convergence on E of a series I:=1 an (x)bn (x) whose terms are products of complex-valued functions an : X n-+ C and realvalued functions bn : X -+ IR is that either of the following pairs of hypotheses be satisfied: k ) the partial sums s k (x) I: an (x) of the series I: a n (x) are unin=1 n=1 formly bounded on E; /h) the sequence of functions bn (x) tends monotonically and uniformly to zero on E; or ) the series I: an (x) converges uniformly on E; n=1 /h) the sequence of functions bn (x) is monotonic and uniformly bounded on E. Proof. The monotonicity of the sequence bn (x) allows us to to write an esti mate analogous to (16.8) for each x E E: 00
=
o: I
00
o:2
I k=�n ak (x)bk (x) l :S n- 1
where we take
4
max
sk (x) - Sn- 1 (x) as Ak (x).
max
)
378
16 Uniform Convergence and Basic Operations of Analysis
If the hypotheses o: 1 ) and (31 ) hold, then, on the one hand, there exists a constant M such that I A k ( x) I ::; M for all k E N and all x E E, while on the other hand, for any number c > 0 we have max { l bn (x) l , l bm (x) l } < 4� for all sufficiently large n and m and all x E E. Hence it follows from (16.8) that a k (x)b k (x) < c for all sufficiently large n and m and all x E E, that is,
I k�n
l
the Cauchy criterion holds for this series. In the case of hypotheses o: 2 ) and f32 ) the quantity max { l bn (x) l , l bm (x) l } is bounded. At the same time, by the uniform convergence of the series 00 I: an (x) and the Cauchy criterion, for every c > 0 we have I A k (x) l =
n= 1 S l s k (x) - n - 1 (x) l
< c for all sufficiently large n and k > n and all x E E. Taking this into account, we again conclude from (16.8) that the Cauchy criterion for uniform convergence holds for this series. D Remark 4. In the case when the functions an and bn are constants Propo sition 3 becomes the Abel-Dirichlet criterion for convergence of numerical series. Example 7. Let us consider the convergence of the series
(16.9) Since
1 1 inx l _ 1
e (16.10) - n"' ' n"' the necessary condition for uniform convergence does not hold for the series (16.9) when o: ::; 0, and it diverges for every x E R Thus we shall assume o: > 0 from now on. If o: > 1, we conclude from the Weierstrass M-test and (16.10) that the series (16.9 ) converges absolutely and uniformly on the entire real line R To study the convergence for 0 < o: ::; 1 we use the Abel-Dirichlet test, setting a n (x) = e i nx and bn (x) = n1a . Since the constant functions bn (x) are monotonic when o: > 0 and obviously tend to zero uniformly for x E JR, it 00 remains only to investigate the partial sums of the series I: ei nx .
n= 1
n
For convenience in citing results below, we shall consider the sums I: ei kx , k =O which differ from the sums of our series only in the first term, which is 1. Using the formula for the sum of a finite geometric series and Euler ' s formula, we obtain successively for x =/= 2nm, m E Z, · 1 x -2 x ei( n + 1 ) x - 1 sin n.=_+2 _ e' � . x ei - 1 Sin � e' 2 __
sin n +2 1 X . sin �
---=--,----- e ' 2 n
•
x -
n x = sin n +2 1 ( cos -x sin �
X
2
+
n i sin -x 2
)
(
16.11 )
379
16.2 Uniform Convergence of Series of Functions
Hence, for every n E
N
1_ ei k x l < _ L sm 2 I k=O n
-
X
•
(16.12)
'
from which it follows by the Abel-Dirichlet criterion that for 0 < a $ 1 the series (16.9) converges uniformly on every set E C lR on which inf I sin i I > x EE 0. I n particular the series (16.9) simply converges for every x =/= 2 rr m , m E Z. If x = 2 rrm , then ei n 2 7r m = 1, and the series (16.9) becomes the numerical 00 series 2: ..1., , which diverges for 0 < a $ 1. n= l We shall show that from what has been said, one can conclude that for 0 < a $ 1 the series (16.9) cannot converge uniformly on any set E whose closure contains points of the form 2 rr m , m E Z. For definiteness, suppose 00 0 E E. The series 2: ..1., diverges for 0 < a $ 1. By the Cauchy criterion,
n= l
there exists c > 0 such that for every N E N, no matter how large, one can find numbers m 2: n > N such that ..1., + · · · + �" > co > 0. By the continuity of the functions ei k x on JR, it follows that one can choose a point x E E close enough to 0 so that
I
I
lei nx eimx > co . + · · · + -n <>
m <>
I
But by the Cauchy criterion for uniform convergence this means that the series (16.9) cannot converge uniformly on E. To supplement what has just been said, we note that , as one can see from
(16.10), the series (16.9) converges nonabsolutely for 0 < a $ 1. Remark 5. It is useful for what follows t o remark that, separating the real and imaginary parts in (16.11), we obtain the following relations: n n+l (16.13) L: cos kx = cos !!2 x · sin 2 k=O n
L: sin kx = k=O
which hold for x =/= 2 rrm , m
E Z.
X
sin i
n+l sin !!x 2 · sin 2 X sin i
(16.14)
As another exmaple of the use of the Abel-Dirichlet test we prove the following proposition.
If a power series n=O L c ( converges at a point ( E C, then it converges uniformly on the closed interval with endpoints and ( . Proposition 4. (The so-called second Abel theorem on power series. ) 00
..
z
-
z0 ) n
zo
16 Uniform Convergence and Basic Operations of Analysis
380
Proof. We represent the points of this interval in the form zo + (( - zo)t, where 0 ::; t ::; 1 . Substituting this expression i n the power series, we obtain 00 00 the series L en (( - zo) n t n . By hypothesis, the numerical series L en (( -
=O n =O and the sequence of functions t n is monotonic andnuniformly z0) n converges,
bounded on the closed interval [0, 1] . Hence conditions a 2 ) and /32 ) in the Abel-Dirichlet test are satisfied, and the proposition is proved. D 1 6 . 2 . 4 Problems and Exercises 1 . Investigate the nature of the convergence on the sets E C lR for different values of the real parameter a in the following series:
a) E co::x . =l CXl
n
b) 2: si�:x . CXl
2.
n= l Prove that the following series converge uniformly on the indicated sets: /_ 1 \n a) I: �x n for 0 � x � 1 . n= l n b) E ( -� ) e - n x for 0 � X � +oo. = l n f _ l \n c) I: '* for 0 � x � +oo. = l n CXl
CXl
Show that if a Dirichlet series I: � converges at a point xo E JR, then it n= l converges uniformly on the set x � xo and absolutely if x > xo + 1 . 3.
CXl
= n-1 2 Verify that the series I: ( (ft2)'':' converges uniformly on JR, and the series
n= l I: ( l ;:2 ) n converges on JR, but nonuniformly. n= l 5 . a) Using the example of the series from Problem 2, show that the Weierstrass M-test is a sufficient condition but not a necessary one for the uniform convergence of a series. b) Construct a series I: a n (x) with nonnegative terms that are continuous on n= l the closed interval 0 � x � 1 and which converges uniformly on that closed interval, while the series I: Mn formed from the quantities M = maxl i a n (x) l diverges. O � :z: � n=l 4.
CXl
CXl
CXl
6. a) State the Abel-Dirichlet test for convergence of a series mentioned in Re mark 4. b) Show that the condition that {bn } be monotonic in the Abel-Dirichlet test can be weakened slightly, requiring only that the sequence {bn } be monotonic up to corrections {.Bn } forming an absolutely convergent series.
16.3 Functional Properties of a Limit Function
381
7. As a supplement to Proposition 4 show, following Abel, that if a power series converges at a boundary point of the disk of convergence, its sum has a limit in that disk when the point is approached along any direction not tangential to the boundary circle.
1 6 . 3 Functional Properties of a Limit Function 1 6 . 3 . 1 Specifics of the Problem
In this section we shall give answers to the questions posed in Sect . 16.1 as to when the limit of a family of continuous, differentiable, or integrable functions is a function having the same property, and when the limit of the derivatives or integrals of the functions equals the derivative or integral of the limiting function of the family. To explain the mathematical content of these questions, let us consider, for example, the connection between continuity and passage to the limit. Let fn (x) -+ f(x) on lR as n -+ oo , and suppose that all the functions in the sequence {fn : n E N} are continuous at the point xo E R We are interested in the continuity of the limit function f at the same point xo. To answer that question, we need to verify the equality xlim --+ x o f(x) = f(xo), which in terms of the original sequence can be rewritten as the relation or, taking account of the given continuity lim lim f = lim f x --+ xo ( n --+oo n (x)) n --+oo n (x0), of fn at x0, as the following relation, subject to verification:
--+oo fn (x)) x --t x o ( nlim lim
= nlim --too ( xlim --t x o fn (x) ) .
(16.15)
On the left-hand side here the limit is first taken over the base n -+ oo , then over the base x -+ x0, while o n the right-hand side the limits over the same bases are taken in the opposite order. When studying functions of several variables we saw that Eq. (16.15) is by no means always true. We also saw this in the examples studied in the two preceding sections, which show that the limit of a sequence of continuous functions is not always continuous. Differentiation and integration are special operations involving passage to the limit. Hence the question whether we get the same result if we first differentiate (or integrate) the functions of a family, then pass to the limit over the parameter of the family or first find the limit function of the family and then differentiate (or integrate) again reduces to verifying the possibility of changing the order of two limiting passages. 16.3.2 Conditions for Two Limiting Passages to Commute
We shall show that if at least one of two limiting passages is uniform, then the limiting passages commute.
16 Uniform Convergence and Basic Operations of Analysis
382
Let { Ft ; t E T} be a family of functions Ft : X --+ C depending on a parameter t; let Bx be a base in X and Br a base in T. If the family converges uniformly on X over the base Br to a function F : X --+ C and the limit lim Ft ( x ) = At exists for each t E T, then both repeated limits Bx lim ( lim Ft ( x )) and lim ( lim Ft ( x )) exist and the equality BT Bx Bx BT Theorem 1 .
(16.16) holds. This theorem can be conveniently written as the following diagram: Ft ( x ) ======* F ( x )
Bx 1 / At
Br / / /
__l_ Br
31
Br
(16.17)
A
in which the hypotheses are written above the diagonal and the consequences below it . Equality (16.17) means that this diagram is commutative, that is, the final result A is the same whether the operations corresponding to passage over the upper and right-hand sides are carried out or one first passes down the left-hand side and then to the right over the lower side. Let us prove this theorem.
Proof. Since Ft :::t F on X over Br, by the Cauchy criterion, for every c: > there exists Br in Br such that
0
(16.18) for every h, t 2 E Br and every x E X. Passing to the limit over Bx in this inequality, we obtain the relation
(16.19) which holds for every h , t 2 E Br . By the Cauchy criterion for existence of the limit of a function it now follows that At has a certain limit A over Br . We now verify that A = lim F ( x ) . Bx Fixing t 2 E Br, we find an element Bx in Bx such that
(16 . 20)
for all x E Bx . Keeping t 2 fixed, we pass to the limit in respect to h . We then find
(16.18) and (16.19) over Br with
383
16.3 Functional Properties of a Limit Function
(16.21) I F(x) - Ft2 (x) l :::; E , (16.22) l A - A t2 1 :::; E ' and (16.21) holds for all x E X. Comparing (16.20)-(16.22), and using the triangle inequality, we find I F(x) - A I < 3E for every x E B We have thus verified that A = lim F( x). 0 Bx Remark 1 . As the proof shows, Theorem 1 remains valid for functions Ft x.
X ---+ Y with values in any complete metric space.
If we add the requirement that the limit lim A t = A exists to the Br hypotheses of Theorem 1, then, as the proof shows, the equality lim F(x) = A Bx can be obtained even without assuming that the space Y of values of the functions Ft : X ---+ Y is complete.
Remark 2.
16.3.3 Continuity and Passage to the Limit
We shall show that if functions that are continuous at a point of a set converge uniformly on that set, then the limit function is also continuous at that point.
Let {ft; t E T} be a family of functions ft : X ---+ C depending on the parameter t; let B be a base in T. If ft f on X over the base B and the functions ft are continuous at x0 E X, then the function f : X C is also continuous at that point. Proof. In this case the diagram (16.17) assumes the following specific form:
Theorem 2.
:4
--t
Here all the limiting passages except the vertical passage on the right are defined by the hypotheses of Theorem 2 itself. The nontrivial conclusion of Theorem 1 that we need is precisely that lim f ( x) = f ( x0). 0
x -+ x o
Remark 3. We have not said anything specific as to the nature of the set X . I n fact it may b e any topological space provided the base x ---+ x0 i s defined in it. The values of the functions ft may lie in any metric space, which, as follows from Remark 2, need not even be complete.
384
16 Uniform Convergence and Basic Operations of Analysis
If a sequence of functions that are continuous on a set con verges uniformly on that set, then the limit function is continuous on the set. Corollary 2. If a series of functions that are continuous on a set converges uniformly on that set, then the sum of the series is also continuous on the set. Corollary 1 .
As an illustration of the possible use of these results, consider the follow ing.
Example 1. Abel 's method of summing series. Comparing Corollary 2 with Abel ' s second theorem ( Proposition 4 of Sect. 16.2), we draw the following conclusion. 00 Proposition 1 . If a power series :L cn ( z - z0 ) n converges at a point (, it n=O converges uniformly on the closed interval [z0 , (] from z0 to (, and the sum of the series is continuous on that interval. In particular, this means that if a numerical series :L Cn converges, then 00
n=O n the power series :L Cn X converges uniformly on the closed interval n=O 00 n of the real axis and its sum = L Cn X is continuous on that interval. n=O 00 00 Since = :L cn , we can thus assert that if the series :L Cn converges, n=O n=O 00
0 :-::; x :-::; 1
s(x)
s(1)
then the following equality holds: 00
L Cn
n=O
=
00
(16.23)
L Cn X n . x-+ 1 -0 lim
n=O
It is interesting that the right-hand side of Eq. (16.23) may have a meaning even when the series on the left diverges in its traditional sense. For example, the series 1 - 1 + 1 - · · · corresponds to the series x - x 2 + x 3 - · · · , which converges to x/(1 + x) for l x l < 1. As x --+ 1, this function has the limit 1/2. The method of summing a series known as Abel summation consists of ascribing to the left-hand side of (16.23) the value of the right-hand side if it 00 is defined. We have seen that if the series :L Cn converges in the traditional n=O sense, then its classical sum will be assigned to it by Abel summation. At the 00 same time, for example, Abel ' s method assigns to the series L ( - 1) n , which n=O
diverges in the traditional sense, the natural average value 1/2. Further questions connected with Example 1 can be found in Problems 5-8 below.
16.3 Functional Properties of a Limit Function
385
Example 2. Earlier, when discussing Taylor ' s formula, we showed that the following expansion holds: ( 1 + x)" = a a ( a - 1 ) 2 . . . a ( a - 1 ) · · · ( a - n + 1) n + . . . ( 16_24) x + + = 1+ x+ x 1! 2! n! We can verify that for a > 0 the numerical series a (a - 1 ) · · · (a - n + 1 ) + a a (a - 1 ) 1+ + +···+ ··· 1! 2! n! converges. Hence by Abel ' s theorem, if a > 0, the series ( 16.24) converges uniformly on the closed interval 0 � x � 1 . But the function ( 1 + x)" is continuous at x = 1, and so one can assert that if a > 0, then Eq. ( 16.24) holds also for x = 1 . In particular, we can assert that for a > 0 ( - 1) 4 . . . ( 1 - t2 ) a = 1 - a t2 + a a t 1! 2! . . . + ( _ 1) n . a ( a - 1 ) · · · ( a - n + 1) t 2 n + . . . ( 1 6_25) n! 2 and this series converges to ( 1 - t ) " uniformly on [- 1 , 1] . Setting a = ! and t 2 = 1 - x 2 in ( 16.25) , for l x l � 1 we find
l x l = 1 - 1!! ( 1 - x 2 ) + H !2!- 1) ( 1 - x 2 ) 2 - . . .
( 16.26) ' and the series of polynomials on the right-hand side converges to l x l uniformly on the closed interval [- 1 , 1] . Setting Pn (x) := Bn (x) - Bn (O), where Bn (x) is the nth partial sum of the series, we find that for any prescribed tolerance c > 0 there is a polynomial P(x) such that P(O) = 0 and max
- l :S x ::;!
l l x l - P(x) l < c .
( 16.27)
Let us now return to the general theory. We have shown that continuity of functions is preserved under uniform passage to the limit. The condition of uniformity in passage to the limit is, however, only a sufficient condition in order that the limit of continuous func tions also be a continuous function ( see Examples 8 and 9 of Sect. 16. 1 ) . At the same time there is a specific situation in which the convergence of con tinuous functions to a continuous function guarantees that the convergence is uniform. Proposition 2. ( Dini ' s 2 theorem ) . If a sequence of continuous functions on
a compact set converges monotonically to a continuous function, then the convergence is uniform. 2 U. Dini (1845-1918) - Italian mathematician best known for his work in the theory of functions.
386
16 Uniform Convergence and Basic Operations of Analysis
Proof. For definiteness suppose that fn is a nondecreasing sequence converg ing to f. We fix an arbitrary E > 0, and for every point x of the compact set K we find an index n x such that 0 :S f ( x) - fn x ( x) < E. Since the func tions f and fn x are continuous on K, the inequality 0 :S f(�) - fn x (�) < E holds in some neighborhood U(x) of x E K. From the covering of the compact set K by these neighborhoods one can extract a finite covering U(x i ), . . . , U(x k ) and then fix the index n(s) = max {nx , , . . . , nx k }. Then for any n > n( E ) , by the fact that the sequence { !n ; n E N} is nondecreasing, we have 0 :S f(O - fn (�) < E at every point � E K. 0 00
If the terms of the series I:= an ( x) are nonnegative functions n l set K and the series converges an : K � that are continuous on a compact to a continuous function on K, then it converges uniformly on K . n Proof. The partial sums Sn ( x) = I:= a k ( x) of this series satisfy the hypothe k l ses of Dini ' s theorem. 0 Example 3. We shall show that the sequence of functions fn (x) = n(l -x 1 1n ) tends to f(x) = ln � as n ---+ + oo uniformly on each closed interval [a, b] contained in the interval 0 < x < oo. Proof. For fixed x > 0 the function x t = et x is convex with respect to t, so that the ratio x t =� (the slope of the chord) is nonincreasing as t + 0 and tends to ln x. Hence fn (x) /' ln � for x > 0 as n ---+ + oo. By Dini ' s theorem it now follows that the convergence of fn ( x) to ln � is uniform on each closed interval [a, b] c]O, + oo[. o Corollary 3 . ---+
t
0
ln
---+
We note that the convergence is obviously not uniform on the interval
0 < x :S 1 , for example, since ln � is unbounded in that interval, while each of the functions fn ( x) is bounded (by a constant depending on n). 1 6 . 3 . 4 Integration and Passage to the Limit
We shall show that if functions that are integrable over a closed interval converge uniformly on that interval, then the limit function is also integrable and its integral over that interval equals the limit of the integrals of the original functions. ---+
Let Ut ; t E T} be a family of functions ft : [a, b] C defined on a closed interval a :S x :S b and depending on the parameter t E T, and let B be a base in T. If the functions of the family are integrable on [a, b] and ft f on [a, b] over the base B, then the limit function f : [a, b] ---+ C is also Theorem 3. :::::t
16.3 Functional Properties of a Limit Function
387
integrable on [a, b] and b
b
J f(x) dx = ffl J ft (x) x li
a
d .
a
Proof. Let p = (P, �) be a partition P of the closed interval [a, b] with dis tinguished points � = { 6 , . . . , �n }. Consider the Riemann sums Ft (p) = n 2:::: ft (�i ) iJ.x i , t E T, and F(p) = 2:::n : f(�i )iJ.x i . Let us estimate the difference
i= l i= l F(p) - Ft (p). Since ft f on [a, b] over the base B, for every E > 0 there exists an element B of B such that [f(x) - ft (x) [ < b :_ a at any t E B and any point x E [a, b]. Hence for t E B we have �
B,
and this estimate holds not only for every t E but also for every parti tion p in the set P = { (P, �)} of partitions of the closed interval [a, b] with distinguished points. Thus Ft � F on P over the base B. Now, taking the traditional base >.. ( P) -+ 0 in P, we find by Theorem 1 that the following diagram is commutative:
n i= l ft(�i )iJ.xi =: Ft (P) .2::::
>. ( P ) -+ 0
b
l
n F(p) := .2:::: f(�i )iJ.x i / i= l /
=====*
/
-
/
3
l >. (P) -+0 b
J ft(x) dx =: A t A := aJ f(x) dx a which proves Theorem 3. ()() Corollary 4. If the series .2:::: fn ( x) consisting of integrable functions on a n = l uniformly on that closed interval, then its closed interval [a, b] lR converges sum is also integrable on [a, b] and D
C
J ( � fn (x) ) dx = � � fn (x) dx . b
a
()()
b
()()
a
When we write si� x in this example, we shall assume that this ratio equals 1 when x = 0. X We have noted earlier that the function Si ( x) = J si� t dt is not an ele
Example 4-
o
mentary function. Using the theorem just proved, we can nevertheless obtain a very simple representation of this function as a power series.
388
16 Uniform Convergence and Basic Operations of Analysis
To do this, we remark that sin t
t
=
� (-1) n t 2n ' � (2n 1) !
(16.28)
+
and the series on the right-hand side converges uniformly on every closed interval [ - a , a ] C R The uniform convergence of the series follows from the Weierstrass M-test , since d��:: ) ! ::; ( 2�:1 ) ! for l t l ::; a, while the numerical
2n
series 2: 2 �+ 1 ! converges. n =O ( ) By Corollary 4 we can now write 00
Si (x) =
J0 ( nf=O (2n(-1) n1 ) 1. t2n ) t = oo oo n 2n 1 n - 2: ( J (2n(-1) 1) ! t 2n ) - 2: (2n(-1)1) !x(2n+ 1) n=O n =O X
d
+
X
0
+
dt
-
+
+
The series just obtained also turns out to converge uniformly on every closed interval of the real line, so that , for any closed interval [ a , b] of variation of the argument x and any preassigned absolute error tolerance, one can choose a polynomial - a partial sum of this series - that makes it possible to compute Si (x) with less than the given error at every point of the closed interval [a , b] . 1 6 . 3 . 5 Differentiation and Passage to the Limit
Let Ut ; t E T} be a family of functions ft : X --+ C defined on a convex bounded set X (in IR, C, or any other normed space) and depending on the parameter t E T; let B be a base in T. If the functions of the family are differentiable on X, the family of derivatives {fl; t E T} converges uniformly on X to a function : X --+ C, and the original family Ut ; t E T} converges at even one point x0 E X, then it converges uniformly on the entire set X to a differentiable function f : X --+ C, and f ' = Proof. We begin by showing that the family Ut ; t E T} converges uniformly
Theorem 4.
cp
cp .
on the set X over the base B. We use the mean-value theorem in the following estimates:
l ft, (x) - ft 2 (x) l ::; ::; I ( ft , ( x) - ft 2 ( x)) - (ft , ( Xo) - ft 2 ( Xo)) I + I ft , ( xo) - ft 2 ( xo) I ::; ::; sup l fl, (�) - fl2 (�) 1 1 x - xo l l ft , (xo) - ft 2 (xo) l = L1(x , t 1 , t 2 ) . �E [x o, x ) +
16.3 Functional Properties of a Limit Function
389
By hypothesis the family {f£; t E T} converges uniformly on X over the base B, and the quantity ft(x0) has a limit over the same base as a function of t, while l x - x0 1 is bounded for x E X. By the necessity part of the Cauchy criterion for uniform convergence of the family of functions f£ and the existence of the limit function ft (x0), for every s > 0 there exists B in l3 such that L1(x, t 1 , t 2 ) < s for any t 1 , t 2 E B and any x E X. But , by the estimates just written, this means that the family of functions {ft; t E T} satisfies the hypotheses of the Cauchy criterion and consequently converges on X over the base l3 to a function f : X -+ C. Again using the mean-value theorem, we now obtain the following esti mates:
I (ft 1 (x + h) - ft 1 (x) - 1;1 (x )h) - (ft 2 (x + h) - ft2 (x) - 1;2 (x )h ) I = = I Ut 1 - fto ) (x + h) - (fh - ft2) (x) - ( ft 1 - ft J ' (x)h l ::::; ::::; sup I Ut 1 - ft2) ' (x + B h) l l h l + I Ut 1 - ft2) ' (x) l l h l = 0 <11< 1 sup 1 1; 1 (x + B h) - J;2 (x + B h) l + 11; 1 (x) - J;2 (x) 1 ) lhl . = ( 0 <11< 1 These estimates, which are valid for x, x + h E X show, in view of the uniform convergence of the family {f£; t E T} on X, that the family { Ft ; t E T} of functions ft(x) - J; (x)h ' t (h) = ft (x + h) - lhl which we shall consider with a fixed value of x E X, converges over the base l3 uniformly with respect to all values of h =I 0 such that x + h E X. We remark that Ft(h) -+ 0 as h -+ 0 since the function ft is differentiable at the point x E X; and since ft -+ f and f£ -+ cp over the base B, we have Ft ( h ) -+ F( h ) = f(x+h) - (�� ) -
_ .
F,
1
p
�
. _
1
16 Uniform Convergence and Basic Operations of Analysis
390
=
=
2::: f� ( x) converges uniformly on X, then 2::: f ( x) also converges uniformly non= 1X, its sum is differentiable on X, and n= 1 n fn (x) ' (x) f:, (x) .
(� )
=
�
This follows from Theorem 4 and the definitions of the sum and uni form convergence of a series, together with the linearity of the operation of differentiation.
Remark 4. The proofs of Theorems 3 and 4, like the theorems themselves and their corollaries, remain valid for functions ft : X --+ Y with values in any complete normed vector space Y. For example, Y may be IR, e, !Rn , e n , C[a , b], and so on. The domain of definition X for the functions ft in Theorem 4 also may be any suitable subset of any normed vector space. In particular, X may be contained in IR, e, !R n , or e n . For real-valued functions of a real argument (under additional convergence requirements) the proofs of these theorems can be made even simpler (see Problem 11). As an illustration of the use of Theorems 2-4 we shall prove the following proposition, which is widely used in both theory and in specific computations. C
Let K e be the convergence disk for a power series n 2::: c (z - z0) . If K contains more than just the point zo, then the sum nof=Othen series f(z) is differentiable inside K and (16.29) f'(z) L ncn (z - zo) " - 1 . n= 1 Moreover, the function f ( z) : K --+ e can be integrated over any path "( : [0, 1] --+ K, and if [0, 1] t r--2-..t z(t) E K, z(O) zo, and z(1) z, then � (z - z0) n + 1 . (16 . 30) J f(z) dz � L...n =O n + 1 1 Remark 5. Here J f(z) dz J0 f ( z(t) ) z'(t) dt. In particular, if the equality f(x) 2:::=O an(x - xo) n holds on an interval - R < x - xo < R of the real n Proposition 3 . =
=
3
I
=
=
line IR, then
=
=
=
:=
I
X
J f(t )
xo
dt
=
=
L� (x - xo) n + l . n 1 + =O n
=
16.3 Functional Properties of a Limit Function
391
Since lim ---+ CXl n -� = nlim --+CXJ \lfcJ, it follows from the Cauchyn Hadamard formula (Theorem 2 of Subsect. 16.2.2 that the power series 00 I: ncn (z - z0) n - 1 obtained by termwise differentiation of the power series
Proof.
n= 1 I: ( z z0 ) , has the same convergence disk K as the original power series. n=O Cn - " But by Theorem 2 of Subsect. 16.2.2 the series I: ncn (z - z0) n - 1 converges n= 1 uniformly in any closed disk Kq contained in the interior of K. Since the series I: c n (z - z0) n obviously converges at z = z0 , Corollary 5 is applicable n=O to it, which justifies the equality (16.29). Thus it has now been shown that a 00
00
00
power series can be differentiated termwise. Let us now verify that it can also be integrated termwise. If 1 : [0, 1] --+ K is a smooth path in K, there exists a closed disk Kq such that 1 C Kq and Kq C K. On Kq the original power series converges uniformly, so that in the equality 00
f ( z(t) ) L cn ( z(t) - zo) n n =O =
the series of continuous functions on the right-hand side converges uniformly on the closed interval 0 ::::; t ::::; 1 to the continuous function f ( z(t)) . Multi plying this equality by the function z '(t), which is continuous on the closed interval [0, 1], does not violate either the equality itself nor the uniform con vergence of the series. Hence by Theorem 3 we obtain
1
1
00
J f(z(t) ) z'(t) dt = L J cn ( z(t) - zo) "z'(t) dt . 0
n=O o
But,
1
1
1 +1 J ( z(t) - z(O) ) " z'(t) dt = n +-1 J d ( z(t) - z(0) ) " = 0
0 1 + 1 1 -(z - zo) n + 1 , =n +-1 ( z(1) - z(0) ) " = n+1 and we arrive at Eq. (16.30). 0 Since it is obvious that Co = f(zo) in the expansion f(z) I: c n (z-zo) n , n=O n applying Eq. (16.29) successively, we again obtain the relation Cn = t< nl \. zo) , which shows that a power series is uniquely determined by its sum and is the =
Taylor series of the sum.
00
392
16 Uniform Convergence and Basic Operations of Analysis
Bessel function Jn (x) , n E N, is a solution of Bessel 's3 x 2 y " + xy' + (x2 - n2 )y 0 . Let us attempt to solve this equation, for example, for n 0, as a power 00 series y 2::: ck x k . Applying formula (16 . 29) successively, after elementary Example 5. equation =
The
=
=
k =O
transformations, we arrive at the relation 00
C 1 + :�:.)k 2 c k + Ck - 2 ) X k - l k =O
=
0,
from which, by the uniqueness of the power series with a given sum, we find
From this it is easy to deduce that
=
k 2, 3, . . . c2 k - l 0, k E N, =
( - 1) k ( k ! )?2 2 k . If we assume Jo(O) 1, we arrive at the solution =
k
00
2 '""" - 1) k X Jo(x) 1 + L.) (k!) 2 22 k =
k= l
and
c2 k
.
This series converges on the entire line lR (and in the entire plane C), so that all the operations carried out above in order to find its specific form are now justified.
Example 6. In Example 5 we sought a solution of an equation as a power series. But if a series is given, using formula (16.29), one can immediately check to see whether it is the solution of a given equation. Thus, by direct computation, one can verify that the function introduced by Gauss a(a + 1) · · · (a + n - 1) ,6 ( ,6 + 1) · · · (,6 + n - 1) x n F(a, j), ')', x) - 1 + � 6 ')'('/' + 1) · · · ('/' + n - 1) n= l
(the hypergeometric series) is well-defined for called hypergeometric equation
lxl
<
1 and satisfies the so
x(x - 1)y" - b - (a + ,6 - 1)x] · y' + aj) · y 0 . In conclusion we note that, in contrast to Theorems 2 and 3, the hy potheses of Theorem 4 require that the family of derivatives, rather than the original family, converge uniformly. We have already seen (Example 2 of Sect . 16.1) that the sequence of functions fn (x) � sin n2 x converges to the differ entiable function f(x) 0 uniformly, while the sequence of derivatives f� (x) does not converge to f' ( x). The point is that the derivative characterizes the =
=
3 F.W. Bessel ( 1 784-1846) - German astronomer.
=
16.3 Functional Properties of a Limit Function
393
rate of variation of the function, not the size of the values of the function. Even when the function changes by an amount that is small in absolute value, the derivative may formally change very strongly, as happens in the present case of small oscillations with large frequency. This is the circumstance that lies at the basis of Weierstrass ' example of a continuous nowhere-differentiable funcoo tion, which he gave as the series f(x) = I: an cos(bn nx) , which obviously n =O converges uniformly on the entire line lR if 0 < a < 1. Weierstrass showed that if the parameter b is chosen so as to satisfy the condition a · b > 1 + � n, then on the one hand f will be continuous, being the sum of a uniformly convergent series of continuous functions, while on the other hand, it will not have a derivative at any point x E R The rigorous verification of this last assertion is rather taxing, so that those who wish to obtain a simpler example of a continuous function having no derivative may see Problem 5 in Sect. 5 . 1 . 16.3.6 Problems and Exercises 1. Using power series, find a solution of the equation y " ( x ) - y ( x ) the conditions a) y(O) = 0, y(1) = 1 ; b ) y(O) = 1 , y ( 1 ) = 0 . 2. 3.
CXl
Find the sum of the series L nC n + l ) n= l n-1
=
0 satisfying
·
a) Verify that the function defined by the series Jn (x) =
( x ) 2 k+ n
(- 1)k
� k!(k + n) ! oo
2
is a solution of Bessel's equation of order n 2: 0 from Example 5. b) Verify that the hypergeometric series in Example 6 provides a solution of the hypergeometric equation. Obtain and justify the following expansions, which are suitable for computation, for the complete elliptic integrals of first and second kind with 0 < k < 1 .
4.
K (k) =
E (k)
=
rr/ 2
rr/ 2
/ }1 - dr.pk2 sin2 0
J V1 - k2 s m2 0
•
'P
1n r
= �
d 'P =
2
( (
(
) ) �) )
1) ! ! 2 k 2 n 1 + � (2n - ! ! (2n) � -1 n
(
� (2n - 1) ! ! 2 1-L 2 ( 2n) ! ! 2n - 1 =l
�
n
;
394 5.
16 Uniform Convergence and Basic Operations of Analysis
Find a) f r k e i k
k =O
b) f r k cos k
k =O
c) f r k sin k
k =O
Show that the following relations hold for l r l
d ) f r k e i k
<
1:
= - r 2rp r2 ' - .!2 . 1 -2 r1cos e) .!2 + 2::: r k cos k
l sin
·
·
·
n
n
Let S n = 2::: a k and O' n = l 2::: S k · The series is Cesaro4 summable, more k=1 k=1 = A. In that case we write 2::: ak precisely ( c, 1 )-summable to A, if nlim <7 n -+ = n =l A(c, 1 ) . a ) Verify that 1 - 1 + 1 - 1 + = � (c, 1 ) . 7.
·
·
·
b ) Show that O'n = f ( 1 - k � 1 ) a k .
k= 1 = c) Verify that if 2::: a k = A in the usual sense, then 2::: a k = A(c, 1 ) . k= 1 k= 1 d) The (c, 2)-sum of the series f a k is the quantity lim � ( <7 1 + + O'n ) if n--+ oo k=l this limit exists. In this way one can define the ( c, r )-sum of any order r. Show that = = if 2::: a k = A(c, r) , then 2::: a k = A(c, r + 1 ) . k= 1 k=1 = e) Prove that if 2::: a k = A(c, 1 ) , then the series is also Abel summable to A. k=1 4 E. Cesaro ( 1859-1906) - Italian mathematician who studied analysis and geom etry. =
·
·
·
16.3 Functional Properties of a Limit Function
395
8. a) A "theorem of Tauberian type " is the collective description for a class of theorems that make it possible, by introducing various extra hypotheses, to judge the behavior of certain quantities from the behavior of certain of their means. An example of such a theorem involving Cesaro summation of series is the following proposition, which you may attempt to prove following Hardy. 5
�
�
If a n = A( c, 1 ) and a n = 0 ( � ) , then the series a n converges in the n l n l ordinary sense and to the same sum. b) Tauber's6 original theorem relates to Abel summation of series and consists of the following. =
Suppose the series L
an x
n
converges for 0
<
x
<
=
1 and x �lim L l -O
an x
n
= A.
If J�� a , + 2 a 2 � · · + n a n = 0, then the series L a n converges to A in the ordinary n= l sense. n= l
=
n= l
9. It is useful to keep in mind that in relation to the limiting passage under the integral sign there exist theorems that give much freer sufficient conditions for the possibility of such a passage than those made possible by Theorem 3. These theo rems constitute one of the major achievements of the so-called Lebesgue integral. In the case when the function is Riemann integrable on a closed interval [a, b] , that is, f E R[a, b] , this function also belongs to the class .C[a, b] of Lebesgueb integrable functions, and the values of the Riemann integral (R) I f(x) dx of f and
a
b
the Lebesgue integral (L) I f(x) dx are the same. a
In general the space .C[a, b] is the completion of R[a, b] (more precisely, R[a, b] ) b with respect to the integral metric) , and the integral (L) I is the continuation of b
a
the linear functional (R) I from R[a, b] to .C[a, b] . a The definitive Lebesgue "dominated convergence" theorem asserts that if a sequence {fn ; n E N} of functions fn E .C [a, b] is such that there exists a non negative function F E .C[a, b] that majorizes the functions of the sequence, that is, l fn ( x ) l :S F(x) almost everywhere on [a, b] , then the convergence fn --t f at almost all points of the closed interval [a, b] implies that f E .C[a, b] and b b lim (L) f f (x) d x = (L) I f(x) dx . n n --+ oo a
a
a) Show by example that even if all the functions of the sequence {fn ; n E N} are bounded by the same constant M on the interval [a, b] , the conditions fn E R[a, b] , n E N and f --t f pointwise on [a, b] still do not imply that f E R[a, b] . (See n Example 5 of Sect. 16. 1.)
5 G.H. Hardy (1877-1947) - British mathematician who worked mainly in number theory and theory of functions. 6 A. Tauber (b. 1866, year of death unknown) - Austrian mathematician who worked mainly in number theory and theory of functions.
16 Uniform Convergence and Basic Operations of Analysis
396
b
b) From what has been said about the relation between the integrals (R) I a
b
and (L) J and Lebesgue's theorem show that, under the hypotheses of part a) , if a
b
b
=
it is known that f E 'R.[a, b] , then (R) I f (x ) dx lim (R) I f (x ) dx. This is a n --+ oo a n a significant strengthening of Theorem 3. c) In the context of the Riemann integral one can also state the following version of Lebesgue's monotone convergence theorem. If the sequence {fn ; n E N} of functions fn E 'R.[a, b] converges to zero mono tonically, that is, 0 � fn+1 � fn and fn -t 0 as n -t oo for every x E [a, b] , then b (R) I fn (x) dx -t 0. a
Prove this assertion, using where needed the following useful observation.
1
d) Let f E 'R.[a, b] , I f I � M, and I f (x ) dx 0
1
2:
a
>
0. Then the set
E
= {x E
[0, 1] f (x ) 2: a/2} contains a finite number of such intervals the sum of whose lengths (l) is at least a/(4M) . Prove this, using, for example, the intervals of a partition P of the closed interval 1 [0, 1] for which the lower Darboux sum s(f, P) satisfies the relation 0 � I f(x) dx 0 s(f, P) < a/4. 10. a) Show by the examples of Sect. 16. 1 , that it is not always possible to extract a subsequence that converges uniformly on a closed interval from a sequence of functions that converge pointwise on the interval. b) It is much more difficult to verify directly that it is impossible to extract a subsequence of the sequence of functions {fn ; n E N}, where fn (x ) = sin nx , that converges at every point of [0, 21r] . Prove that this is nevertheless the case. (Use 2 11" the result of Problem 9b) and the circumstance that I (sin n k x - sin n k +1 x ) 2 dx 0 21r =I 0 for n k < n k+1 · ) c) Let {in ; n E N} be a uniformly bounded sequence of functions fn E 'R.[a, b] . Let X
=
Fn (x )
= J fn (t) dt a
(a � X � b) .
Show that one can extract a subsequence of the sequence {Fn ; n E N} that converges uniformly on the closed interval [a, b] . 1 1 . a) Show that if f , fn E 'R.([a, b] , JR.) and fn � f on [a, b] as n -t oo, then for every E > 0 there exists an integer N E N such that
ju - fn ) (x ) dx b
for every n > N.
a
<
t:: ( b - a) .
16.4 *Subsets of the Space of Continuous Functions
397
b ) Let fn E cC ll ( [a, bJ , lR ) , n E N. Using the formula fn (x) = fn (xo) + J f� ( t ) dt, show that if f� :::; 'P on [a, b] and there exists a point xo E [a, b] for which X
XQ
the sequence {fn (xo); n E N} converges, then the sequence of functions { fn ; n E N} converges uniformly on [a, b] to some function f E c < ll ( [a, b] , lR) and f� :::; f' = 'P ·
16.4 * Compact and Dense Subsets of the Space of Continuous Functions The present section is devoted to more specialized questions, involving the space of continuous functions, which is ubiquitous in analysis. All these ques tions, like the metric of the space of continuous functions 7 itself, are closely connected with the concept of uniform convergence. 16.4. 1 The Arzela-Ascoli Theorem Definition 1. A family F of functions f : X --+ Y defined on a set X and assuming values in a metric space Y is uniformly bounded on X if the set of values V = {y E Y l �� E F �x E X (y = f ( x) ) } of the functions in the family is bounded in Y.
For numerical functions or for functions f : X --+ JR. n , this simply means that there exists a constant M E JR. such that I f ( x) I ::; M for all x E X and all functions f E B. Definition 1 ' . If the set V C Y of values of the functions of the family F is totally bounded (that is, for every c; > 0 there is a finite c-grid for V in Y), the family F is totally bounded. For spaces Y in which the concept of boundedness and total boundedness are the same (for example, for JR.,
X and Y be metric spaces. A family F of functions f : Y equicontinuous on X if for every > 0 there exists o > 0 such that dy ( f(x l ), f(x 2 )) < E for any function f in the family and any X 1 , X 2 E X such that dx(x 1 , x 2 ) < o. Example 1 . The family of functions { xn ; n E N} is not equicontinuous on [0, 1], but it is equicontinuous on any closed interval of the form [0, q] where O < q < l. Definition 2. Let X --+ is
c;
7 If you have not completely mastered the general concepts of Chap. 9, you may assume without any loss of content in the following that the functions discussed always map lR into lR or C into C, or lRm into lR n .
398
16 Uniform Convergence and Basic Operations of Analysis
Example 2. The family of functions {sin nx; n any nondegenerate closed interval [a, b] C R
E N} is not equicontinuous on
Example 3. If the family {!a : [a, b] -+ lR; E A} of differentiable functions fa is such that the family { !�; E A} of their derivatives is uniformly bounded by a constant, then l fa(x 2 ) - fa(xi) I :S: M l x 2 - x 1 l , as follows from the mean-value theorem, and hence the original family is equicontinuous on the closed interval [a, b]. o:
o:
The connection of these concepts with uniform convergence of continuous functions is shown by the following lemma.
Let K and Y be metric spaces, with K compact. A necessary condition for the sequence Un ; n E N} of continuous functions fn : K -+ Y to converge uniformly on K is that the family Un ; n E N} be totally bounded and equicontinuous. Proof. Let fn f on K. By Theorem 2 of Sect. 16.3, we conclude that f E C(K, Y). It follows from the uniform continuity of f on the compact set K that for every E > 0 there exists {J > 0 such that ( dK(x 1 , x 2 ) < {J dy ( f(xi), f(x 2 )) < E ) for all x 1 , x 2 E K. Given the same E > 0 we can find an index N E N such that dy (f ( x), fn ( x) ) < E for all n > N and all x E X. Combining these inequalities and using the triangle inequality, we find that dK(x 1 , x 2 ) < fJ implies dy ( fn (xi), fn (x 2 )) < 3E for every n > N and x 1 , x 2 E K. Hence the family Un ; n > N } is equicontinuous. Adjoining to this family the equicontinuous family {h , . . . , fN } consisting of a finite number of functions continuous on the compact set K, we obtain an equicon tinuous family { !n ; n E N}. Total boundededness of F follows from the inequality dy (! ( X ) ' fn ( X )) < N E, which holds for x E K and n > N , and the fact that f(K) and U fn (K) n= l are compact sets in Y and hence totally bounded in Y. 0 Lemma 1 .
==l
==::}
Actually the following general result is true. Theorem 1. (Arzela-Ascoli) . Let F be a family of functions f : K -+ Y de fined on a compact metric space K with values in a complete metric space Y. A necessary and sufficient condition for every sequence Un E F ; n E N} to contain a uniformly convergent subsequence is that the family F be totally bounded and equicontinuous. Proof. N e c e s s i t y. If F were not a totally bounded family, one could obvi ously construct a sequence {fn ; n E N} of functions fn E F that would not
be totally bounded and from which (see the lemma) one could not extract a uniformly convergence subsequence. If F is not equicontinuous, there exist a number Eo > 0, a sequence of functions Un E F; n E N}, and a sequence {(x�, x�); n E N} of pairs
16.4 *Subsets of the Space of Continuous Functions
399
(x�, x�) of points x� and x� that converge to a point x0 E K as n --+ oo , but d y ( fn (x�), fn (x�) ) � c:0 > 0. Then one could not extract a uniformly convergent subsequence from the sequence {fn ; n E N}: in fact , by Lemma 1, the functions of such a subsequence must form an equicontinuous family. S u ff i c i e n c y. We shall assume that the compact set K is infinite, since the assertion is trivial otherwise. We fix a countable dense subset E in K a sequence {x n E K; n E N}. Such a set E is easy to obtain by taking, for example, the union of the points of finite c:-grids in K obtained for c: = -
1, 1/2, . . . , 1/n, . . .. Let { in ; n E N} be an arbitrary sequence of functions of :F. The sequence Un (x 1 ); n E N} of values of these functions at the point x 1 is totally bounded in Y by hypothesis. Since Y is a complete space, it is possible to extract from it a convergent subsequence Unk (xi); k E N}. The functions of this sequence, as will be seen, can be conveniently denoted f� , n E N. The superscript 1 shows that this is the sequence constructed for the point x 1 . From this subsequence we extract a further subsequence {f� k ; k E N} which we denote {f�; n E N} such that the sequence {f� k (x 2 ); k E N} con verges. Continuing this process, we obtain a series { !!; n E N}, k 1, 2, . . . of sequences. If we now take the "diagonal" sequence {gn = f;:; n E N}, it will converge at every point of the dense set E K, as one can easily see. We shall show that the sequence { 9n ; n E N} converges at every point of K and that the convergence is uniform on K. To do this, we fix c: > 0 and choose 8 > 0 in accordance with Definition 2 of equicontinuity of the family :F. Let E1 { 6 , . . . , �k } be a finite subset of E forming a 8-grid on K. Since the sequences {gn (�i ); n E N}, i = 1, 2, . . . , k, all converge, there exists N such that d y (gm (�i ), gn (�i ) ) < c: for i = 1, 2, . . . , k and all m, n � N . For each point x E K there exists �j E E such that dK( X , �j ) < 8. By the equicontinuity of the family :F, it now follows that dy (gn (x) , gn (�j) ) < c: for every n E N. Using these inequalities, we now find that dy (gm (x) , gn (x) ) :S dy (gn (x) , gn ( �j)) + dy ( gm ( �j), gn ( �j ) ) + + dy (grn (x), gm ( �j)) < + + = 3c: for all m , n > N . But x was an arbitrary point of the compact set K, so that , by the Cauchy criterion the sequence {gn ; n E N} indeed converges uniformly on K. 0 =
C
=
C:
16.4.2
C:
C:
The Metric Space C(K, Y )
One of the most natural metrics on the set C(K, Y) of functions f : K --+ Y that are continuous on a compact set K and assume values in a complete metric space Y is the following metric of uniform convergence.
400
16 Uniform Convergence and Basic Operations of Analysis
d ( f, g )
=
max dy ( f ( x ) , g ( x )) ,
x EK
where f, g E C ( K , Y), and the maximum exists, since K is compact. The name metric comes from the obvious fact that d ( fn , f) --+ 0 <=? fn � f on K. Taking account of this last relation, by Theorem 2 of Sect. 16.3 and the Cauchy criterion for uniform convergence we can conclude that the metric space C ( K , Y) with the metric of uniform convergence is complete. We recall that a precompact subset of a metric space is a subset such that from every sequence of its points one can extract a Cauchy ( fundamental ) subsequence. If the original metric space is complete, such a sequence will even be convergent . The Arzellt-Ascoli theorem gives a description of the precompact subsets of the metric space C ( K , Y). The important theorem we are about to prove gives a description of a large variety of dense subsets of the space C ( K, Y). The natural interest of such subsets comes from the fact that one can approximate any continuous function f : K --+ Y uniformly with absolute error as small as desired by functions from these subsets.
Example 4.
The classical result of Weierstrass, to which we shall often return, and which is generalized by Stone ' s theorem below, is the following.
( Weierstrass ) . If f E C ( [a, b], C ) , there exists a sequence { Pn ; N} of polynomials Pn [a, b] --+ C such that Pn f on [a, b] . Here, if f C ( [a, b], IR ) , the polynomials can also be chosen from C ( [a, b], IR ) .
Theorem 2. nE E
:
�
In geometric language this means, for example, that the polynomials with real coefficients form an everywhere dense subset of C ( [a, b] , IR ) .
Example 5. Although Theorem 2 still requires a nontrivial proof ( given be low ) , one can at least conclude from the uniform continuity of any function f E C([a, b] , IR ) that the piecewise-linear continuous real-valued functions on the interval [a, b] are a dense subset of C ( [a, b] , IR ) . Remark 1 .
We note that if E 1 is everywhere dense in E2 and E2 is everywhere dense in E3 , then E1 is obviously everywhere dense in E:� . This means, for example, that to prove Theorem 2 it suffices to show that a piecewise linear function can be approximated arbitrarily closely by a polynomial on the given interval. 16.4.3 Stone's Theorem
Before proving the general theorem of Stone, we first give the following proof of Theorem 2 ( Weierstrass ' theorem ) for the case of real-valued functions, which is useful in helping to appreciate what is to follow.
16.4 *Subsets of the Space of Continuous Functions
401
Proof. We first remark that if f, g E C ( [a, b] , JR) , a E JR, and the functions f and g admit a uniform approximation (with arbitrary accuracy) by polyno mials, then the continuous functions f + g, f · g, and af also admit such an approximation. On the closed interval [ - 1, 1], as was shown in Example 2 of Sect. 16.3, the function lxl admits a uniform approximation by polynomials Pn (x) = n I:; ak x k . Hence, the corresponding sequence of polynomials M · Pn ( x /M) k= l
gives a uniform approximation to lxl on the closed interval lxl ::::; M. If f E C ( [a, b], JR ) and M = max l f(x)l, it follows from the inequality
l lyl - k�l ck yk l <
l
�
l
for IYI ::::; M that l f(x)l ck f k (x) < c for a ::::; x ::::; b. l k Hence if f admits a uniform approximation by polynomials on [a, b], then n I:; ck f k and lfl also admit such an approximation. c
k = l Finally, if f and g admit a uniform approximation by polynomials on the closed interval [a, b], then by what has been said, the functions max{!, g} = H U + g) + I f - g l) and min{f, g} = H U + g) - I f - g l ) also admit such an approximation. Let a ::::; 6 ::::; 6 ::::; b, f(x) 0, g6 6 (x) = �-=t , h (x) 1, Pf. 1 6 = max{f, g6 6 } , and F6 6 = min{ h, tf>6 6 } . Linear combinations of functions of the form F6 6 obviously generate the entire set of continuous piecewise-linear functions on the closed interval [a, b], from which, by Example 5, Weierstrass ' =
=
theorem follows. D
Before stating Stone ' s theorem, we define some new concepts.
A of real- (or complex-)valued functions on a set X is real (or complex) algebra of functions on X if (f + g) E A , (f · g ) E A , (af) E A when f, g E A and a E lR (or E C). Example 6. Let X C C. The polynomials P(z) = co + c 1 z + c2 z2 + · · · + cn z n , n E N, obviously form a complex algebra of functions on X. If we take X [a, b] C JR, and take only polynomials with real coefficients, we obtain a real algebra of functions on the closed interval [a, b]. Example 7. The linear combinations of functions enx , n = 0, 1, 2, . . . with coefficients in lR or C also form a (real or complex respectively) algebra on any closed interval [a, b] C JR. The same can be said of linear combinations of the functions { e i n x ; n E Z} . Definition 4. We shall say that a set S of functions on X separates points on X if for every pair of distinct points x 1 , x 2 E X there exists a function f E S such that f(xi ) =/= f(x2 ) · Definition 3. A set
called a
a
=
402
16 Uniform Convergence and Basic Operations of Analysis
Example 8.
The set of functions { e n x ; n E N}, and even each individual function in the set , separates points on JR.. At the same time, the 27!"-periodic functions {ei n x ; n E Z} separates points of a closed interval if its length is less than 271" and obviously does not separate the points of an interval of length greater than or equal to 271".
Example 9. The real polynomials together form a set of functions that sepa rates the points of every closed interval [a, b] , since the polynomial P(x) = x does that all by itself. What has just been said can be repeated for a set X C C and the set of complex polynomials on X. As a single separating function, one can take P(z) = z. :F of functions f X -+ C does not vanish on X (is nondegenerate) if for every point x0 E X there is a function fo E :F such that fo(xo) =f. 0. Example 1 0. The family :F = { 1, x, x 2 , , } does not vanish on the closed interval [0, 1], but all the functions of the family :F0 = { x, x 2 , . . , } vanish at x = O. Lemma 2 . If an algebra A of real {resp. complex) functions on X separates the points of X and does not vanish on X, then for any two distinct points X 1 , x 2 E X and any real {resp. complex) numbers c 1 , c2 there is a function f in A such that f(x l ) = c 1 and f(x 2 ) = c2 . Proof. It obviously suffices to prove the lemma when c 1 = 0, c2 = 1 and when c 1 = 1, c 2 = 0. By the symmetry of the hypotheses on x 1 and x 2 , we consider only the case c 1 = 1, c 2 = 0. We begin by remarking that A contains a special function s separating the points X 1 and x 2 that , in addition to the condition s(x 1 ) =f. s(x 2 ), also satisfies the condition s(x l ) =f. 0. Let g, h E A, g(x l ) =f. g(x 2 ), g(x l ) = 0, and h(x l ) =f. 0. There is obviously a number A E lR \ 0 such that A (h (x l ) - h(x 2 ) ) =f. g(x 2 ). The function s = g + Ah then has the required properties. Now, setting f(x) = 3;i7l-=���l=���) , we obtain a function f in the algebra A satisfying f(x l ) = 1 and f(x 2 ) = 0. Theorem 3 . (Stone8 ) . Let A be an algebra of continuous real-valued func tions defined on a compact set K. If A separates the points of K and does not vanish on K, then A is an everywhere-dense subspace of C(K, JR) . Proof. Let A be the closure of the set A C(K, JR) in C(K, JR), that is, A consists of the continuous functions f E C(K, JR) that can be approximated :
Definition 5 . The family
•
•
•
.
D
C
8
M.H. Stone (1903-1989) - American mathematician who worked mainly in topol ogy and functional analysis.
16.4 *Subsets of the Space of Continuous Functions
403
uniformly with arbitrary precision by functions of A. The theorem asserts that A = C(K, !R). Repeating the reasoning in the proof of Weierstrass ' theorem, we note that if J, g E A and o: E IR, then the functions J + g, f · g, o:J, IJI , max{f, g}, min{!, g} also belong t o A. B y induction we can verify that i n general if JI , f2 , . . . , fn E A, then max { JI , f2 , . . . , fn } and min{ JI , f2 , . . . , fn } also lie in A. We now show that for every function f E C(K, JR), every point x E K, and every number c > 0, there exists a function 9x E A such that 9x (x) = f(x) and 9x (t) > f(t) - c for every t E K. To verify this, for each point y E K we use Lemma 2 to choose a function hy E A such that hy (x) = f(x) and hy ( Y ) = f(y). By the continuity of f and hy on K, there exists an open neighborhood Uy of y such that hy (t) > f(t) - € for every t E Uy . From the covering of the compact set K by the open sets Uy we select a finite covering {Uy1 , Uy 2 , , Uy n }. Then the function function. 9x = max{ h h1 , hy2 , hyn } E A will be the desired Now taking such a function 9x for each point x E K, we remark that by the continuity of 9x and J, there exists an open neighborhood Vx of x E K such that 9x (t) < f(t) + c for every t E Vx . Since K is compact , there exists a finite covering {Vx1 , Vx 2 , , Vx m } by such neighborhoods. The function g = min {gx1 , , 9x m } belongs to A and by construction, satisfies both in equalities •
•
•
•
•
•
•
•
•
•
•
•
f(t) -
c
< g(t) <
f(t) +
c
at every point . But the number c > 0 was arbitrary, so that any function can be uniformly approximated on K by functions in A. 0
f E C(K, !R)
16.4.4 Problems and Exercises 1. A family :F of functions f : X -t Y defined on the metric space X and assuming values in the metric space Y is equicontinuous at Xo E X if for every o: > 0 there exists 8 > 0 such that dx (x, xo) < 8 implies dy ( !(x) , J (xo) ) < o: for every f E :F. a) Show that if a family :F of functions f : X -T Y is equicontinuous at Xo E X, then every function f E :F is continuous at xo , although the converse is not true. b ) Prove that if the family :F of functions f : K -t Y is equicontinuous at each point of the compact set K, then it is equicontinuous on K in the sense of Definition 2. c ) Show that if a metric space X is not compact, then equicontinuity of a family :F of functions f : X -t Y at each point x E X does not imply equicontinuity of :F on X. For this reason, if the family :F is equicontinuous on a set X in the sense of Definition 2, we often call it uniformly equicontinuous on the set. Thus, the relation between equicontinuity at a point and uniform equicontinuity of a family of
404
16 Uniform Convergence and Basic Operations of Analysis
functions on a set X is the same as that between continuity and uniform continuity of an individual function f : X --+ Y on the set X. d) Let w(f ; E) be the oscillation of the function f : X --+ Y on the set E C X, and B(x, 6) the ball of radius 6 with center at x E X. What concepts are defined by the following formulas?
Vc
>
0 :3 6
>
0 Vf E :F w ( t ; B(x, 6) ) < c ,
Vc > 0 :3 6 > 0 Vf E :F Vx E X w ( f ; B(x, 6) ) < c .
e) Show by example that the Arzela-Ascoli theorem is in general not true if K is not compact: construct a uniformly bounded and equicontinuous sequence { fn ; n E N} of functions fn (x) = rp(x + n) from which it is not possible to extract a subsequence that converges uniformly on JR. f) Using the Arzela-Ascoli theorem, solve Problem 10c) from Sect. 16.3. 2 . a) Explain in detail why every continuous piecewise-linear function on a closed interval [a , b] can be represented as a linear combination of functions of the form F�1 � 2 shown in the proof of Weierstrass' theorem. b) Prove Weierstrass' theorem for complex-valued functions f : [a , b] --+ IC.
c) The quantity Mn
J f (x)x n dx is often called the nth moment of the funcb
=
a
tion f : [a, b] --+
16.4 *Subsets of the Space of Continuous Functions i) For which location of the closed interval by rp(x) = x dense in c([a, bJ , IR ) ?
[a, b]
C
405
lR is the algebra generated
a) A complex algebra of functions A is self-adjoint if it follows from f E A that
7 E A, where f(x) is the value conjugate to f(x) . Show that if a complex algebra 4.
A is nondegenerate on X and separates the points of X, then, given that A is self adjoint, one can assert that the subalgebra AR of real-valued functions in A is also nondegenerate on X and also separates points on X . b ) Prove the following complex version o f Stone's theorem. If a complex algebra A of functions f : X --+ C is nondegenerate on X and separates the points of X, then, given that it is self-adjoint, one can assert that it is dense in C(X, C) . c) Let X = {z E tej Jzj = 1 } be the unit circle and A the algebra on X generated by the function e;"' , where rp is the polar angle of the point z E C. This algebra is nondegenerate on X and separates the points of X , but is not self-adjoint.
Prove that the equalities r f ( e;"' ) ei n
1 7 Integrals Dep ending on a Parameter
In this chapter the general theorems on families of functions depending on a parameter will be applied to the type of family most frequently encountered in analysis - integrals depending on a parameter.
1 7 . 1 Proper Integrals Depending on a Parameter
17 1 1 .
An
.
The Concept of an Integral Depending on a Parameter
integral depending on a parameter is a function of the form F(t)
=
j
f(x, t) dx ,
(17.1)
E,
where t plays the role of a parameter ranging over a set T,and to each value t E T there corresponds a set Et and a function 'Pt ( x) = f ( x, t) that is integrable over Et in the proper or improper sense. The nature of the set T may be quite varied, but of course the most important cases occur when T is a subset of � ' C, � n , or e n . If the integral (17.1) is a proper integral for each value of the parameter t E T, we say that the function F in (17.1) is a proper integral depending on
a parameter.
But if the integral in (17.1) exists only as an improper integral for some or all of the values of t E T, we usually call F an improper integral depending
on a parameter. But these are of course merely terminological conventions. When x E � m , Et � m , and m > 1, we say that we are dealing with a multiple (double, triple, and so forth) integral (17.1) depending on a param eter. C
We shall concentrate, however, on the one-dimensional case, which forms the foundation for all generalizations. Moreover, for the sake of simplicity, we shall first take Et to be intervals of the real line � independent of the parameter, and we shall assume that the integral (17.1) over these intervals exists as a proper integral.
408
17 Integrals Depending on a Parameter
1 7. 1 . 2 Continuity of an Integral Depending on a Parameter 1\
Let P = {(x, y) E JR.2 1 a ::::; x ::::; b c ::::; y ::::; d} be a rectangle in the plane IR.2 • If the function f : P --+ JR. is continuous, that is, if f E C ( P, JR.), then the function
Proposition 1.
b
F(y) = J f(x, y) dx
(17.2)
a
is continuous at every point y E [c, d] . Proof. It follows from the uniform continuity of the function f on the compact set P that 'P y (x) := f(x, y) f(x, yo) =: 'Py0 (x) on [a, b] as y --+ Yo , for y, y0 E [c, d]. For each y E [c, d] the function 'Py (x) = f(x, y) is continuous with respect to x on the closed interval [a, b] and hence integrable over that ::::t
interval. By the theorem on passage to the limit under an integral sign we can now assert that
F(yo)
=
b
b
J f(x, Yo) dx = y-tyo lim J f(x, y) dx = lim F(y) . 0 y-tyo a
a
Remark 1 . As can be seen from this proof, Proposition 1 on the continuity of the function (17.2) remains valid if we take any compact set K as the set of values of the parameter y, assuming, of course, that f E C(I x K, JR.), where I = {x E JR.I a ::::; x ::::; b}.
Hence, in particular, one can conclude that if f E C(I x D, JR.), where D is an open set in IR.n , then F E C ( D, JR.), since every point y0 E D has a compact neighborhood K C D, and the restriction of f to I x K is a continuous function on the compact set I x K. We have stated Proposition 1 for real-valued functions, but of course it and its proof remain valid for vector-valued functions, for example, for functions assuming values in c, IR. m , or e m . Example 1 . In the proof of Morse ' s lemma ( see Section 8.6, Part 1) we men tioned the following proposition, called H a d a m a r d ' s l e m m a.
If a function f belongs to the class c ( l > (u, JR.) in a neighborhood U of the point x0, then in some neighborhood of x0 it can be represented in the form (17.3) f(x) = f(xo) + cp(x)(x - xo) , where cp is a continuous function and cp(xo) f'(xo). Equality (17.3) follows easily from the Newton-Leibniz formula =
1
f(xo + h) - f(xo) = J f'(xo + t h ) dt 0
·
h
(17.4)
409
17. 1 Proper Integrals Depending on a Parameter
1 1 applied to the function F( h) J0 f'(x0 + t h) dt. All that remains is to make the substitution h x - x0 and set cp(x) F(x - x0). It is useful to remark that Eq. (17.4) holds for x0, h E m; n , where n is not restricted to the value 1. Writing out the symbol f ' in more detail, and for simplicity setting x0 0, one can write, instead of (17.4) =
and Proposition
=
=
=
and then one should set
n
cp(x)x L 'Pi(x) x i i= 1 1 in Eq. (17.3), where 'P i (x) J i1r (tx) dt. 0 =
=
17. 1 .3 Differentiation of an Integral Depending on a Parameter
Proposition 2. If the function f P IR is continuous and has a contin uous partial derivative with respect to y on the rectangle P {(x, y) E IR 2 1 a ::; X ::; b /\ c ::; y ::; d} then the integral ( 17.2 ) belongs to c ( l ) ( [ c, d], IR ) and ' ' --t
:
F'(y)
=
=
b
J �� (x, y)
(
dx .
a
17.5 )
Formula ( 17.5 ) for differentiating the proper integral ( 17.2 ) with respect to a parameter is frequently calledLeibniz ' formula or Leibniz ' rule. Proof. We shall verify directly that if y0 E [c, d], then F' (yo) can be computed by formula (17.5):
I F(yo + h) - F(yo) - ( J �� (x, Yo) dx) h i b
I j (f(x, Yo + h) - f(x, Yo) - �� (x, Yo)h) x l ::; ::; j l f (x, Yo + h) - f(x, Yo) - �� (x, Yo)h l dx ::; J 0<8< 1 l -;::;-afuy (x, Yo + ()h) - -;::;-ofuy (x, Yo) I dx J h l = cp(yo, h) · I h i . =
b
::;
a
=
b
a
d
b
a
sup
a
17 Integrals Depending on a Parameter
410
By hypothesis interval a ::; x ::;
h --+ 0.
0
%£ E C(P, JR), so that %£ (x, y) :::; %£ (x, y0) on the closed b as y --+ yo, from which it follows that tp(Yo, h) --+ 0 as
f
Remark 2. The continuity of the original function is used in the proof only as a sufficient condition for the existence of all the integrals that appear in the proof. Remark 3. The proof just given and the form of the mean-value theorm used in it show that Proposition 2 remains valid if the closed interval [c, d] is replaced by any convex compact set in any normed vector space. Here one may obviously assume as well that takes values in some complete normed vector space. In particular - and this is sometimes very useful - formula (17.5) is also applicable to complex-valued functions F of a complex variable and to func tions F(y) = F(y l , . . . , y n ) of a vector parameter y = (y 1 , . . . , yn ) E C'. In this case %£ can of course be written coordinatewise as ( *fr , . . . , 1:/n), and then (17.5) yields the corresponding partial derivatives:
f
b
of
8F = I �(x, y 1 , . . . , yn ) dx �(y) uy' uy' of the function
a
F.
7r
u( x) = J0 cos ( ntp - x sin 'P) dtp satisfies Bessel ' s equation x 2 u" + xu' + (x 2 - n 2 )u = 0. Indeed, after carrying out the differentiation with formula (17.5) and mak Example 2.
Let us verify that the function
ing simple transformations we find 7r
7r
-x 2 I in2 tp cos( ntp - x sin tp) dtp + x I sin 'P sin( n tp - x sin 'P) dtp + 0 0 + (x 2 - n2 ) I cos(ntp - x sin tp) dtp = 0 = - I ( ( x2 sin2 tp + n2 - x2 ) cos ( ntp - x sin 'P) 0 - x sin tp sin( n tp - x sin 'P)) dtp = = -( n + x cos 'P) sin ( ntp - x sin tp) I � = 0 . s
7r
7r
411
17. 1 Proper Integrals Depending on a Parameter
Example 3 .
The complete elliptic integrals
7r /2
E(k) =
IJ ()
1 - k 2 sin 2
( 1 7.6)
as functions of the parameter k, 0 < k < 1, called the modulus of the corre sponding elliptic integral, are connected by the relations dE dk
E-K k
dK dk
E K 2 k(1 - k ) k .
Let us verify, for example, the first of these. By formula ( 1 7.5) dE =dk =
7r / 2
I() 1
k
Example 4.
integral. Let
k sin 2
·
(1 - k2 sin2
K E -17r /2( 1 - k2 sm. 2
()
k
()
Formulas ( 1 7.5) sometimes make it possible even to compute the
7r / 2
F(o:) =
I()
ln(o: 2 - sin 2
(o: >
1) .
According to formula ( 17.5)
7r / 2
F ' (o:) =
I()
2o: d
from which we find F(o:) = 1r ln ( o: + .Jo: 2 - 1 ) + c. The constant c is also easy to find, if we note that, on the one hand F(o:) = 7r ln o: + 7r ln 2 + c + o( 1 ) as o: ---+ +oo, and on the other hand, from the definition of F( o:) , taking account of the equality ln( o: 2 -sin 2
2'. Suppose the function f P ---+ JR. is continuous and has a continuous partial derivative U on the rectangle P = { ( y) E JR. 2 1 a � � b c � y � d}; further suppose o:( y ) and {J(y) are continuously differentiable
Proposition 1\
:
x,
x
412
17 Integrals Depending on a Parameter
functions on [c, d] whose values lie in [a, b] for every y E [c, d] . Then the integral {3( y ) F(y) J f(x, y) dx (17.7) =
n (y)
is defined for every y E [c, d] and belongs to CC l l ( [c, d], R) , and the following formula holds: =
·
F'(y) f (f3 (y), y ) f3'(y) - f (a(y), y ) a'(y) + ·
{3( y )
J �� (x, y) dx .
(17.8)
n( y )
Proof. In accordance with the rule for differentiating an integral with respect to the limits of integration, taking account of formula (17.5), we can say that if a, j3 E [a, b] and y E [c, d], then the function cJ>(a, j3, y)
{3
J f(x, y) dx
=
has the following partial derivatives:
acJ> f(/3, y) ' 8(3
{3
8f (x, y) dx . y) J -f(a, ' aa ay ay Taking account of Proposition 1, we conclude that all the partial deriva tives of cJ> are continuous in its domain of definition. Hence cJ> is continuously differentiable. Formula (17.8) now follows from the chain rule for differentia tion of the composite function F(y) cJ> ( a(y), j3(y), y ) . 0 Example 5. Let Fn (x) ( n � 1)! J (x - t) n - l f(t) dt , 0 where n E N and f is a function that is continuous on the interval of inte gration. Let us verify that F�n ) (x) f(x). For n 1 we have F1 (x) J f(t) dt and F{(x) f(x). 0 By formula (17.8) we find for n > 1 that F� (x) = ( n � l) ! (x - x) n - l f(x) + ( n � 2 ) ! (x - t) n - 2 f(t) dt Fn - l (x) . 0 We now conclude by induction that indeed F�n ) (x) f(x) for every n E N. =
8cf>
=
8cf>
=
=
X
=
=
=
X
=
=
j X
=
=
413
17. 1 Proper Integrals Depending on a Parameter 17.1.4 Integration o f a n Integral Depending o n a Parameter -t
Proposition 3. If the function f P � is continuous in the rectangle P {( x, y) E �2 l a :::; x :::; b l\ c :::; y :::; d}, then the integral (17.2) is integrable over the closed interval [c, d] and the following equality holds: :
=
J ( ! f(x, y) dx ) dy J ( ! f(x , y) dy) dx . d
b
a
c
Proof.
=
b
d
(17.9)
c
a
From the point of view of multiple integrals, Eq. (17.9) is an elemen tary version of Fubini ' s theorem. However, we shall give a proof of (17.9) that justifies it independently of Fubini ' s theorem. Consider the functions
cp(u)
=
u
J ( J f(x, y) dx) dy , c
b
a
b
�(u) = J a
( ! f( u
c
)
x , y) dy dx .
Since f E C(P, �) , by Proposition 1 and the continuous dependence of the integral on the upper limit of integration, we conclude that cp and � belong to C([c, d], � ) . Then, by the continuity of the function (17.2), we find that =
b
=
b
cp' (u) J f(x, u) dx, and finally by formula (17.5) that � ' (u) J f(x, u) dx for u E [c, d]. Thus cp' (u) � ' (u), and hence cp(u) �(u) + C on [c, d]. But since cp(c) = �(c) = 0, we have cp(u) �(u) on [c, d], from which relation (17.9) follows for u = d. 0 a
=
=
=
a
17. 1 . 5 Problems and Exercises
a ) Explain why the function F(y) in (17.2) has the limit J 'P( x ) dx if the family a of functions 'f'y (x) = f(x, y) depending on the parameter y E Y and integrable over the closed interval a :::; x :::; b converges uniformly on that closed interval to a function 'P( x) over some base B in Y ( for example, the base y --t yo) . b ) Prove that i f E is a measurable set in JR m and the function f : E x r --t lR defined on the direct product E x I n = { (x, t) E JR m +n l x E E 1\ t E r } of the set E and the n-dimensional interval I n is continuous, then the function F defined by (17.1) for Et = E is continous on r . c ) Let P = { (x, y) E JR 2 a :::; x :::; b 1\ c :::; y :::; d}, and let f E C(P, JR) , o:, f3 E c([c, d], [a , bJ ) . Prove that i n that case the function ( 17.7) i s continuous on the closed interval [c, d]. b
1.
1
414
1 7 Integrals Depending on a Parameter
C(.IR, .IR) , then the function F(x) = 21a f f (x + t) dt is not -a only continuous, but also differentiable on JR. b) Find the derivative of this function F(x) and verify that F E C ( ll (JR, .IR) . 2. a) Prove that if f E
3. Using differentiation with respect to the parameter, show that for lrl 7r
F(r) =
<
1
I ln ( 1 - 2r cos x + r2 ) dx = 0 . 0
4. Verify that the following functions satisfy Bessel ' s equation of Example
2:
"
a) u = x n J cos(x cos
+1 b) Jn (x) = ( 2n "' � ) !!w J ( 1 - t 2 ) n - 1 / 2 l cos xt dt . -1 c) Show that the functions Jn corresponding to different values of n E N are connected by the relation Jn+1 = Jn - 1 - 2J� . 5 . Developing Example 3 and setting k show, following Legendre, that a) .fk (EK + EK - K K ) = 0. b) EK + EK - KK = rr/2.
:=
v'f"=k"2 , E(k) := E ( k ) , K(k) : = K(k) ,
6. Instead of the integral (17.2) , consider the integral b
:F(y) =
I f(x, y)g(x) dx , a
where g is a function that is integrable over the closed interval [a, b] (g E R[a, b] ) . By repeating the proofs of Propositions 1-3 above verify succesively that a) if the function f satisfies the hypotheses of Proposition 1 , then :F is contin uous on [c, d] (:F E C[c, d] ) ; b ) if f satisfies the hypotheses o f Proposition 2, then :F is continuously differ entiable on [c, d] (:F E C ( 1 ) [c, d] ) , and b
:F' (y) =
I �� (x, y) g(x) dx ; a
c) if f satisfies the hypotheses of Proposition 3, then :F is integrable over [c, d] (:F E R[c, d] ) and
1 c
:F(y) dy =
j (1 a
c
)
f (x , y)g(x) dy dx .
415
1 7.2 Improper Integrals Depending on a Parameter
7. Taylor's formula and Hadamard 's lemma.
a) Show that if f is a smooth function and f (O) = 0, then f(x) = x�(x ) , where � is a continuous function and �(0) = f ' (O) . b) Show that if f E c < nl and / (k ) (O) = 0 for k = 0, 1 , . . . , n - 1 , then f(x) = n x �(x) , where � is a continuous function and �(0) = � / ( n ) (0) . c) Let J be a c < nl function defined in a neighborhood of 0. Verify that the following version of Taylor ' s formula with the Hadamard form of the remainder holds: f(x)
= f (O) + � f' (O)x + · · · +
(
n
� 1 ) / n - 1 ) (O)x n - 1 + x n �(x) ,
where � is a funciton that is continuous on a neighborhood of zero, and �(0) = (n) � n . J (O) . d) Generalize the results of a) , b) , and c) to the case when f is a function of several variables. Write the basic Taylor formula in multi-index notation: n- 1 f (x) = L f(O)x " + L x " � a (x) , a. l <> l = n 1 <> 1 = 0 and note in addition to what was stated in a) , b) , and c) , that if f E c < n+ p ) , then �a E C ( P l .
�D"
1 7 . 2 Improper Integrals Depending on a Parameter
17.2.1 Uniform Convergence of an Improper Integral with Respect to a Parameter a. Basic Definition and Examples Suppose that the improper integral
F(y)
=
w
J f(x, y) dx a
(17.10)
over the interval [a, w] c IR converges for each value y E Y. For definiteness we shall assume that the integral (17.10) has only one singularity and that it involves the upper limit of integration (that is, either w = +oo or the function f is unbounded as a function of x in a neighborhood of w). Definition. We say that the improper integral (17.10) depending o n the parameter y E Y converges uniformly on the set E C Y if for every s > 0 there exists a neighborhood U[ a, w [(w) of w in the set [a, w[ such that the estimate
I ] f ( x, y) dx I < s b
(17.11)
416
17 Integrals Depending on a Parameter
for the remainder of the integral every y E E.
(17.10) holds for every b E U[a,w[ (w) and
If we introduce the notation
H (y)
:=
b
Ia f(x, y) x
(17.12)
d
for a proper integral approximating the improper integral ( 17.10), the basic definition of this section can be restated ( and, as will be seen in what follows, very usefully ) in a different form equivalent to the previous one: uniform convergence of the integral (17.10) on the set E C Y by definition means that (17.13) Fb ( Y ) � F(y) on E as b --+ w , b E [a, w[ . Indeed,
F(y)
=
w
I J(x, y) dx a
:=
b
lim
I f(x, y) dx
bEb--+w [a,w[ a
=
lim
b--+w bE[a,w[
Fb ( Y ) ,
(17.11) can be rewritten as (17.14) I F(y) - Fb ( Y ) I < E: . This last inequality holds for every b E U[a,b[ (w) and every y E E, as shown in (17.13). Thus, relations (17.11), (17.13), and (17.14) mean that if the integral (17.10) converges uniformly on a set E of parameter value::;, then this im proper integral (17.19) can be replaced by a certain proper integral (17.12) depending on the same parameter y with any preassigned precision, simulta neously for all y E E. Example 1 . The integral l+oo 2 dx 2 x y 1 converges uniformly on the entire set � of values of the parameter y E � ' since for every y E � and therefore relation
+
provided b >
1/E:.
Example 2 .
17.2 Improper Integrals Depending on a Parameter
The integral
417
+ oo
J e- xy dx ,
obviously converges only when every set { y IRI y 2': Yo > 0}. Indeed, if y 2': y0 > 0, then
E
0 y > 0.
Moreover it converges uniformly on
+ oo e - bYo --+ 0 0 J e - xy dx = �Y e - by ::=:; � Yo ::=:;
as
b --+ +oo.
At the same time, the convergence is not uniform on the entire set JR + = E on a set E means that
b
{ y IRI y > 0}. Indeed, negating uniform convergence of the integral (17.10)
3co > 0 VB E [a, w[ 3b E [B, w[ 3y E E
( I / f x, y x l ) (
) d > Ea
In the present case Eo can be taken as any real number, since
+ oo
J e- xy dx = � e - by --+ +oo , as y --+ +0 , b
bE
for every fixed value of [0, +oo [ . Let us consider a less trivial example, which we shall be using below.
Example 3.
Let us show that each of the integrals
+ oo
cJ> ( x ) =
J x"' y"'+f3 + 1 e -( l + x ) y dy ,
F (y) =
J x"' y"'+f3 + 1 e -( l +x ) y dx ,
0 + oo 0
in which a and (3 are fixed positive numbers, converges uniformly on the set of nonnegative values of the parameter. For the remainder of the integral cJ> ( x ) we find immediately that
0 ::::;
+ oo a J x"' y +f3+ 1 e -( l +x ) y dy = b + oo J ( xy ) "' e - xy yf3 + l e - y dy < Ma b
17 Integrals Depending on a Parameter
418
where Ma = max uae - u . Since this last integral converges, it can be made O: 0 for sufficiently large values of b E JR. But this means that the integral
0 :S
+oo
J xa ya+f3+l e -(l +x) y dx = b
= yf3 e - y Since
+oo a J ( xy) e- XY y dx = yf3 e-y J Ua e- u du . b � +oo
+oo
J ua e - u du < +oo , 0
for y 2 0 and yf3 e - Y --+ 0 as y --+ 0, for each c > 0 there obviously exists a number y0 > 0 such that for every y E [0, y0 ] the remainder of the integral will be less than c even independently of the value of b E [0 , +oo[. And if y 2 y0 > 0, taking account of the relations M13 = max yf3 e - Y <
O:'O y <+oo +oo +oo a +oo and 0 :S I u e - u du :S I u"'e - u du --+ 0 as b --+ +oo, we conclude byo by that for all sufficiently large values of b E [0, +oo [ and simultaneously for all y 2 y0 > 0 the remainder of the integral F (y) can be made less than c.
Combining the intervals [0 , y0 ] and [y0 , +oo[, we conclude that indeed for every c > 0 one can choose a number such that for every b > and every y 2 0 the corresponding remainder of the integral F(y) will be less than c .
B
B
b . The Cauchy Criterion for Uniform Convergence of an Integral
A necessary and sufficient condition for the improper integral (17.10) depending on the parameter y E Y to converge uniformly on a set E Y is that for every 0 there exist a neighborhood U[a, w [(w) of the point w such that
Proposition 1 . (Cauchy criterion) . c
c
b2
J f(x, y) dx
b,
>
(17.15)
for every b 1 , b2 E U[a, w [(w) and every y E E . Proof. Inequality ( 17.15) i s equivalent t o the relation 1Fb2 (y) - H2 (y) I < so that Proposition 1 i s an immediate corollary o f the form (17.13) for the definition of uniform convergence of the integral (17.10) and the Cauchy cri terion for uniform convergence on E of a family of functions Fb ( Y ) depending on the parameter b E [a, w [ . D c,
17.2 Improper Integrals Depending on a Parameter
419
As an illustration of the use of this Cauchy criterion, we consider the following corollary of it , which is sometimes useful.
If the function f in the integral (17.10) is continuous on the set [a, w[ x [c, d] and the integral (17.10) converges for every y E]c, d[ but diverges for y = c or y = d, then it converges nonuniformly on the interval ]c, d[ and also on any set E c]c, d[ whose closure contains the point of divergence. Proof. If the integral (17.10) diverges at y = c, then by the Cauchy criterion for convergence of an improper integral there exists > 0 such that in every neighborhood U[a , w [(w) there exist numbers b 1 , b2 for which Corollary 1.
Eo
j f(x, c) x > b2
d
bt
The proper integral
Eo
.
(17.16)
b2
J f(x, y) dx
bt
is in this case a continuous function of the parameter y on the entire closed interval [c, d] (see Proposition 1 of Sect . 17.1), so that for all values of y sufficiently close to c, the inequality
j f(x, y) dx > b2
bt
E
will hold along with the inequality (17.16). On the basis of the Cauchy criterion for uniform convergence of an im proper integral depending on a parameter, we now conclude that this integral cannot converge uniformly on any subset E c]c, d[ whose closure contains the point c. The case when the integral diverges for y = d is handled similarly. D
Example 4.
The integral
+ oo
J e-tx2 dx 0
converges for t > 0 and diverges at t = 0, hence it demonstrably converges nonuniformly on every set of positive numbers having 0 as a limit point . In particular, it converges nonuniformly on the whole set { t E lR I t > 0} of positive numbers.
420
17 Integrals Depending on a Parameter
In this case, one can easily verify these statements directly:
+ co
Jb set
2
1
e - t x dx = Vt
+ co
J 2
e - u du -+
+oo
as
t -+ +0 .
hli
We emphasize that this integral nevertheless converges uniformly on any { t E JR I t :2': t0 > 0} that is bounded away from 0, since
c. Sufficient Conditions for Uniform Convergence of an Improper Integral Depending on a Parameter
Proposition 2. ( The Weierstrass test ) . Suppose the functions f(x, y) and g(x, y) are integrable with respect to x on every closed interval [a, b] [a, w[ for each value of y E Y. If the inequality I f ( x, y) I :::; g( x, y) holds for each value of y E Y and every x E [a, w[ and the integral g(x, y) dx C
w
J a
converges uniformly on Y, then the integral w
j f(x, y) dx a
converges absolutely for each y E Y and uniformly on Y. Proof. This follows from the estimates
b2 b2 J f(x, y) dx :::; j l f(x, y) l dx :::; j g(x, y) dx b2
b,
and Cauchy ' s criterion for uniform convergence of an integral ( Proposition
D
1).
The most frequently encountered case of Proposition 2 occurs when the function g is independent of the parameter y. It is this case in which Propo sition 2) is usually called the Weierstrass M-test for uniform convergence of
an integral.
Example 5.
17.2 Improper Integrals Depending on a Parameter
The integral
00
J0
421
cos a x dx 1 + x2
converges uniformly on the the whole set < 1 and t he mtegra cos · · smce l
� of values of the parameter a,
J0oo l +dxx2 converges. I l +xax2 I l +x2, Example In view of the inequality I sin x e - tx 2 1 e - tx 2 , the integral 00 J sinxe-tx2 dx , _
�
6.
0
as follows from Proposition 2 and the results of Example 3, converges uni formly on every set of the form { t E �� t � t0 > 0}. Since the integral diverges for t = 0, on the basis of the Cauchy criterion we conclude that it cannot converge uniformly on any set having zero as a limit point.
Proposition 3. ( Abel-Dirichlet test ) . Assume that the functions f(x, y) and g(x, y) are integrable with respect to x at each y E Y on every closed interval [a, b] [a, w[. A sufficient condition for uniform convergence of the integral C
j (! g)(x, y) dx w
·
a
on the set Y is that one of the following two pairs of conditions holds: a 1 ) either there exists a constant M E � such that b
J f(x, y) dx a
<
M
for any b E [a, w[ and any y E Y and (31 ) for each y E Y the function g(x, y) is monotonic with respect to x on the interval [a, w[ and g(x, y) 0 on Y as x w, x E [a, w[, or a2 ) the integral f(x, y) dx =l
--t
j w
a
converges uniformly on the set Y and
422
17 Integrals Depending on a Parameter
/h ) for each y E Y the function g(x, y) is monotonic with respect to x on the interval [a, w[ and there exists a constant M E lR such that l g(x, Y ) l < M for every x E [a, w[ and every y E Y. Proof. Applying the second mean-value theorem for the integral, we write �
J (f g)(x, y) ·
b,
�
e
dx
J
J
= g(b 1 , y) f(x, y) dx + g(b2 , y) f(x, y) dx , b,
e
where � E [b 1 , b 2 ]. If b 1 and b2 are taken in a sufficiently small neighborhood U[a , w[ (w) of the point w, then the right-hand side of this equality can be made smaller in absolute value than any prescribed c > 0, and indeed simultane ously for all values of y E Y. In the case of the first pair of conditions a i ), /31 ) this is obvious. In the case of the second pair a 2 ), /32 ), it becomes ob vious if we use the Cauchy criterion for uniform convergence of the integral (Proposition 1 ) . Thus, again invoking the Cauchy criterion, we conclude that the original integral of the product f · g over the interval [a, w[ does indeed converge uniformly on the set Y of parameter values. D
Example
7.
The integral
+=
J1
sin x dx ' xa
as follows from the Cauchy criterion and the Abel-Dirichlet test for conver gence of improper integrals, converges only for a > 0. Setting f ( x, a) = sin x, g(x, a) = x - a , we see that the pair a 1 ), /31 ) of hypotheses of Proposition 3 holds for a ?: a0 > 0. Consequently, on every set of the form {a E JR. I a ?: a0 > 0} this integral converges uniformly. On the set {a E lR I a > 0} of positive values of the parameter the integral converges nonuniformly, since it diverges at a = 0.
Example 8 .
The integral
converges uniformly on the set
+=
-x y dx J0 --e X sin x
·
JRI
{y E y ?: 0} .
Proof. First of all, on the basis of the Cauchy criterion for convergence of the improper integral one can easily conclude that for y < 0 this integral diverges. Now assuming y ?: 0 and setting f(x, y) = s i � x , g(x, y) = e -x y , we see that the second pair a 2 ), /32 ) of hypotheses of Proposition 3 holds, from which it follows that this integral converges uniformly on the set {y E JR. y ?: 0}. D
I
17.2 Improper Integrals Depending on a Parameter
423
Thus we have introduced the concept of uniform convergence of an im proper integral depending on a parameter and indicated several of the most important tests for such convergence completely analogous to the correspond ing tests for uniform convergence of series of functions. Before passing on, we make two remarks. Remark 1. So as not to distract the reader ' s attention from the basic con cept of uniform convergence of an integral introduced here, we have assumed throughout that the discussion involves integrating real-valued functions. At the same time, as one can now easily check, these results extend to integrals of vector-valued functions, in particular to integrals of complex-valued func tions. Here one need only note that , as always, in the Cauchy criterion one must assume in addition that the corresponding vector space of values of the integrand is complete (this is the case for � ' c, � n , and e n ) ; and in the Abel-Dirichlet test , as in the corresponding test for uniform convergence of series of functions, the factor in the product f · g that is assumed to be a monotonic function, must of course be real-valued. Everything that has just been said applies equally to the main results of the following subsections in this section. Remark 2. We have considered an improper integral whose only sin gularity was at the upper limit of integration. The uniform convergence of an integral whose only singularity is at the lower limit of integration can be defined and studied similarly. If the integral has a singularity at both limits of integration, it can be represented as
(17.10)
C
W2
W2
j f (x, y) dx = j f (x, y) dx + j f (x, y) dx , c
where c E]w 1 , w2 [, and regarded as uniformly convergent on a set E C Y if both of the integrals on the right-hand side of the equality converge uniformly. It is easy to verify that this definition is unambiguous, that is, independent of the choice of the point c E]w l , w2 [. 17.2.2 Limiting Passage under the Sign of an Improper Integral and Continuity of an Improper Integral Depending on a Parameter
Let f(x, y) be a family of functions depending on a param eter y E Y that are integrable, possibly in the improper sense, on the interval a :::; < w, and let By be a base in Y. If a) for every b E [a, w[ f(x, y) ip(x) on [a, b] over the base By and Proposition 4. x
�
424
17 Integrals Depending on a Parameter
b) the integral I f( x , y) dx converges uniformly on Y, a then the limit function cp is improperly integrable on [a, w[ and the follow ing equality holds: w w lA� I f( x , y) dx I cp ( x ) dx . (17.17) w
=
a
Proof.
a
The proof reduces to checking the following diagram:
Fb ( Y )
:=
I f(x, y) dx b
a
1 ( ) By
---
I cp x dx b
a
I f(x, y) dx = : F(y) w
b""* bE[t:
__-
_j_ __- a
1( )
By
I cp x dx . w
� [a,w [
a
bE
The left vertical limiting passage follows from hypothesis a) and the the orem on passage to the limit under a proper integral sign ( see Theorem 3 of Sect. 16.3). The upper horizontal limiting passage is an expression of hypothesis b) . By the theorem on the commutativity of two limiting passages it follows from this that both limits below the diagonal exist and are equal. The right-hand vertical limit passage is what stands on the left-hand side of Eq. (17.17), and the lower horizontal limit gives by definition the improper integral on the right-hand side of (17.17).
D
The following example shows that condition a ) alone is generally insuffi cient to guarantee Eq. (17.17) in this case.
Example 9.
Let Y
=
{y E JR I y > 0} and 1/y , if 0 $ f(x, y) 0 , if y
0
=
{
Obviously, f(x, y) :4 on the interval time, for every y E Y ,
+ oo
Y
0
$
x
<
X
$
y,
< X .
+oo as y -+ +oo. At the same Y
I f(x, y) dx I f(x, y) dx I t dx 0
=
0
=
0
=
1,
( 17 .17) does not hold in this case. Using Dini ' s theorem ( Proposition 2 of Sect . 16.3), we can obtain the following sometimes useful corollary of Proposition 4. and therefore Eq.
17.2 Improper Integrals Depending on a Parameter
425
Corollary 2 . Suppose that the real-valued function f(x, y) is nonnegative at each value of the real parameter y E Y JR. and continuous on the interval a ::; x w. If a) the function f(x, y) is monotonically increasing as y increases and tends to a function �.p(x) on [a, w[, b) E C ( [a, w[, JR. ) , and c) the integral J �.p( x) dx converges, then Eq. ( 17.17) holds. Proof. It follows from Dini ' s theorem that f(x, y) ::::; �.p(x) on each closed interval [a, b] [a, w[. It follows from the inequalities 0 ::; f(x, y) ::; �.p(x) and the Weierstrass M-test for uniform convergence that the integral of f(x, y) over the interval a ::; x w converges uniformly with respect to the parameter y. Thus, both hypotheses of Proposition 4 hold, and so Eq. (17.17) holds. C
<
'P
w
a
C
<
0
Example 1 0. In Example 31 ofn Sect . 16.3 we verified that the sequence of fn (x) = n(1 - x 1 ) is monotonically increasing on the interval 0 Hence, x ::; 1 , and fn (x) / ln � as n --+ +oo. by Corollary 2 1 1 1 n lim n ( 1 - x 1 ) d x = ln � dx . n-+oo 0 0 functions <
j
j
X
5. If a) the function f(x, y) is continuous on the set {(x, y) E JR. 2 1 a ::; x w c ::; y ::; d}, and b) the integral F(y) = J f(x, y) dx converges uniformly on [c, d] , then the function F(y) is continuous on [c, d] . Proof. It follows from hypothesis a) that for any b E [a, w[ the proper integral b Fb ( Y ) f(x, y) dx
Proposition 1\
<
w
a
=
j a
is a continuous function on [c, d] (see Proposition 1 of Sect. 17.1 ) . By hypothesis b) we have Fb ( Y ) ::::; F(y) on [c, d] as b --+ w, b E [a, w[, from which it now follows that the function F(y) is continuous on [c, d] . 0
426
17 Integrals Depending on a Parameter
Example 1 1 .
It was shown in Example
F(y) =
+ oo
8 that the integral
J si� x e -xy dx 0
(17.18)
converges uniformly on the interval 0 ::; y < +oo. Hence by Proposition 5 one can conclude that F(y) is continuous on each closed interval [0, d] C [0, +oo[, that is, it is continuous on the entire interval 0 ::; y < +oo. In particular, it follows from this that . 1 1m
y ---+ + 0
+ oo
J0
-- e -xy X
sin x
(17.19)
dx =
1 7. 2 . 3 Differentiation of an Improper Integral with Respect to a Parameter
If a) the functions f ( x, y) and f� ( x, y) are continuous on the set { ( x, y) E ffi:.2 J a ::; X w A c ::; y ::; d}, b) the integral
w
a
w
w
F'(y) = J f�(x, y) dx . Proof.
By hypothesis a ) , for every
a
b E [a, w [ the function b Fb ( Y ) = J f(x, y) dx a
is defined and differentiable on the interval
b
c ::; y ::; d and by Leibniz ' rule
( H )'(y) = J f�(x, y) dx . a
17.2 Improper Integrals Depending on a Parameter
427
By hypothesis b) the family of functions (Fb )'(y) depending on the pa rameter b E [ [ converges uniformly on [c, d] to the function P( y ) as b -+ bE [ [ as b -+ b E [ [ ByIt follows hypothesis c)thisthe(seequantity Fb (Y4 oof) hasSect.a limit from Theorem 16. 3 ) that the fami ly of func tions F ( ) itself converges uniformly on [c, d] to the limiting function F(y) Y asandb -+theb equality b E [ [ the function F is differentiable on the interval c :::; y :::; d, proved. D F'(y) = P(y ) holds. But this is precisely what was to be Example For a fixed value a > 0 the integral +oo J xa e -xy dx 0 converges uniformly with respect to the parameter y on every interval of the form { y E �� y 2 Yo > 0}. This follows from the estimate 0 :::; x"'e - xy x a e - xyo e - x l!f, which holds for all sufficiently large x E Hence, by Proposition 6, the function a, w
w,
a, w .
w,
w,
a, w .
a, w ,
12.
<
R
<
+oo
F(y) = J e -xy dx 0 is infinitely differentiable for y > 0 and p ( n ) (y) = ( - I t
F(y) that = t ' and therefore p canButconclude +oo
+oo J xn e -xy dx . ()
( n ) (y )
= (- l ) n Y ���
J xn e -xy dx = � y n+l . 0
In particular, for y = 1 we obtain +oo J xn e-x dx = n! . 0 Example Let us compute the Dirichlet integral 1 3.
and consequently we
428
17 Integrals Depending on a Parameter
To do this we return to the integral (17.18), and we remark that for y > 0 + oo F (y) = - J sinxe- xy dx , (17. 20) 0 since the integral (17. 20) converges uniformly on every set of the form {y E JR. I Y 2: Yo > 0}. (17. 20)is that is easily computed from the primitive of the inte grand,Theandintegral the result 1 F (y) -1 +-y2 for y > 0 , from which it follows that F(y) = - arctany + c for y > 0 . (17. 2 1) We have F(y) --+(17.21) 0 as ythat --+ +oo, as can be seen from relation (17.18), so that itthatfollows from c = 1r /2. It now results from (17.19) and (17. 2 1) F(O) = 1r /2. Thus, + oo (17. 2 2) J0 sinxX dx �2 . We2 2) remark thatimmediate the relation F(y)of--+Proposition 0 as y --+ +oo" used in deriving (17. is not an corollary 4, since si� x e - xy 0 ofon theintervals form {xof theE JRIform x ;:::0: x0 >x 0},x while as yconvergence --+ +oo onlyis onnot intervals the uniform 0 : for s i n x e - xy --+ 1 as x --+ 0. But for xo > 0 we have xo + oo oo si n x si n x -xy xy J0 X e dx - J0 X e dx + xJo sinX x e-xy dx and, and given > 0 we firstxochoose x0 so closexoto 0 that sin x 2: 0 for x E [0 , x0] 0 J sinX x e - xy dx J sinX x dx �2 0 0 for every y > 0. Then, after fixing x0 , on theoverbasis[x of Proposition 4, by letting +oo, we can make the integral yabsolute tend tovalue. 0 , +oo[ also less than c:/2 in '
'
=
=
"
<
X
--
E
_
<
--
--
<
<
<
=l
17.2 Improper Integrals Depending on a Parameter
429
17.2.4 Integration of an Improper Integral with Respect to a Parameter
If the function f(x, y) is continuous on the set {(x, y) E IR 2 1 a :S x w 1\ c :S y :S d} and b) the integral F(y) = J f(x, y) dx converges uniformly on the closed interval [c, d], then the function F is integrable on [c, d] and the following equality holds:
Proposition 7. a)
<
w
a
( 17.23) J dy J f(x, y) dx = J dx J f(x, y) dy . Proof. For b E [ [ by hypothesis a) and Proposition 3 of Sect. 17.1 for improper integrals one can write w
d
c
w
a
d
c
a
a, w ,
d
b
d
b
J dy J f(x, y) dx = J dx j f(x, y) dy . ( 17.24) Usingan integral hypothesissign,b) weandcarry Theorem 3 of Sect. 16.3 on passage to the limit under out a limiting passage on the left-hand side ofthe(17.24) as b -+ b E [ [ and obtain the left-hand side of (17.23). By very definition of an improper integral, the right-hand side of (17.23) is the limit of the right-hand si d e of (17.24) as b -+ b E [ [ Thus, by hypothesis b) we obtain ( 17.23) from (17.24) as b -+ b E [ , [. D Theof followi ng example shows that,integrals, in contrast to thea)reversibility of the order integration with two proper condition alone is in general not sufficient to guarantee (17.23). Example 14. Consider the function f(x, y) = (2 - xy)xye - xy on the set {(x, y) E IR 2 1 0 :S x +oo 1\ 0 :S y :S 1}. Using the primitive u 2 e - u of the function (2 - u)ue-u, it is easy to compute directly that + oo + oo 0 J dy J (2 - xy)xye - xy dx =f. J dx J (2 - xy)xye - xy dy 1 . c
a
w,
a
c
a, w
w,
a, w .
w,
=
<
1
0
0
a w
1
()
=
0
If a ) the function f ( x, y) is continuous on the set a :S x w 1\ c :S y :S d} and b) nonnegative on P, and Corollary 3. <
P
{(x, y) E JR 2 1
430
17 Integrals Depending on a Parameter w
c) the integral F (y) = J f(x, y) dx is continuous on the closed interval [c, d] as a function of y, then Eq. (17. 2 3) holds. Proof. It follows from hypothesis a) that for every b E [a, w[ the integral a
b Fb ( Y ) = J f(x, y) dx is continuous with b)respect to ,y(y)on theF closed interval [c, d]. ItByfollows from that F � b2 (y) for b 1 � b2 . b ' s theorem and hypothesis c) we now conclude that Fb F on Dini [c, d] as b -+ w, b E [a, w[. the hypotheses Proposition (17.Thus 2 3) indeed holds in theof present case.7 are satisfied and consequently Eq. Corollary 3 shows that Example 14 results from the fact that the function f ( x, y) is not of constant sign. gralsIntoconclusion commute.we now prove a sufficient condition for two improper inte a
=l
D
If a) the function f(x, y) is continuous on the set {(x, y) E IR2 1 a � x < w 1\ c � y < w}, b) both integrals
Proposition 8.
w
w
lf> (x) = j f(x, y) dy
F(y) = j f(x, y) dx ,
c
a
c
converge uniformly, the first with respect to y on any closed interval [c, d] [c, w], the second with respect to x on any closed interval [a, b] [a, w[, and c) at least one of the iterated integrals C
w
w
w
w
j dy j l f l (x, y), dx , J dx J l f l (x, y) dy c
a
a
c
converges, then the following equality holds: w
w
w
w
J dy J f(x, y) dx = J dx J f(x, y) dy . c
a
a
c
(17. 2 5)
17.2 Improper Integrals Depending on a Parameter
431
For definiteness suppose that the second of the two iterated integrals inProof. c)Byexists. a)holdsandforthethefirstfunction conditionf forin b)everyonedcanE [c,sayw[.by Proposition 7 thatIfEq.wecondition (17. 2 3) that the right-hand side of (17. 23) tends to the right-hand side ofthe(17.left-hand 25)show as dside -+ w, d E [c, w[, then Eq. (17.25) will have been proved, since thendefinition also existofandan improper be the limitintegral. of the left-hand side of Eq. (17. 2 3) by thewillvery Let us set q] (x) := j f(x, y) dy; . function is defined for each fixed d E [c, w[ and, since f is contin uous,The is continuous on the interval ::; x By theclosed secondinterval of hypotheses b) we have q] (x) q](x) as d -+ w, d E [c, w[ on each [ , b] [ [ Since l q] 0uniformly shows at with the same timeto thethatparameter for everyyfixed value of setA >of real 0 it respect on the entire numbers Thebysame thingd:ccould have been said about the integral obtained from this one replacing with dy. The values of these integrals happen to differ only in sign. A direct computation shows that d
d
c
q)d
qJ d
< w.
a
a
w
d
C a, w .
�
w
a
c
w
a
w
d -tw d E [c,w]
d
a
1 5.
1 <
JR.
7r 4
A
A
A
A
432
17 Integrals Depending on a Parameter
For a > 0 and f3 > 0 the iterated integral + co + co + co + co y x l l y J dy J x<> y<> +.B- e -( + ) dx = J y.B e - dy J (xy) <>e-(xy) y dx 0 0 0 0 of a nonnegative continuous function+ coexists, as this identity shows: it equals zero for y = 0 and J0+ co y.Be-Y dy J0 u<> e - u du for y > 0. Thus, in this case a) and c) of Proposition 8 hold. The fact that both conditions of bwehypotheses ) hold for this integral was verified in Example 3. Hence by Proposition 8 have the equality Example 1 6.
·
+ co
+ co
+ co
+ co
J dx J x<> y<> +.B+ l e ( l +x) y dy . 0 0 0 0 Justcorollary as Corollary followed from lowing from 3Proposition 8. Proposition 7, we can deduce the fol J dy J x<> y<> +.B+ l e-( l +x) y dx =
-
If a) the function f(x, y) is continuous on the set P = {(x, y) E lR2 i a � x < w A c � y � w} , and b) is nonnegative on P, and c) the two integrals
Corollary 4.
w
F(y) = j f(x, y) dx , a
w
cJ>(x) = j f(x, y) dy c
are continuous functions on [a, w[ and [c, w[ respectively, and d) at least one of the iterated integrals w
w
w
w
j dy j f(x, y) dx , j dx j f (x, y) dy , a
c
a
c
exists, then the other iterated integral also exists and their values are the same. Reasoning as in the proof of Corollary 3, we conclude from hypotheses ainProof. ) , b ) , and c ) and Dini ' s theorem that hypothesis b ) of Proposition 8 holds this case. Since f 2:: 0, hypothesis d ) here is the same as hypothesis c) of
Proposition 8. Thus all the hypotheses of Proposition 8 are satisfied, and so Eq. (17. 24) holds. D
17.2 Improper Integrals Depending on a Parameter
433
Remark As pointed out in Remark 2, an integral having singularities at both limits of integration reduces to themakes sum itofpossible two integrals, eachthe ofproposi which has a singularity at only one limit. This to apply tions and corollaries just proved to integrals over intervalson] wclosed Here 1 , w2 [ intervals naturally the hypotheses that were satisfied previously [a ,b] [a,w [ must now hold on closed intervals [a , b] c ] w1 ,w2 [. letExample us show thatBy changing the =order of integration in two improper integrals, f0 e_x2 dx � J7r . (17.26) This is the famous Euler-Poisson integral. Proof. We first observe that for y > 0 += += 2 J e -u du y J e - (xy) 2 dx , 0 0 and that the value interval of the integral inor(17.the2 6)openis the same]O,whether it is taken over the half-open [ 0 , +oo[ interval +oo[ . Thus, += += += += J ye-Y2 dy J e- (xy) 2 dx J e_Y2 dy J e - u2 du = .:72 ' 0 0 0 0 y extends over the interval ] O, +oo [ . andAsweweassume that the integration on permissible to reverse the order of integration over this verify, iteratedit isintegral, and therefore x and y inshall dx 7r 1 + x2 4 ' fromLetwhiuschnowEq.justify (17. 2 6)reversing follows. the order of integration. The function += 1-2 J ye - ( l+x 2 ) y2 dy = �2 1 + x 0 is continuous for x 0, and the function += J ye- ( l+x2 ) y2 dx e_Y2 3.
C
C
1 7.
=
J
:=
=
=
�
0
=
.J
R
17 Integrals Depending on a Parameter
434
is continuous for > 0. Taking account of the general Remark 3, we now conclude indeed legal.from DCorollary 4 that this reversal in the order of integration is y
1 7 . 2 . 5 Problems and Exercises
a = ao < a1 < · · · < a n < < w. We represent the integral (17. 10) as the oo an sum of the series L 'P n ( y ) , where 'P n ( Y ) = J f ( x , y ) dx . Prove that the integral n= l an - 1 converges uniformly on the set E C Y if and only if to each sequence { a n } of this form there corresponds a series L
·
·
·
00
2 . a ) In accordance with Remark 1 carry out all the constructions in Subsect . 17.2. 1 for the case of a complex-valued integrand f. b ) Verify the assertions in Remark 2. 3 . Verify that the function
� y' + y = 0.
Jo ( x ) =
+ oo
J
4. a Starting from the equality
)
( 2n-3 ) !! 1 ( 2n - 2 ) !! . x 2n - 1
•
0
x
1
J c� dt satisfies Bessel ' s equation y" + 1 - t2 0
v
2�y
7r
2"
2
2n 3 !! dy ) n 2 (( 2n -- 2 )) !! 1 + ( y 2 /n ) ( 1 + ( y 2 j n )) - n \., e - y 2 on lR + oo
b) Verify that I 0
c ) Show that
.!. 7r
(
.!!:
+ oo
. l 1m
n ->+ oo
I 0
+ oo
a) J e 0
2 x
( 2n - 3 ) ! ! lim -+ oo ( 2n - 2 ) ! ! n
(17.26) , show that
cos 2xy dx =
� yTie-y 2 .
2 2 b ) J e - x sin 2xy dx = e - y +oo 0
n.
as
dy ( 1 + ( y 2 j n )) n -
d ) Obtain the following formula of Wallis: 5 . Taking account of Eq.
. r,;; V
Y
J et 2 dt. 0
n -+ +oo and that
+ oo
I 0
2 e - y dY
= _y7i1_ .
·
6. Assuming t
17.2 Improper Integrals Depending on a Parameter
>
0, prove the identity +
= e� t
x
I 1 + x2 0
dx
=
+
I t
= sin(x -
x
435
t) dx '
using the fact that both of these integrals, as functions of the parameter t, satisfy the equation jj + y 1/t and tend to zero as t --+ +oo.
=
7. Show that
where K(k)
1
rr / 2
I K(k) dk = I sin 0
0
=
rr / 2
J 0
>
'P
is the complete elliptic integral of first kind.
d .
V 1 - k 2 sm 2 r.p
8. a) Assuming that a
_5!_ dip
0 and b > 0 and using the equality
0
a
0
compute this last integral. b) For a > 0 and b > 0 compute the integral +
I 0
=e
�x a
_ X
e�b
x cos x dx .
-----
c) Using the Dirichlet integral (17.22) and the equality +
b
I �I 0
=
compute this last integral. 9. a) Prove that for
+
I 0
a
sin xy dy
= I+ = cos ax - cos bx dx , X
0
2
k>0
= � kt . e sm t dt
+
I 0
= � tu 2 e du
=
I du I e�(k+u2)t sintdt . + 0
=
+
=
0
b) Show that the preceding equality remains valid for the value k c) Using the Euler�Poisson integral (17.26) , verify that
1
vt
2 + =e �tu 2 du .
fi
I 0
= 0.
17 Integrals Depending on a Parameter
436
d) Using this last equality and the relations
l Sill. + oo
X
2
0
l sinVtt dt , l cos x dx = 1 l COS../it dt , + oo
1
dx = 2
+ oo
+ oo
2
0
2
0
obtain the value ( 4 �) for the Fresnel integrals
0
I sin x dx , I cos x dx . + oo
+ oo
2
10.
2
0
0
a) Use the equality
I sin x dx I e - xy dy + oo
+ oo
0
0
0 0 {zr. I 0, 0 0,
and, by justifying a reversal in the order of integration in the iterated integral, obtain once again the value of the Dirichlet integral (17.22) found in Example 13. b) Show that for a > and j3 > + oo
4
sin ax cos j3x dx = --
""
x
0
if /3 < a ,
2 '
, if
j3 = a ,
if /3 > a . This integral is often called the Dirichlet discontinuous factor. c) Assuming a > and j3 > verify the equality
I
+ oo
sin ax sin /3x
o
X
X
dx =
{
!r.2 j3 '
j3 '5:. Q ' if Q '5:. j3 .
if
2a , 1r
d) Prove that if the numbers a, a 1 , . . . , a n are positive and a > L: a; , then + oo
I 0
11.
sin ax sin a 1 x
sin
On X -- --- · · · ---
X
X
X
n
dx =
1r
i= l
- Q 1 02 · · · Q n .
2
Consider the integral
:F(y) = I f(x, y)g(x) dx , a w
g is a locally integrable function [a, w[ (that is, for each b E [a, w[ g[[a ,bJ E R[a, b] ) . Let the function f satisfy the various hypotheses a) of Propositions 58. If the integrand f(x, y) is replaced by f(x, y) g(x) in the other hypotheses of these propositions, the results are hypotheses under which, by using Problem 6 of where
·
17.3 The Eulerian Integrals
437
Sect. 17. 1 and repeating verbatim the proofs of Propositions 5-8, one can conclude respectively that a) :F E C[c, d] ; b ) :F E C (l ) [c, dj , and
� �� (x, y)g(x) dx ; w
:F' (y) =
a
c ) :F E R[c, d] and
j :F(y) dy = ] ( j f(x, y)g(x) dy) dx ; c
c
a
c ) :F is improperly integrable on [c, w[, and
J :F(y) dy = ] (J f(x, y)g(x) dy) dx .
Verify this.
c
c
a
17.3 The Eulerian Integrals
In this section we shallintegrals illustrateof theimportance applicationin analysis of the theory developed aboveandtothesomenextspecific that depend on a parameter. Following Legendre, we define the Eulerian integrals of first and second kinds respectively as the two special functions that follow: 1 (17. 2 7) B(a,/3) J x"' - 1 (1 - x),B - 1 dx , :=
0
+ oo
(17.28) T(a) · - J x"' - 1 e - x dx . 0 The first of these is called the beta function and the second the gamma function. 17.3.1 The Beta Function
necessary and sufficient condition for the con vergence of the integral (17.27) limit is that a > 0. Similarly, convergence at 1 occurs if and onlyat theif (3 lower > 0.
a. Domain of definition A
438
17 Integrals Depending on a Parameter
Thus the function B( a, /3) is defined when both of the following conditions hold simultaneously: a > 0 and f3 > 0 . Remark. We are regarding a and f3 as real numbers here. However, it should beand keptgamma in mindfunctions that theandmostthe complete picture applications of the properties of theinvolve beta most profound of them their extension into the complex parameter domain. b. Symmetry Let us verify that B(a,/3) = B(f3,a) . (17 . 29) Proof. It suffices to make the change of variable x = 1 - t in the integral (17. 2 7). D c. The Reduction Formula If a > 1, the following equality holds: B(a,/3) = a +a -f3 -1 1 B(a - 1, /3). (17. 30) forProof.a >Integrating 1 and f3 > 0,bywepartsobtainand carrying out some identity transformations 1 B(a, /3) = - � xa - 1 (1 - x) f3 1 � + a ; 1 · j x"'- 2 (1 - x) f3 dx = 0 1 = a ; 1 j xa - 2 ((1 - x) f3- 1 - (1 - x) f3- 1 x) dx = 0 a --B(a 1 a-1 =13 - 1, /3) - -13-B(a, /3) , from which the reduction formula (17. 30) follows. D Taking account of formula (17. 29), we can now write the reduction formula B(a,/3) = a +/3 f3- -1 1 B(a,/3 - 1) (17. 30') on the parameter /3,immediately assuming, from of course, that f3 > of1. the beta function that It can be seen the definition B(a, 1) = � ' and so for n E N we obtain n-1 n-2 n - (n - 1 ) B(a,n) = --a +_n_-'-..,.-(n_.. --'-....,.1). B (a ' 1) = a+n-1 a+n-2 _ (n - 1)! (17. 3 1 ) a(a + 1) · . . . · (a + n - 1) · In particular, for m, n E N - 1)! B(m, n) = (m( m- +1)!(n (17.32) n - 1)!
17.3 The Eulerian Integrals d. Another Integral Representation of the Beta Function
lowing representation of the beta function is sometimes useful:
439
The fol
(17. 3 3) Proof. This representation can be obtained from (17. 27) by the change of van. ables x - 1 y +Y .
D
17.3.2 The Gamma Function
It canconverges be seen byatformula the inte gral defining the gamma function zero only(17.for28)athat > 0, while it converges atfactor infinitye - xfor. all values of a E IR, due to the presence of the rapidly decreasing Thus the gamma function is defined for a > 0. b. Smoothness and the Formula for the Derivatives The gamma func tion is infinitely differentiable, and (17. 34) r ( n ) ( a ) = J x"' - 1 lnn xe - x dx . 0 Proof.toWethefirstparameter verify thatontheeachintegral (17. 34) converges uniformly withfixedre spect for each closed interval [a, b] c] +oo[ valueIf 0of n aE :::;N.a, then (asince 2 lnn x 0 as x 0)O, there exists > 0 Cn x"'l + such that a. Domain of Definition
+ oo
<
�
�
forconclude 0 x that :::; en .theHence by the Weierstrass M-test for uniform convergence we integral J x"' - 1 lnn x e- x dx 0 converges withforrespect If a :::; uniformly b then x 1,to a on the interval [a, +oo[. <
c,
<
+oo,
?:
440
17 Integrals Depending on a Parameter
and we conclude similarly that the integral +oo J x"' 1 lnn xe- x dx converges uniformly with respect weto afindon that the interval ] O, b] .(17. 34) converges Combining these conclusions, the integral unifBut ormlyunder on every closed intervaldifferentiation [a, b] c ] O, +oo[.under the integral sign in these conditions (17. 2 7)open is justified. Hence, ontheanygamma such function closed interval, and hence on the entire interval 0 a, is infinitely differentiable and formula (17. 34) holds. D c . The Reduction Formula The relation r(a + 1) = ar(a) (17. 35) holds. It is known as the reduction formula for the gamma function. Proof. Integrating by parts, we find that for a > 0 +oo +oo oo r(a + 1) := J x"'e- x dx = -x"'e- x i � + a J x"' - 1 e - x dx = -
Cn
<
0
0
+oo
= a J x"' - 1 e- x dx = aF(a) . D 0
Since F(1) = +ooJ0 e - x dx = 1, we conclude that for n E N r(n + 1) = n! . (17. 3 6) Thus the gamma function number-theoretic function n!. turns out to be closely connected with the d . The Euler-Gauss Formula This is the name usually given to the fol lowing equality: r(a) = nli---+moo n"' . a(a + 1) (n· . . -. · 1)!(a + n - 1) (17.37) Proof. To prove this formula, we make the change of variable x = ln � in the integral (17. 2 8), resulting in the following integral representation of the gamma function: 1 r(a) = J ln"' 1 (�) du . (17. 38) -
0
441
17.3 The Eulerian Integrals
It was shown in Example 3 of Sect. 16. 3 that the sequence of functions = n(1 u 1 f n ) increases monotonically and converges to ln ( � ) on the interval 010 0. 0 e. The Complement Formula For 0 < a < 1 the values a and 1 - a of the argument of the gamma function are mutually complementary, so that the equality 7f (0 < a < 1) (17. 4 0) T(a) · r(1 - a) = -Slll. 7fa is called the complement formula for the gamma function. Proof. Using the Euler-Gauss formula (17. 3 7) and simple identities, we find that r(a)r(1 - a) = nbm-+. oo (na a (a + 1 ) (n. . . -. . 1)!(a + n - 1 ) X x n 1 - a (1 - a)(2 -(na)- ·1)!. . . · (n - a) ) = nl-+1. moo (n a(1 + -f) · . .1. · ( l + n� 1 ) x x (1 - -f) (1 - � ) · . -� · (1 - nS )(n - a) = = -a1 nli-+moo (1 - � ) ( 1 - � ) 1· . . . (1 - � 1 2 ) (n ) fn (u)
-t oo.
=
·
=
2
2
·
2
442
17 Integrals Depending on a Parameter
Hence for 0 a 1 12. r(a)F(1 - a) = -a1 n=II00l -(17. 4 1) 1-a But the following expansion is classical: 00 ( 1 - �22 ) . sin 1ra = 1ra n=l (17.42) II (We shallasnota simple take theexample time toofprove thisofformula just now, sincewhenit willwe bestudyobtained the use the general theory Fourier series. 6 of Sect. 18. 2 . ) Comparing relationsSee (17Example . 4 1) and (17. 4 2), we obtain (17.40). D It follows in particular from (17.40) that (17.43) We observe that r G ) = J x - 112 e- x dx = 2 f e- u2 du , 0 0 and thus we again arrive at the Euler-Poisson integral <
<
Ti2
+ oo
+ oo
1 7. 3 . 3 Connection Between the Beta and Gamma Functions
Comparing formulas (17. 3 2) and (17. 3 6), one may suspect the following connection: r(a) . r({3) ( B a, {3) = (17.44) r(a + {3) between the beta and gamma functions. Let us prove this formula. Proof. We remark that for > 0 r(a) = Ya J Xa - J e -xy dx , y
+ oo
0
443
17.3 The Eulerian Integrals
and therefore the following equality also holds:
using which, taking account of (17. 3 3), we obtain +oo a l . T(a {3) B( a , {3) = J r (1( a y{3))a+y f3- dy = +
+
+
0
+oo 1 a ( J y J xa +f3- l e - ( l+y) x dx) dy � +oo +oo � J ( J y a - l x a+ f3 - l e -( l+ y ) x dy ) dx = +oo +oo l ( J xf3- e-x J (xy) a - l e -xY x dy) dx = +oo +oo J (xf3- l e -x J ua - l e - u du) dx = r (a ) r ({3) +oo 0
0
0
0
0
0
0
0
0
0
Allpoint. that But remains isis toexactly explainwhatthe was equality distinguished by oftheSect. exclama tion that done in Example 16 17.2. D 17.3.4 Examples
In conclusion us considerB and a small group of interconnected examples in which the speciallet functions r introduced here occur. Example 7r /2 (17. 4 5) J sina - l cpcosf3- l cpdcp = � B (� , �) . 0 Proof. To prove this, it suffices to make the change of variable sin2 cp = x in the integral. D Usingfunction. formula In(17.particular, 44), we cantaking expressaccount the integral in terms of the gamma of (17. 4(17.3), 4we5) obtain 7r /2 7r /2 a (17. 4 6) J sin - l cpdcp = j cosa - l cpdcp .fii2 rr( a( �+l) ) . 2 0 0 1.
=
17 Integrals Depending on a Parameter
444
A one-dimensional ball of radius r is simply an open interval and itsExample ( one-dimensional ) volume V1 (r) is the length (2r) of that interval. Thus V1 (r) = 2r . If we assume that ther is((expressed n - ! )-dimensional ) volume of the ( n - ! ) dimensional ball of radius by the formula Vn _ 1 (r) = Cn - 1 T n - t , then, integrating over sections (see Example 3 of Sect. 11. 4 ), we obtain 2.
Vn (r) =
r
J
-r
n --1 Cn - 1 (r 2 X 2 ) -2 -
(
dx =
7f /2
Cn - 1
J
- 7f / 2
COSn
) cpdcp
7f /2 Cn = 2Cn- 1 COSn
J cpdcp . By relations (17. 46) we can rewrite this last equality as 0
r(�) C , Cn - Vir r ( � ) n- 1
so that
r(3) r ( n+ 1 ) r( !!: ) en = ( Vir) n - 1 r ( ) . r ( 2 ) . . . . . r ( ,_2 ) · c 1 � �
or, more briefly, But C1 = 2, and
�
- r(� Cn = n 2 1 r ( n )2 C 1 . t) r (� ) = �r(� ) = � fo, n cn = -;--: r ( �..) .,.-,-7T
so that
7T 2
Consequently,
or, what is the same,
n Vn (r) = r r n . � (�)
(17.47) Example It is clear from geometric considerations that d Vn ( r) Sn - 1 ( r) dr, where Sn - 1 ( r) is the ( n 1 ) -dimensional surface area of the sphere the n-dimensional ball of radius r in JR.n . Thusbounding Sn _ 1 (r) = ddrn (r) , and, taking account of (17. 4 7), we obtain 3.
7T 2
-
17.3
The Eulerian Integrals
445
17.3.5 Problems and Exercises 1.
Show that a) B(1 /2 , 1 /2)
= 1r; b) B(o:, 1 - o:) = I dx; 1 c ) �� ( a: , ;3) = I x" - 1 (1 - x l 1 ln x dx; 00
0
d)
e) f)
� ( a +bx q ) r
I Ioo Ioo d
+ oo 0
+
0
+
0
0
x" _ , 1+x
-
=
a-r q
( ) B ( z±l r - p + 1 ) · £ b
E.±.!_ q
q
'
'
q
"- ____ * '. �l +x n _Q_£_ -
n sin 21r
x _ . 1 + x 3 - 3 .,/:3 '
h) Ioo x " -l '+Inx +
0
n
dx =
x
do: n ( sm. 1ro: )
--'.!..::_
"-
-
(0
<
0:
<
1)·'
i ) the length of the curve defined in polar coordinates by the equation rn nip, where n E N and a > 0, is aB ( � , � ) . 2
an cos
Show that a) F ( 1 ) = F( 2) ; b) the derivative F' of F is zero at some point Xo E ] 1, 2[; c ) the function r' is monotonically increasing on the interval ] O, +oo[; d ) the function r is monotonically decreasing on ]0, xo] and monotonically in creasing on [xo , +oo[; 2.
1
1
e ) the integral I ( In � ) x - In In � du equals zero if x = xo ; 0
F(o:) ± as o: --+ +O; + n g) n--+ lim Ioo e - dx = 1. oo f)
"' 0
3.
x
Euler's formula E := ff r ( �)
k= 1 a) Show that E 2 = X( r ( � ) r ( n ;: k ) . n1 b) Verify that E 2 = sin n" sin 2 :;;n · ...- · s in n - 1 ""n · ( )
17 Integrals Depending on a Parameter -1 2 c) Starting from the identity z;_-/ = nTI ( z - e i �" , let z tend to 1 to obtain ) k=1 n- 1 · 2 k 1r n = II ( 1 - e' " ) , k=1 and from this relation derive the relation n- 1 . n = 2 n - 1 II s1n k7r . n k=1 446
d) Using this last equality, obtain Euler's formula. 4.
Legendre 's formula r(a)r ( a: + � ) = 2 2'{Ji 1 r(2o:) . 1 /2 2 a- 1 dx. a) Show that B(o:, o:) = 2 I (t - ( � - x ) ) 0
b) By a change of variable in this last integral, prove that B(o:, o:) 2;_ 2 1 B ( � , a: ) c) Now obtain Legendre's formula. ·
5. Retaining the notation of Problem 5 of Sect. 17. 1 , show a route by which the second, more delicate part of the problem can be carried out using the Euler inte grals. a) Observe that k = k for k = � and
?r / 2
E=E=
I J1 � sin2 cp dcp , -
0
b) After a suitable change of variable these integrals can be brought into a form from which it follows that for k = 1 I .;2 1 1 K = rn B(114, 112) and 2E - K = rn B(3I4, 11 2 ) . 2v 2 2v 2 c) It now results that for k = 1 I .J2
EK + EK - KK = 7rl2 .
1 6. Raabe 's 1 integral I ln r(x) dx. 0
Show that 1 1 a) I ln r(x) dx = I ln r( 1 - x) dx. 0
0
1 • /2 b) I ln r(x) dx = � ln 1r - � I ln sin x dx. 0
0
1 J.L. Raabe (1801-1859) - Swiss mathematician and physicist.
n/2
17.3 The Eulerian Integrals
n/2
447
c) I0 ln sin x dx = I0 ln sin 2x dx - � ln 2. n/2
d) I0 ln sin x dx = - � ln 2. e) I0 ln r ( x ) dx = ln .;21r.
1
7.
Using the equality
_!,_ X8
_
1_ r(s)
_
I y s - 1 e - x y dy + oo 0
and justifying the reversal in the order of the corresponding integrations, verify that + oo n a "' - 1 a) I0 � x a dx = 2 r ( a ) co s ¥ (0 < a < 1) ·' s i n bx dx = b) I0 --;;r r + oo
n b/3 - 1 2 r ( {3 ) s i n ii,j
( 0 < {3 < 2).
c) Now obtain once again the value of the Dirichlet integral I0 s i: x dx and the + oo
+ oo
+ oo
value of the Fresnel integrals I0 cos x 2 dx and I0 sin x 2 dx. ·
8.
Show that for a > 1 + oo
I 0
x "'- 1 dx = r(a) . ((a) ' ex - 1
where ((a) = L n1"' is the Riemann zeta function. n= l 00
9.
Gauss ' formula. In Example 6 of Sect. 16.3 we exhibited the function � a( a + 1) · · · (a + n - 1){3({3 + 1) · · · ({3 + n - 1) x n F(a, f3, 'Y, x) ·. - 1 + � ' n!'Y('Y + 1 ) · · · ('Y + n - 1 )
which was introduced by Gauss and is the sum of this hypergeometric series. It turns out that the following formula of Gauss holds: . F(a, {3, 'Y, 1 ) = r('Y) r('Y. - a - {3) . r('Y - a) F('Y - {3) a) By developing the function (1 - t x) - 13 in a series, show that for a > 0, 'Y - a > 0, and 0 < x < 1 the integral P(x) =
1
I t "' 1 ( 1 - t ) 0
-
-y - a -
1 ( 1 - t x) - !3 dt
448
17 Integrals Depending on a Parameter
can be represented as P(x) =
00
Pn · X n , L n=O
'y a w h ere pn - !3(13+ 1 ) ..n!·(/H n - 1 ) · r ( a+n)·r r ( 'Y+n( ) - ) b) Show that r("y - a: ) . a: ( a: + 1) ( a: + n - 1) ,8 ( ,8 + 1) (,B + n - 1) Pn = r(o:) r("y) �----�--��--���--�--�------� n !"Yh + 1) ("y + n - 1) c) Now prove that for a: > 0, "Y - a: > 0, and 0 < x < 1 r(o:) . r("y - a: ) P(x) = · F(o:, ,B, "'f, X) . T("y) d) Under the additional condition "Y - a: - ,B > 0 justify the possibility of passing to the limit as x ---+ 1 - 0 on both sides of the last equality, and show that r(o:) . r("y - a: - ,B) = r(o:) . r("y - a:) F(o:, ,B, "'f, 1) , r("y - ,B) r("y) from which Gauss' formula follows. 10. Stirling 's2 formula. Show that 2 m for lxl < 1 '· x_ a) ln 11 -+ x = 2x "' L..J _ 2m+l _
0
0
0 0 0
0 0
0 0 •
00
x m=O b) ( n + �) In ( 1 + � ) = 1 + L 2n : 1 ) 2 + t ( 2n : 1 ) 4 + L2n:1 ) 5 + . . · ; c) 1 < ( n + �) ln ( 1 + �) < 1 + 12n (�+1 ) for n E N ; ( 1 + n ) n + 1 /2 eiTn1 d) 1 <
_l
<
e
e 1 2(n + l )
e) a n = n < ;:-;;7 2 ) is a monotonically decreasing sequence; f) b n = a n e - 1 21 n is a monotonically increasing sequence; g) n ' = cn n+1 1 2 e- n+ � , where 0 < (}n < 1 , and c = nlim a n = nlim bn , --+ oc;. --+ CXJ h) the relation sin 1rx = 1rx ( 1 - S ) with x = 1/2 implies Wallis' formula ·
·
J:t
. (n !?2 2n . 1r,:; ; ,fir = n-+ hm oo ( 2n ) '. y n i) Stirling 's formula holds: n ! = � ; e � ' 0 < (jn < 1 ; ••
(r
y'2;;; ( � r as X ---+ + oo. Show that r(x) f= ( �!�n � + I e- 1 e-t dt. This relation makes it possible 1 n=O
j) T(x + 1) 11.
rv
=
°
to define r(z) for complex z E
2 J . Stirling (1692-1 770) - Scottish mathematician.
17.4 Convolution and Generalized Functions
449
17.4 Convolution of Functions and Elementary Facts about Generalized Functions 17.4. 1 Convolution in Physical Problems (Introductory Considerations)
AcarryvariOEt ety oftheirdevices and systems in theto living and nonliving natural world functions responding a stimulus f with an appropriate signal f. In other words, each such device or system is an operator A that transforms the incoming signal f into the outgoing signal ! = A f . Naturally, each such operator has its own domain ofthem perceivable signals (domain of def inition) and i t s own form of response to (range of values). A convenient mathematical a large class of actual processes and machines is a linear operatormodel A thatforpreserves translations. Definition 1 . Let A be a linear operator acting on a vector space of real- or complex-valued functionsacting definedon theon JR.sameWespace denoteaccording by Tt0 theto the shift operator or translation operator rule (Tt0 f)(t) := f(t - to) . The operator A is translation-invariant (or preserves translations) if for every function f in the domain of definition of the operator A. If t is time, the properties relation Aof theTt0device can be interpreted as the = Tt0A areA time-invari assumption that the a nt: the reaction ofto the devcenothing to the signals f ( t) and f ( t - t0) differ only in a shift by the amount inFortime, more. every device --!f ofthethefollowi ng totwoanfundamental problems arise: first, to predict the �action device arbitrary input f; second, knowing the Atoutput f, to determine, if possible, the input signal f. this point shall solve the afirstnt linear of theseoperator two problems in application to awetranslation-invari A. � is aheuristically simple, but very important fact that in order to describe the response f of such a device A to any input signal f, it suffices to know the response E of A to a pulse o. Definition 2. The response E(t) of the device A to a unit pulse o is called the of theengineering). device (in optics) or the transient pulse function of thesystem devicefunction (in electrical As a rule,going we shall usedetailthejust brieferyet,term "system function. " can be imi Without into we shall say that a pulse tated, for example, by the function 00(t) shown in Fig. 17.1, and this imitation is assumed to become closer as the duration a of the "pulse" gets shorter, o
o
450
17 Integrals Depending on a Parameter
�L C¥ 1
I I I I I I I I I I
0
t
a
Fig. 1 7 . 1 .
preserving theusingrelation � 1. Instead of step functions, one may imi tate a pulse smooth functions ( Fig. 17. 2 ) while preserving the natural conditions: 1 1Q ( t) dt 1 , I fet (t) dt 1 as 0 , whereTheU(O) is an ofarbitrary neighborhood of the point t 0. response the device A to an ideal unit pulse ( denoted, following Dirac, byofthetheletter 8) should be regarded as a function E(t) to which the response device to an continuity input approximating 8 tends as the imitation improves. Naturally a Acertain of the operator A is assumed ( not made preciseunderas yeta continuous ) , that is, continuity of the change in the response J of the For device change in {..:1the (t)input f. example, if we take a sequence n } of step functions ..:1 n (t) 81; n (t) ( Fig. 17.1), then, setting A..:1 n En , we obtain A8 E li--+oo m En n lim A..:1 n . n --+oo Let usfunction now consider the input signal f in Fig. 17. 3 andf astheh piecewise constant lh (t) 'L f( Ti ) 8h (t - Ti ) h. Since lh 0, one i must assume that lh Alh Af f as h 0 . a ·
=
--+
=
lR
U ( O)
a --+
=
:=
=:
=
=
Fig. 1 7 . 2 .
--+
=
--+
--+
Fig. 1 7 . 3 .
:= =
=
--+
17.4 Convolution and Generalized Functions
451
But if the operator A is linear and preserves translates, then lh ( t) = L ! h )Eh (t - Ti )h i where Eh = Aoh . Thus, as h --+ 0 we finally obtain J( t) = I f (r ) E ( t - r ) dr . (17. 48) Formula (17.48) solves the first of the two problems indicated above. It represents the response j( t) of the device A in the form of a special integral depending on the parameter t. This integral is completely determined by the input signal f(t) and the system function E ( t) of the device A. From the mathematical point of view the device A and the integral (17. 4 8) are simply identical. that the problem of determining the input signal fromWethenoteoutputincidentally J now reduces to solving the integral equation (17. 4 8) for f. Definition 3. The convolution of th e functions u : lR --+ C and v : lR --+ C is the function u * v lR --+ C defined by the relation (17. 4 9) (u * v)(x) := I u(y)v(x - y) dy , provided this improper integral exists for all x E JR. Thus formula (17.to48)anasserts that theby response of a flinear device A that preserves translates input given the function is the convolution f * E of the function f and the system function E of the device A. '
R
:
R
17.4.2 General Properties of Convolution
Now pointletof usview.consider the basic properties of convolution from a mathematical a. Sufficient Conditions for Existence We first recall certain definitions andLetnotation. a real- or complex-valued function defined on an open set The G cffunction R: G --+ Cf beis locally on G if every point x E G has a neigh borhood U(x) G in whichintegrable the function fl u ( x ) is integrable. In particular, ifequiGv=alentJR, the condition of local integrability of theclosed function f is obviously to the relation ! I [a,b] E R[a, b] for every interval [a, b]. support of the function f ( denoted supp f) is the closure in G of the set The {x E G l f(x) =f- 0}. C
17 Integrals Depending on a Parameter
452
function f is of compact support (in G) if its support is a compact set. The set of functions f : G --+ C having continuous derivatives in G up to order m (0 :::; m :::; oo) inclusive, is usually denoted C (m) (G) andm) the subset of it consisting of functions of compact support is denoted Cr� (G) . In the case when G = � ' minstead ofmC (m ) (�) and c6m) (�) it is customary to use the Weabbreviation C ( ) and c6 l respectively. most frequently encounteredwithout cases ofdifficulty. convolution of functions,nowinexhibit which ithets existence can be established A
Each of the conditions listed below is sufficient for the ex istence of the convolution u * v of locally integrable functions u : � --+ C and v : � -+ C. 1) The functions l u l 2 and l v l 2 are integrable on R 2) One of the functions l u i , l v l is integrable on � and the other is bounded on R 3) One of the functions u and v is of compact support. Proof. 1) By the Cauchy-Bunyakovskii inequality Proposition 1 .
( j l u (y ) v( x - y)l dyr :::; j l u l 2 (y ) dy j l v l 2 (x - y) dy , IR
IR
IR
from which it follows that the integral (17. 49) exists, since + oo
+ oo
J l v l 2 (x - y ) dy = J l v l 2 (y ) dy .
- oo
- oo
2) If, for example, l u i is integrable on � and l v l :::; M on � ' then J l u (y ) v (x - y )l dy :::; M J l u l (y ) dy < +oo . 3) Suppose supp u [a, b] Then obviously IR
IR
C
C
R
b
J u (y)v(x - y) dy = J u(y) v (x - y) dy . u and v are locally integrable, this last integral exists for every value of xSince E case whenby thethe change functionofofvariable compact support is v reduces to the one justTheconsidered x - y = z. D IR
R
a
17.4 Convolution and Generalized Functions
453
b. Symmetry
If the convolution u * v exists, then the convolution v * u also exists, and the following equality holds:
Proposition 2.
(17. 50)
Making the change of variable x - y z in (17. 49), we obtain u * v(x) := J u(y)v(x - y) dy = J v(z)u(x - z) dz v * u(x) . D <X) <X) invariance c. Translation Invariance Suppose, as above, that Tx 0 is the shift operator, that is, (Tx0 )f(x) = f(x - xo). =
Proof.
+=
+=
-
=:
-
If the convolution u * v of the functions u and v exists, then the following equalities hold: (17. 5 1) Tx 0 ( U * V ) Tx 0 U * V U * Tx 0 V . Proof. If we recall the physical meaning of formula (17. 48 ) , the first of these
Proposition 3.
=
=
equalities becomes obvious,Nevertheless, and the second can then be obtained fromof the symmetry of convolution. let us give a formal verification the first equality: (Tx 0 )(u * v)(x) := (u * v)(x - xo) += j u(y)v(x - x o - y) dy = :=
=
-
CXl
=
+=
j u(y - xo)v(x - y) dy
-
+=
CXl
J (Tx0 u)(y)v(x - y) dy
-=
=:
=
( (Tx 0 U) * v) (x) .
D
The convolutionof ofit functions is an integral depending on a parameter, and differentiation can be carried out in accordance with thehypotheses general ruleshold.for differentiating such integrals, provided of course suitable The conditions under which the convolution (17.49) of the functions u and v is continuously differentiable are demonstrably satisfied if, for example, u issupport. continuous and v is a smooth function and one of the two is of compact Proof. Indeed, if we confine the variation of the parameter to any finite in terval, then under theseclosed hypotheses theindependent entire integral (17. 4 9) reduces to the integral over a finite interval of x. Such an integral can beruledifferentiated respect to a parameter in accordance with the classical of Leibniz. with D d. Differentiation o f a Convolution
454
17 Integrals Depending on a Parameter
In general the following proposition holds: Proposition 4. If u is a locally integrable function and v is a c6 m l function of compact support (0 ::::; m ::::; +oo), then ( u * v ) E C ( m ) , and? (17. 52) Proof. When u is a continuous funcction, the proposition follows immediately from what was just proved above. In its general form it can be obtained if we also keep in mind the observation made in Problem 6 of Sect. 17.1. D Remark In view of the commutativity of convolution (formula (17. 50)) Proposition of course remains valid if u and v are interchanged, preserving the Formula left-hand4(17.side of Eq. (17.52). 5 2)it commutes shows that with convolution commutes with(17.the5 1)). differentiation operator, just as translation (formula But while (in17.51) is symmetric in u and v, one cannot in general interchange u and v fail to have the corresponding the right-hand sidethatof the(17.convolution 5 2), since u may derivative. The fact u * v, as one can see by (17. 5 2), may turn out to 4bearea differentiable might suggest that the hypotheses ofstillProposition sufficient, butfunction, not necessary for differentiability of the convolution. Example Let f be a locally integrable function and ba the "step" function shown in Fig. 17 .1. Then 1.
1.
+=
X
(17. 53) J f ( y) ba (x - y) dy = �x-Ja f(y) dy , and consequently point of (smoothing) continuity ofproperty f the convolution f * ba is differentiable, dueatto every the averaging of the integral. The4 conditions for completely differentiability of theforconvolution stated incasesPropo sition are, however, sufficient practically all the one encounters in which formula (17. 5 2) is applied. For that reason we shall not attempt to refine them any further, preferring to illustrate some beautiful new possibilities just discovered.that open up as a result of the smoothing action of convolution (J * ba ) (x) =
-=
17.4.3 Approximate Identities and the Weierstrass Approximation Theorem
We that the integral in (17. 5 3) gives the average value of the func tionremark f on the interval [x x] , and therefore, if f is continuous at x, the ·-
a,
3 Here D is differentiation, and, as usual D k v = v ( k ) .
17.4 Convolution and Generalized Functions
455
relation ( ! * 8a ) (xconsiderations ) f (x) obviously holds17.4.1 as that0.gave In accordance with the introductory of Subsect. a picture of 8-function, we would like to write this last relation as the limiting equalitythe ( ! * 8) (x ) f(x) , if f is continuous at x . (17.54) This equalityelement showswith thatrespect the 8-function can be Equality interpreted(17.54) as thecanidenbe tity (neutral) to convolution. regarded astomaking perfect sense if it same is shown that every family of fami functions converging the 8-function has the property as the special ly 8 a of (17.53). Let us now pass to precise statements and introduce the following useful definition. 4. The family { Lla ; E A} of functions Lla lR lR depending onDefinition the parameter E A forms an approximate identity over a base in A if the following three conditions hold: a) all the functions in the family are nonnegative (Lla 2: 0); b) for every function Lla in the family, IRI Lla (x) dx 1; c) for every neighborhood U of 0 E JR, liBm UI Lla(x) dx 1. Takivalent ng account the firstlimtwoI conditions, we see that this last condition is equi to the ofrelation Lla(x) dx 0. B IR\U The original family of "step" functions 8a considered in Example 1 of Subsect. 17.4.1 is an approximate identity as 0. We shall now give other examples of approximate identities. Example 2. Let lR lR be an arbitrary nonnegative function of compact support that is integrable over lR and satisfies IRI c.p(x) dx 1. For > 0 we ) construct antheapproximate functions Lla(x) obviously identity as� e.p ( � . +0The(seefamiFig.ly of17.2).these functions is Example 3. Consider the sequence of functions for l xl :S 1 , for l x l > 1 . To that establish that c)thisof fami ly is an4 holds approximate identity weandneedb). only verify condition Definition in addition to a) But for every E]O, 1] we have ----+
a ----+
=
:
a
a
=
----+
B
=
=
a ----+
c
:
----+
:=
=
a ----+
a
17 Integrals Depending on a Parameter
456
0 :S J (1 - x2 t dx :S J (1 - c-2 t dx 1
1
At the same time,
=
1- . /(0 1 - x2 ) n dx > /(1 - x t dx n +1 0 Therefore condition c) holds. Example 4- Let Lln (x) cos2n (x) / -''P1rj 2 cos2n (x) dx for l x l :S 7r/2 , for l x l > 7r /2 . 0 firstAsof ialln Example that 3, it remains only to verify condition c) here. We remark 7r/ 2 r ( ) > __L r( ) J0 cos2n xdx -B21 (n + -21 , -)21 -21 r (nF(+n) ) __L n 2n On the other hand, for E]O, / 2 [ 7r / 2 7r /2 J cos 2n xdx :S J cos2n c- dx � (cos c- ) 2n . Combining the two inequalities just obtained, we conclude that for every E]O, / 2] , 7r / 2 J Lln (x) dx -+ 0 as n -+ from which it follows that condition c) of Definition 4 holds. Definition 5 . The function f : G -+ 0 there exists p > 0 such that l f(x) - f(y) l for every x E E and every y E G belonging to the p-neighborhood U[';(x) of x in G. if E G weon simply back theof definition. definition of a function thatInisparticular, uniformly continuous its entiregetdomain 1
1
=
{
=
=
1
2ri:
=
E
·
1
1
·
1r
<
E
7r
oo ,
C
E
< E
=
17.4 Convolution and Generalized Functions
457
We now prove a fundamental proposition. Proposition 5 . Let f : IR ---+
�
C
w; .
(f * Ll "' )(x) f(x) on E as a ---+ Thus it is asserted that the family of functions f * Ll"' convergesif Euniformly toof only f on a set E on which it is uniformly continuous. In particular, consists one point, the condition of uniform continuity of f on E reduces to that f be continuous at x, and we find that U * Ll<>)(x) ---+ f(x) astheacondition ---+ Previously this fact served as our motivation for writing relation w.
(17.Let54).us now prove Proposition 5. Supposewithl f(x)Definition l M on R Given a number E > 0, we choose > 0 iProof. nU(O).accordance 5 and denote the p-neighborhood of 0 in IR by ng account the symmetry of convolution, twoTaki estimates, whichofhold simultaneously for all x E weE: obtain the following p
:::;;
= I J f(x - y)Ll"' (y) dy - f(x) l I J (f(x - y) - f(x) )Ll"' (y) dy l :::;; J l f(x - y) - f(x) IL1"' (y) dy J l f(x - y) - f(x) I Ll"' (y) dy J Ll"' (y) dy J Ll<> (Y) dy J Ll"' (y) dy :::;;
j (f * Ll "' )(x) - f(x) j
=
=
IR
:::;;
IR
<
+
U(O)
IR\U(O)
<
As
a
---+
E + 2M
+ 2M
E
U(O)
w,
IR\U(O)
0
IR\U{O)
this last integral tends to zero, so that the inequality l (f * Ll "' (x) - f(x) l
<
2E
holds for all5.x ED E from some point a6 on. This completes the proof of Proposition Corollary 1 . Every continuous function of compact support on IR can be uniformly approximated by infinitely differentiable functions. Proof. Let verify that cg= ) is everywhere dense in Co in this sense.
458
17 Integrals Depending on a Parameter
We let, for example,
{
exp ( - 1 _lx2 ) for lxl 1 , for lxl 1 , 0 where k is chosen so that J r.p(x ) dx = 1. Thethefunction r.p is of compact support and infinitely differentiable. In that case, fami l y of infinitely differentiable functions Ll0+0. If�fr.p E( �C) , asitobserved inthatExample 2, is an approximate identity as 0, is clear f * Ll0 E C0. Moreover, by Proposition 4 we have f * Ll0 E C0. Finally, it follows from Proposition 5 that f * Ll0 f on lR as +0. D Remark If the function f E Co belongs to C�m ) , then for every value n E {0, 1, . . . , m} we can guarantee that ( ! * Ll a ) ( n ) f ( n ) on lR as +0. Proof. Indeed, in this case ( ! * Ll a ) ( n ) f ( n ) * Ll0 (see Proposition 4 and Remark 1). All that now remains is to cite Corollary 1. D Corollary 2. (The Weierstrass approximation theorem). Every continuous r.p( x )
=
<
k·
�
IR
:::::t
2.
=
a -+
a -+
:::::t
=
a -+
function on a closed interval can be uniformly approximated on that interval by an algebraic polynomial. Proof. Since polynomials map to polynomials under a linear change of vari
able preserved, while the continuity and uniformity of the2 onapproximation of functions are it suffices to verify Corollary any convenient interval we shall assume 0 a b 1 and let [a ,=b] min{a, 1For- b}.thatWereason continue the given function f E C[a, b] to a func tion F that is continuous on lR by setting F(x) 0 for x E IR\]0, 1 [ and, for example, letting F be a linear function going from 0 to f(a) and from f(b) to 0Ifonwethenow intervals , a] and [b, 1] identity respectively. [0approximate take the Ll of Example 3, we can conclude from Propositionconsisting 5 that ofF *theLlnfunctions f = F ln on [a , b] as n oo. But for x E [a , b] [0 , 1] we have 1 00 F * Ll n (x) : = J F ( y )Ll n (x - y) d y = J F(y ) Ll n (x - y) d y p
C
R
=
-+
1
-
<
<
<
:::::t
C
= j F(y)pn 0
=
0
oo
·
(
1-
(x - y) 2 r
[a , b]
1
dy = j F(y) ( L=O ak (y)xk ) dy 0
2n
k
2n
1
=
)
= L (j F ( y )a k (y) dy x k . k =O o
is a polynomial P2n (x ) of degree 2n and we have shownThisthatlastP2expression n f on [a, b] as n oo. D :::::t
-+
17.4 Convolution and Generalized Functions
459
Remark By a slight extension of this reasoning one can show that Weier strass compacttheorem subset remains of R valid if the interval [a, b] is replaced by an arbitrary Remark 4. It is also not difficult to verify that for every open set G in lR and every function f E c< m ) ( G ) there exists a sequence { Pk } of polynomials such n ) ( that P� f n) on every compact set K G for each n E {0, 1, . . . , m} as k ---+If in addition the set G is bounded and f E c<m l (G) , then one can even get P�n) f ( n) on G as k ---+ oo. Remark Just as the approximate identity of Example 3 was used in the proof of211'Corollary 2, one canonuselR can the besequence fromapproximated Example 4 tobyprove that every periodic function uniformly trigono metric polynomials of the form n Tn (x) = L a k cos kx + bk sin kx . k=O Wesupport have used onlyHowever, approximate identities madein mind up of that functions of com pact above. it should be kept approximate identities of functions not oftwocompact in many cases. We shallthatgiveareonly examples.support play an important role Example The family of functions Ll y (x) = � x2 �Y2 is an approximate identity on lR as y ---+ +0, since Lly > 0 for y > 0, f00 Lly (x) dx = ..!:_11' arctan (�)y l +oo= oo = 1 , x - oo and for every > 0 we have f Lly (x) dx = -11'2 arctan -y ---+ 1 , whenIf fy ---+lR +0.---+ lR is a bounded continuous function then the function , +oo (17. 5 5) u(x, y) = :;;:1 f (x -f(�)y 02 + y 2 d� , - oo whiAsch isonethecanconvolution f * Lly , is defined for all x E lR and y > 0. easily verify the Weierstrass M-test, the integral (17.55), which is called the Poisson using integral for the half-plane, is a bounded infinitely 3.
C
�
00 .
�
5.
·
5.
p
p
-p
:
p
460
17 Integrals Depending on a Parameter
2 differentiable in thesign, half-plane tiating it underfunction the integral we verifyIR� that{(x,fory)y E> IR0 1 y > 0}. Differen =
that is, u is a harmonic function. By Proposition 5 one can also guarantee that u(x, y) converges to f(x) asbounded y -+ 0. Thus, the integral (17. 5 5) solves the problem of constructing a function values that isf harmonic scribed boundary on oiR� . in the half-plane IR� and assumes pre Example The family of functions Llt 2 � e - is an approximate identity on IR as t -+ +0. Indeed, we certainly have Llt > 0 and +ooJ Llt(x) 1, since -+ooJoo e - v dv ,j1i ( the Euler-Poisson integral) . Finally, for every > 0 we have e - v dv -+ 1 , as t -+ +0 . 6.
2 �.
=
=
2
p
=
2
If f is a continuous and, for example, bounded function on IR, then the function +oo 1 - d� , (17. 56) u ( x , t) 2v nt J f(�) e which is the convolution f * Llt , is obviously infinitely differentiable for t > 0. By differentiating under the integral sign for t > 0, we find that =
c;
(x - 0 2
-
oc
4t
that is,condition the function u satisifes the one-dimensional heat equation with the initial u(x, 0) as the5. which should followsbefrominterpreted Proposition limiting relation u (x , t) -+f(xf(x)). This as t -+last+0,equality =
17.4.4 *Elementary Concepts Involving Distributions of Generalized Functions In Subsect. 17. 4 .1 of this section wea.us Definition derived the formula (17.48)ofona linear the heuristic level. ThisA toequation enabled to determine the response transformation an input signal f given that we know the system function E of the device A. In determining
17.4 Convolution and Generalized Functions
461
the system function ofaction a device wethe made essentialthatusedescribes of a certain intuitive idea of a unit pulse and £�" function it. It issenseclear,of however, that the £�" function is really not a function i n the classical the term, since it must have the following properties, which contradict the classical point of view: J(x) 0 on JR; J(x) = 0 for x =f. 0, J J(x) dx = 1. Thethe concepts connectedof with linearacquire operators, convolution, the £�"-function, and system function a device a precise mathematical descrip tion in the so-called theory of generalized functions or the theory of distribu tions.everWemore are now explain theofbasic principles and the elementary, but widelygoingusedtotechniques this theory. Example Consider a point mass m that can move along the axis and is attached to oneelastic end ofconstant an elasticof the springspring. whoseSuppose other endthatis fixed at the origin; letforcek bef(t)the a time-dependent begins to act on the point resting at the origin, moving it along the axis. By Newton ' s law, (17. 5 7) mx + kx = f ' where x( t)atistime the coordinate of the point (its displacement from its equilibrium position) t. Underf,these conditions thex(t)function x(t) is uniquely determined by the function and the solution of the differential (17. 5 7)with is obvi ously a linear function of the right-hand side f. Thusequation we are dealing the linear operator f � x inverse to the differential operator x � f (where k) that connects x(t) and f(t) by the relation Bx = f. Since the B = m/;xA+obviously operator commutes with translations over time, it follows from (17. 4 8) that i n order to find the response x(t) of this mechanical system to the function suffices tofundamental find its response J, that is, it suffices to knowf(t),theit so-called solutionto aEunit of thepulseequation (17. 58) mE + kE = J . RelationHowever (17. 5 8)Eq.would not raise anyclear. problems if J actually denoted as function. (17. 5 8) is not yet But being formally unclear i quiteexplain a differentthe thing fromofbeing actually false. In the present case one need only meaning (17. 5 8). to such anidentity explanation is already familiar to us:andwe can interpretof Jclassical asOnean route approximate imitating the delta-function consisting functions Lla(t); we interpret E as the limit to which the solution Ea(t) of the equation (17. 5 7') tends as the parameter changes suitably. �
7.
a
IR
462
17 Integrals Depending on a Parameter
seconda fundamental approach to this problem,ofonethethatideahasof asignificant is toAmake enlargement function. advantages, It proceeds from the remark that in general objects of observation are characterized bya their interaction with other ("test") objects. Thus we propose regarding function not onas other a set (test) of valuesobjects at different points,manner. but rather astryantoobject that can act in a certain Let us make this statement, which as of now is too general, more specific. Example Let f E C(JR, JR) . As our test functions, we choose functions in C0 (continuous functions of compact support on JR) . A function f generates the following functional, which acts on C : 8.
0
(17.59) approximate identities0consisting of functions of compact support, oneUsing can easily see that (!, cp) on C0 if and only if f(x) 0 on R Thus, each function f E C(JR, JR) generates via (17. 59) a linear functional At : C0 lR and, we emphasize, different functionals A and A correspond to different functions(17.h59)andestablishes h. an embedding (injective mapping) of Hence formula the setconsequently of functionsevery C ( JR, JR ) into the set .C ( C0 ; JR) of linear functionals on C0 , and function f E C (JR, JR) can be interpreted a certain functional At E .C(Co ; JR) . IfC (weJR, JRconsider the class of locally integrable functions on lR instead of the setmapping ) of continuous functions, we obtain by the same formula (17. 5 9) a of this set into the space .C(C0 ; JR) . Moreover ( (!, cp) 0 on Co ) {::} ( f(x) = 0 at all points of continuity of f on JR, that is, f(x) = 0 almost everywhere on lR) . Hence in this case we obtain an embedding of equivalence classes of functions i n to .C ( C0 ; lR ) if each equivalence class contains locally integrable functions that differ functions only on a fsetonoflRmeasure zero. equivalence Thus, the locally integrable (more precisely, classes of such functions) can be interpreted via (17.59) as linear functionals At E .C(Co ; lR) . The mapping f H At = (!, ·) provided by (17.59) of locally integrable functions intofunctions .C(C0 ; JR) is not a mapping onto all of .C(C0 ; JR) . Therefore, interpreting as elements of .C(interpreted C0 ; JR ) (that is, as func tionals) we obtain, besides the classical functions asnofunctionals ofin the form (17.59), also new functions (functionals) that have pre-image the classical functions. Example The functional E .C ( C0 ; JR) is defined by the relation (17. 60) (J, cp) := J(cp) := cp(O) ' which must hold for every function E C0 . (!, cp) : = J f(x)cp(x) dx . IR
=
--t
h
=
h
as
=
9.
c5
'P
17.4 Convolution and Generalized Functions
463
can verify (see Problemo in7)thethatformno locally integrable function f on IR canWe represent the functional (17.59). Thus we have embedded the set of classical locallyfunctionals integrable arefunctions into a larger set of linear functionals. These linear called generalized functions or distributions (a precise definition is given below). The widely used term "distribution" has its origin in physics. Example Suppose a unit mass (or unit charge) is distributed on If this distribution sufficientlydensity regular,p(x)in the it has, offortheexample, continuous oris integrable on IR,sensethe that interaction mass Ma with other objects described by functions 'Po E Cboo) can be defined as a functional M( tp) J p(x)tp(x) dx . Ifat thea single distribution is singular, for example, the whole mass M is concentrated point, then by "smearing" the mass and intepreting thedistribu limiting point situation using an approximate identity made up of regular tions, we find the interaction mass with the other objects mentioned abovethatshould be expressedofbythea formula M( tp ) tp (O) , which shows(17.that60) such o-function on R.a mass distribution on IR should be identified with the These preliminary considerations give some sense to the following general definition. Definition 6. Let P be a vector space of functions, which will be called the space of test functions from now on, on which there is defined a notion of convergence. TheP'space of generalized functions or distributions on P is the vector space of continuous (realor complex-valued) linear functionals on P. Here it is assumed that each element f E P generates a certain functional A (!, ·) E P' and that the mapping f A is a continuous embedding ofconvergence P into P'ofiffunctionals, the convergence that is,in P' is introduced as weak ("pointwise") 1 0.
JR.
=
lR
Jl;f
=
f =
f--t
f
Let us make this definition more precise in the particular case when P is the vectorin space Cboo) ( G, C) of infinitely differentiable functions of compact support G, where G is an arbitrary open subset of IR (possibly IR itself). Definition 7 . (The spaces V and V') . We introduce convergence in Cb00 ( G, C) as follows: A sequence { 'Pn } of functions 'Pn E Cb00 \ G) con verges to 'P E C1�oo) ( G, C) if there exists a compact set K G that contains )
C
17 Integrals Depending on a Parameter
464
the supports of all the functions of the sequence { cpn } and cp�m ) cp ( m ) on K (and hence also on G) as forwayall mwith= 0,this1, 2,convergence . . .. The vector space obtained i n this is usually denoted V(G), and when G = JR. , simply V. We denote the space of generalized functions (distributions) corresponding to this space of basic (test) functions by V' (G) or V' respectively. In thisother sectionthanandthetheelements one following we(G)shall notintroduced. consider For any that generalized functions of V' just reason weof V'shall(G)usewithout the termsayingdistribution or generalized function to refer to elements so explicitly. asDefinition 8 . A distribution F E V' (G) is regular if it can be represented F(cp) = J f(x)cp(x) dx , cp E V(G) whereNonregular f is a locally integrable function in G. distributions will be called singular distributions or singular =t
n --+ oo
,
G
generalized functions.
In accordance with this definition the 8-function of Example 9 is a singular generalized function. The action of a generalized function (distribution) F on a test function cp, that is, the pairing of F and cp will be denoted, as before, by either of the equivalent expressions F(cp) or ( F, cp) . Before passing to the technicalformachinery connected withthatgeneralized functions, which was our motive defining them, we note the con cept of a generalized function, like the majority of mathematical concepts, had aofcertain periodof ofmathematicians. gestation, during which it developed implicitly in the workPhysicists, a number Dirac,1930smadeandactive use ofwiththe singular 8-functiongeneralized as early asfunctions the latewithout 1920sfollowing and early operated worrying about the absence of the necessary mathematical theory. Thelaididea ofmathematical a generalizedfoundations function wasof thestatedtheory explicitly by S. L . Sobolev4, who the of generalized functions intionsthewas mid-1930s. The current state of the machinery of the theory of 5 What has just been said distribu largely the work of L. Schwartz explainsto why, for example, the space V' of generalized functions is often referred as the Sobolev-Schwartz space of generalized functions. 4 S L . Sobolev (1908-1989) - one of the most prominent Soviet mathematicians. 5 L . Schwartz ( 1915-2002) - well-known French mathematician. He was awarded .
the Fields medal, a prize for young mathematicians, at the International Congress of Mathematicians in 1950 for the abovementioned work.
17.4 Convolution and Generalized Functions
465
We shall nowTheexplain certain elements of theofmachinery ofthisthemachin theory oferydistributions. development and extension the use of even today, mainlythein equations connectionofwith the requirements the theorycontinues of differential equations, mathematical physics,offunc tionalTo analysis, andnotation their applications. simplify the weproperties, shall consider below onlyfrom generalized functions inandV',proofs, although all of their as will be seen their definitions remain validof for distributions of any class V' (G), where G is an arbitrary open subset Operations withvalidistributions arefunctions, defined that by starting with thegeneralized integral relations that are d for classical is, for regular functions. b. Multiplication of a Distribution by a Function If f is a locally integrable function on lR and g E C ( oo ) , then for any function rp E Cboo ) , oo ) onequality the one hand grp E Cb and, on the other hand, we have the obvious J U · g)(x) rp (x) dx = j f(x)(g · rp)(x) dx or, in other notation (f · g, rp) = (f,g · rp) . Thibasiss relation, which nisg vali d for regular generalized functions, provides the for the followi definition of the distribution F · g obtained by multiplying the distribution F E V' by the function g E C ( 00 l : (17. 6 1) (F · g , rp) := (F, g · cp) . right-handFsideonof any Eq. function (17. 6 1) is defined, and thus defines the value ofitselftheTheisfunctional E V, that is, the functional F · g · g rp defined. Example Let us see how the distribution · g acts, where g E C ( oo ) . In accordance with the definition (17. 6 1) and the definition of we obtain ( o . g, rp ) : = ( o, g . cp) : = ( g . rp )( o ) = g ( o ) . rp ( o ) . c. Differentiation of Generalized Functions If f E C ( l ) and rp E Cb oo ) , integration by parts yields the equality J f'(x) rp (x) dx = - J f(x) rp'(x) dx . (17. 6 2) Thisofequality is the point of departure for the following fundamental def inition differentiation of a generalized function F E V': (17. 6 3) ( F', cp) := - ( F, cp') . R
�
�
11.
0
�
�
0,
466
17 Integrals Depending on a Parameter
Example 1 2. If f E C ( l ) , the derivative of f in the classical sense equals its
derivative inwith the distribution sense ( provided, naturally, the classical function isfollows identified the regular generalized function corresponding totheit ) .right This from a comparison of relations (17. 6 2) and (17. 6 3), in which hand sides are equal if the distribution F is generated by the function f. Example Take the Heaviside6 function { 0 for 0 , H( ) 1 for 0 , sometimes called theH unit step. Regarding it as a generalized function, let us find the derivative ' of this function, which is discontinuous in the classical sense.From the definition of the regular generalized function H corresponding to the Heaviside function and relation (17. 6 3) we find + oo + oo H( )tp'( ) d (H', 'P) -(H, tp') I tp'( ) d tp(O) , I 0 - oo 00 00 since H' = 'P8. E c6 Thus ( H', 'P) ( 8, 'P), for every function 'P E c6 Hence Example Let us compute ( 8', tp) : - (8, tp1 ) -tp'(O) . ( 8', 'P) It is natural that in the theory of generalized functions, as in the the ory of classical functions, the higher-order derivatives are defined by setting p ( n+ l ) ( F ( nJ ) ' . Comparing the results of the last two examples, one can consequently write (H", tp) -tp'(O) . Example Let us show that ( 8 ( n l , tp) (-1) n 'P ( n ) (O). Proof. For 0 this is the definition of the 8-function. We seen init Example 14 thatassuming this equality for established 1. for We have now prove by induction, that itholds has been a fixed value E N. Using definition (17. 63), we find ( 8 ( n + l ) , 'P) (( 8 ( n ) )', 'P) ( 8 ( n ) , tp') n n nl nl 1 3.
x
x
=
x
:= -
:=
).
14.
x
x
<
�
x
= -
x
x
=
).
=
:=
=
:=
=
1 5.
n = n
6
:=
:=
_
=
=
n =
=
-(- l) ('P') ( ) (O) (-1) + 'P ( + l (O) . =
0
0 . Heaviside ( 185Q-1925) - British physicist and engineer, who developed on the symbolic level the important mathematical machinery known as the operational calculus.
17.4 Convolution and Generalized Functions
467
Example Suppose the function f lR C is continuously differentiable for x function 0 and forexistx >at0,0.andWe suppose thequantity one-sidedf(limits f(f(-0) and f(saltus +0) ofor the denote the +0) the -0), jump ofofthef infunction at 0, by If(O), and bythef'distribution and { !'} respectively the derivative the distribution sense and defined by the function equal to isthenotusualdefined, derivative of fisfornotx important 0 and x for> 0.theAtintegral x=0 this last function but that through which it1 defines thethat regular distribution { !'}. In Example we noted if f E C ( l ) , then f' = { ! '}. We shall show that in general this is not the case, but rather the following important formula holds: (17. 64) f' = { !'} + If(O) · 8 . Proof. Indeed, + oo (!', IP) = - (!, tp' ) = - J f(x) tp'(x) dx = -+oooo = - ( j + j ) (f(x) tp'(x)) dx = - ( U · tp (x) l �= - oo - oo + oo - J f'(x) tp (x) dx + (f · tp)(x) l � 00 - J f'(x) tp (x) dx) = - oo + oo = (f(+O) - f(-O)) tp (O) + j j'(x) tp (x)dx = - 00 = (If(O) · 8, tp) + ( { !'}, tp) If all xderivatives up> 0,to and ordertheym are of thecontinuous functionandf have lR C exist on the iatntervals 0 and x x = 0, then, repeating the reasoning used to derive (17. 64),one-sided we obtainlimits j ( m ) = {j( m )} + Ij(O) . 8 ( m - l ) + Ij' (0) . 8 ( m - 2 ) + . . . · · · +It< m - I l (o) · 8 . (17. 6 5) We now exhibit some properties of the operation of differentiation of gen eralized functions. Proposition 6. ) Every generalized function F E is infinitely differen tiable. b) The differentiation operation D : is linear. <
:
1 6.
--+
<
0
0
0
0
:
<
a
D' --+ D'
D'
--+
. D
17 Integrals Depending on a Parameter
468 c
)
holds:
If F E V' and g E c ( = l , then (F · g) E V', and the Leibniz formula ( F( k ) . g (m - k ) . (F . g) ( m ) =
� �)
d) The differentiation operation D V' V' is continuous. e) If the series I:00= fk (x) S(x) formed from locally integrable functions k l uniformly on each compact subset of IR, then it can fk IR C converges be differentiated termwise any number of times in the sense of generalized functions, and the series so obtained will converge in V'. Proof. a) (F ( m l , cp) - (F ( m - l ) , cp') ( - l) m (F, cp ( m ) ). =
--+
:
:
:=
b) Obvious. c) Let us verify the formula for :=
:=
--+
:=
m =
1: =
=
( (F · g)', cp) -(Fg, cp') - (F, g · cp') -(F, (g · cp)' - g ' · cp) (F', gcp) + (F, g' · cp) (F' · g, cp) + (F · g', cp) (F' · g + F · g', cp) . =
=
=
In the general case we can obtain the formula by induction. Fm F in V' as V (Fd) , Let cp) (F, cp) as oo. Then oo, that is, for every function cp E m
--+
--+
m --+ :=
m --+
--+
=:
-(Fm , cp') -(F, cp') (F', cp) . e) Under these conditions the sum S(x) of them series, being the :uniform limit of locally integrable functions Sm (x) kI:= l fk (x) on compact sets, is locally integrable. It remains to observe that for every function cp E V (that (F:,, cp)
=
is, of compact support and infinitely differentiable) we have the relation (Sm , cp) J Sm (x)cp(x) dx --+ J S(x)cp(x) dx (S, cp) . Weoo.nowDconclude on the basis of what was proved in d) that s:, S' as Wemostseeimportant that the operation of differentiation of generalizedwhile functions retainsa the properties of classical differentiation acquiring of remarkable new properties that opencase up abecause great deal of presence freedom ofofnumber operation, which di d not exist i n the classical of the nondifferentiable functions the instability (lack of continuity) of classical differentiation under there limitingandprocesses. =
m --+
IR
IR
=
--+
17.4 Convolution and Generalized Functions
469
d. Fundamental Solutions and Convolution We began this subsection with intuitive7 ideas of the unit pulse and themechanical system function of thenaturally device. Ingenerates Example we exhibited an elementary system that a linear operator preserving time shifts. Studying it, we arrived at Eq.We(17.58), which thethissystem functionbyEreturning of that operator mustto satisfy. shall conclude subsection once again these ques tiscription ons, butinnowthe with the goal of illustrating an adequate mathematical de language of generalized functions. We begin by making sense of Eq. (17.58). On its right-hand side is the gen eralized function 8, so that relation (17.58) should be interpreted as equality oferalized generalized functions. Sinceoperations we know theon operations of differentiating gen functions and linear distributions, it follows that the left-hand side of Eq. (17.58) is now also comprehensible, even if interpreted in theLetsense of generalized functions. us now attempt to solve Eq. (17.58). At times t 0 the system was in a state of rest. At t 0 the point received at >unit0 there pulse,arethereby acquiring aacting velocityon thev =system, v(O) such that mv 1. For no external forces and its law of mostion x x( t ) is subject to the usual differential equation <
=
=
=
(17.66) which are to be solved with the initial conditions x(O) = 0, ±(0) = v = 1/m. mx + kx = 0 ,
Such a solution is unique and can be written out immediately: 1 . [£ , t 0 . x( t ) = v. rr=: m km sm -t thatSince in the present case theH system is at rest for t 0, we can conclude , tE�, (17.67) E(t) v� km sin y{kt -;;;, whereLetHusisnowtheverify, Heaviside function (seeforExample 13). using the rules differentiating generalizedE tfunctions and the results of the examples studied above, that the function ( ) defined by Eq. (17.67) satisfies Eq. (17.58). To simplify the writing we shall verify that the function (x) H ( x ) sinwx (17.68) w satisfies (in the sense of distribution theory) the equation d22 + w2 ) = 8 . (17.69) ( dx �
<
=
e
=
e
4 70
1 7 Integrals Depending on a Parameter
Indeed, (d2 + w) e - d2 (H -sinwx ) + w2 (H -sinwx ) dx2 sin wwx w dx2 • = H -+ w + 2H coswx - wH (. x) smwx + wH(x) sinwx = 8,sinwx -w + 215 coswx . Further, for every function
_
1
11
1) ,
,
-
1
:
1 7.
m
0
--+
17.4
Convolution and Generalized Functions
471
17.4.5 Problems and Exercises
a) Verify that convolution is associative: u * ( v * w ) = ( u * v ) * w. b) Suppose, as always, that r(a) is the Euler gamma function and H(x) is the Heaviside function. We set 1.
Hf (x )
a- 1
:= H (x ) ; (a) e >. x , where a > 0 , and .X E C .
Show that Hf * Hf = Hf + f3 . c ) Verify that the function F = H(x) (':,':._�� 1 e >. x is the nth convolution power of f = H (x )e >. , that is, F = f * f * · · · * f. '--v--"
"'
n
The function Gu (x) = " � e- 2.;2" , a > 0, defines the probability density function for the Gaussian normal distribution. a) Draw the graph of Gu (x) for different values of the parameter a. b) Verify that the mathematical expectation (mean value) of a random variable ,with the probability distribution Gu is zero, that is, J xGu (x) dx = 0. .,2
2.
lR
c) Verify that the standard deviation of x (the square root of the variance of x)
is a , that is
( [ x2 Gu (x) dx)
1
1
2
= a.
d) It is proved in probability theory that the probability density of the sum of two independent random variables is the convolution of the densities of the individual variables. Verify that G a * G {3 = G a2 {32 • J + e) Show that the sum of n independent identically distributed random variables (for example, n independent measurements of the same object) , all distributed according to the normal law Gu , is distributed according to the law G u .,;n · From this it follows in particular that the expected order of errors for the average of n such measurements when taken as the value of the measured quantity, equals a / y'ri, where a is the probable error of an individual measurement.
We recall that the function A(x) = L an x n is called the generating function of n =O the sequence ao , a 1 , . . . . Suppose given two sequences {a k } and {bk } · If we assume that a k = b k = 0 for k < 0, then the convolution of the sequences {a k } and {b k } can be naturally defined as the sequence C k = � a m bk- m . Show that the generating function of the convolution of two sequences equals the product of the generating functions of these sequences. 00
3.
{
}
4. a) Verify that if the convolution u * v is defined and one of the functions u and v is periodic with period T, then u * v is also a function of period T.
472
17 Integrals Depending on a Parameter
b) Prove the Weierstrass theorem on approximation of a continuous periodic function by a trigonometric polynomial (see Remark 5) . c) Prove the strengthened versions of the Weierstrass approximation theorem given in Example 4.
5. a) Suppose the interior of the compact set K C lR contains the closure E of the set E in Proposition 5. Show that in that case I f(y)L1 k (x - y) dy � f ( x ) on E. K
b) From the expansion ( 1 - z) - 1 = 1 + z + z 2 + · · · deduce that g(p, O) 1+pe16 2 ( 1 - pe'(j ) = .!2 + pei9 + p2 ei 2 9 + . . . for 0 -< p < 1 . c) Verify that if 0 � p < 1 and Pp(O) : = Re g(p , 0) = 21 + p cos O + p2 cos 20 + , ·
then the function
·
·
Pp(O) has the form
1 - p2 (0) = ! 2 1 - 2p cos 0 + p2 and is called the Poisson kernel for the disk. d) Show that the family of functions Pp(O) depending on the parameter p has . 2 71" - e 2 1r the following set of properties: Pp(O) ;::: 0, � I Pp(O) dO = 1 , I Pp(O) dO � 0 as 0 •>0 p � 1 - 0. e) Prove that if f E C[O, 211'] , then the function 2 71' u(p, 0) = Pp(O - t )f(t ) dt P.P
;I 0
is a harmonic function in the disk p < 1 and u(p, 0) � f(O) as p � 1 - 0. Thus, the Poisson kernel makes it possible to construct a function harmonic in the disk having prescribed values on the boundary circle. f) For locally integrable functions u and v that are periodic with the same period T, one can give an unambiguous definition of the convolution (convolution over a period) as follows: (u � v ) (x) : =
a+ T
I u (y)v(x - y) dy . a
The periodic functions on lR can be intepreted as functions defined on the circle, so that this operation can naturally be regarded as the convolution of two functions defined on a circle. Show that if f(O) is a locally integrable 211'-periodic function on lR (or, what is the same, f is a function on a circle) , and the family Pp(O) of functions depending on the parameter p has the properties of the Poisson kernel enumerated in d) , then (f 2*,.. Pp ) (0) � f(O) as p � 1 - 0 at each point of continuity of f.
17.4 Convolution and Generalized Functions
c,1 L 1 )
473
6. a) Suppose cp(x) := a exp for l xl < 1 and cp(x) : = 0 for l x l 2 1 . Let the constant a be chosen so that J cp(x) dx = 1. Verify that the family of functions lR
0 construct a function e(x) of class c� oo ) such that 0 ::::; e(x) ::::; 1 on R, e(x) = 1 <=> X E I, and finally, supp e c Ie , where IE is the €-neighborhood (or the €-inflation) of the set I in R. (Verify that for a suitable value of a > 0 one can take e(x) to be )(I * 0 there exists a countable set { e k } of functions e k E C� oo ) (an €-partition of unity on R) that possesses the following properties: Vk E N, Vx E R (0 ::::; e k (x) ::::; 1); the diameter of the support supp e k of every function in the family is at most c > 0; every point x E R belongs to only a finite number of the sets supp e k ; l: e k ( x) = 1 on R. k d) Show that for every open covering {U., , 1 E r} of the open set G C R and every function cp E c < oo l (G) there exists a sequence {cp k ; k E N} of functions (G) of regular generalized functions. f) Two generalized functions H and H in D' (G) are regarded as equal on an open set U C G if H (cp) = F2 (cp) for every function cp E D( G ) whose support is contained in U. Generalized functions F1 and H are regarded as locally equal at the point x E G if they are equal in some neighborhood U(x) C G of that point. Prove that ( F1 = H ) <=> ( H = F2 locally at each point x E G) .
Cx 1L 1 )
for l x l < 1 and cp(x) : = 0 for l x l 2 1 . Show that a) Let cp(x) := exp J f(x)cpe (x) dx --+ 0 as c --+ +0 for every function f that is locally integrable on R, 7.
IR
where cpe (x) = cp ( � ) . b) Taking account of the preceding result and the fact that (8, cpe ) = cp(O) =f. 0, prove that the generalized function 8 is not regular. c) Show that there exists a sequence of regular generalized functions (even corresponding to functions of class C� 00 ) ) that converges in D' to the generalized function 8. (In fact every generalized function is the limit of regular generalized functions corresponding to functions in D = C� 00 ) ) In this sense the regular gener alized functions form an everywhere dense set in D' , just as the rational numbers Q are everywhere dense in the real numbers R.) .
a) Compute the value (F, cp) of the generalized function F E D' on the function cp E D if F = sin x8; F = 2 cos x8; F = (1 + x 2 )8. b) Verify that the operation F >-+ 'tj;F of multiplication by the function 1/J E c < oo l is a continuous operation in D' . c) Verify that linear operations on generalized functions are continuous in D' .
8.
474
17 Integrals Depending on a Parameter
that if F is the regular distribution generated by the function f (x) = { x0a) fforShow x -< 0 t hen F' = H , where H t he d 1stnbutwn correspond mg to t he ' ' ' or x > 0 ,
9.
· IS
'
·
Heaviside function. b) Compute the derivative of the distribution corresponding to the function lxl .
a) Verify that the following limiting passages in D' are correct: ax = 1rXu� ; lim � = ln x . a = 1r8 ,· 1 1m . lim l l <> -++O a 2 + x 2 a -++ 0 x 2 + a 2 a -++O x + a b) Show that if f = f ( x) is a locally integrable function on lP1. and fe = f ( x + c) , then fe --+ f i n D' as c --+ 0. c) Prove that if { Ll a } is an approximate identity consisting of smooth functions X as a --+ 0, then Fa = I Ll a (t) dt --+ H as a --+ 0, where H is the generalized oo function corresponding to the Heaviside function. 1 0.
-
1 1 . a) The symbol 8(x - a) usually denotes the 8-function shifted to the point a, that is, the generalized function acting on a function c.p E D according to the rule (8(x - a) , c.p) = c.p(a) . Show that the series 2:= 8(x - k) converges in D' .
kEZ
b) Find the derivative of the function [x] (the integer part of x) . c) A 21r-periodic function on lP1. is defined in the interval ]O, 21r] by the formula f i J 0 , 2 11" ] (x) = ! - 2x1l" . Show that f' = - 2� + 2:= 8(x - 21rk) . kEZ
d) Verify that 8(x - c) --+ 8(x) as c --+ 0. e) As before, denoting the 8-function shifted to the point c by 8(x - c), show by direct computation that � ( 8(x - c) - 8(x) ) --+ -8' (x) = -8'. f) Starting from the preceding limiting passage, interpret -8' as the distribution of charges corresponding to a dipole with eletric moment + 1 located at the point x = 0. Verify that (-8' , 1) = 0 (the total charge of a dipole is zero) and that (-8' , x) = 1 (its moment is indeed 1 ) . g ) A n important property o f the 8-function i s its homogeneity: 8(>-.x) = r 1 8(x) . Prove this equality. a) For the generalized function F defined as (F, c.p) = I y'xc.p(x) dx, verify the 0 following equalities: += c.p(x) . (F ' <.p ) = dx , , 2 v'x 0 += c.p(x) - c.p(O) dx ,. (F" , c.p) 4 x3/ 2 0 3 + = c.p(x) - c.p(O) - xc.p' (O) dx ,. (F "' , c.p) 8 x5 / 2 0 00
12.
�I -� I I
17.4 Convolution and Generalized Functions
()
( F n , 'P )
= X
( - 1 ) n - 1 (2n - 3) ! ! X 2n 'P (x) - 'fJ(O) - X'f! 1 (0) -
/+ oo 0
X
· ·
�
·
475
� 'P ( n - 2 ) ( 0) dX .
-
2
b ) Show that if n - 1 < p < n and the generalized function x:;:P is defined by the relation + oo 'P ( X ) - 'fJ(O) - X'f!' (0) - . . . - (�"_-22) ' 'P ( n - 2 ) (0) (x:;:P , 'fJ ) := dx ,
/
.
xP
0
Then its derivative is the function -px _:;: < P + 1 ) defined by the relation 1 + (p + )
( -px 13.
' tn) .,-
= -p
/+ oo 'P (x) - 'fJ (O) - X'f!' (0)xP-+ 1. . . - (�"_-11') 'P ( n 1 ) (0) dx . -
.
0
The generalized function defined by the equality
is denoted P � . Show that
\P� , 'P) +r
=
X
y-+
I
+
X
X
lY
14. Some difficulties may arise with the definition of the multiplication of gener alized functions: for example, the function lxl - 2/ 3 is absolutely ( improperly ) inte grable on lR; it generates a corresponding generalized function += J l x i 2 ! 3 'P (x) dx, oo but its square lxl - 4 / 3 is no longer an integrable function, even in the improper sense. The answers to the following questions show that it is theoretically impossi ble to define a natural associative and commutative operation of multiplication for any generalized functions. a) Show that f (x) o = f (O) o for every function f E c < = l . b ) Verify that xP � = 1 in D' . c ) If the operation of multiplication were extended to to all pairs of generalized functions, it would at least not be associative and commutative. Otherwise, -
1
1
1
(
1
0 = OP - = (xo(x))P- = (o(x)x)P - = o(x) xP X X X X
)
=
o (x) 1
= lO (x) = o .
476
17 Integrals Depending on a Parameter
1 5 . a) Show that a fundamental solution E for the linear operator A : D' ---+ D' is in general ambiguously defined, up to any solution of the homogeneous equation Af = 0 . b) Consider the differential operator d : = dn d n- 1 x, (x) a + ! dx n- 1 + . . . + a n (x) . dx n dx p
( )
Show that if uo = ua (x) is a solution of the equation P ( x , d: ) uo = 0 that satisfies the initial conditions ua (O) = · · · = u6n -2 ) (0) = 0 and u6n - I ) (O) = 1, then the function E(x) = H(x)ua (x) (where H ( X ) is the Heaviside function) is a fundamental solution for the operator P ( x, ddx ) . c) Use this method to find the fundamental solutions for the following operators:
mEN. d) Using these reulsts and the convolution, find solutions of the equations �:,;: d J , ( d x + a ) = J, where f E C(ffi., ffi.) .
m
=
1 7. 5 Multiple Integrals Depending on a Parameter
In the first two subsections of the present section we shall exhibit proper ties oftotalproper andof these improper multipleis integrals depending on a ofparameter. The result subsections that the basic properties multiple integrals depending on a parameter do not differ essentially from the corre sponding properties of one-dimensional integrals depending on a parameter studied above. Insingularity the thirditself subsection weonshalla parameter, study the case of isanimportant improper integral whose depends which in applications. Finally, in the fourthandsubsection we shall study the convolu tion of functions of several variables some specifically multi-dimensional on generalized closely connected integrals depending onquestions a parameter and the functions classical integral formulas ofwithanalysis. 1 7 . 5 . 1 Proper Multiple Integrals Depending on a Parameter
Let X be a measurable subset of JR.n , for example, a bounded domain with smooth or piecewise-smooth boundary, and let Y be a subset of JR.n . Consider the following integral depending on a parameter: (17. 72) F ( y) = J f(x, y) dx , where function is assumed on X fortheeach fixedfvalue of y E toY .be defined on the set X x Y and integrable X
17.5 Multiple Integrals Depending on a Parameter
The following propositions hold. Proposition
1.
then F E C(Y) .
477
If X x Y is a compact subset of JRn + m and f E C(X x Y),
2.
If Y is a domain in JRm , f E C(X x Y), and *!r E C(X x Y), then ithe function F is differentiable with respect to yi in Y, where y (y l , . . . , y , . . . , ym ) and oF of 0 . (y) = J 0 . (x, y) dx . (17.73) y' Y' Proposition
=
X
If X and Y are measurable compact subsets of JRn and JRm respectively, while f E C(X x Y), then F E C(Y) R(Y), and J F(y) dy := j dy j f(x, y) dx j dx j f(x, y) dy . (17.74)
Proposition 3.
Y
Y
=
X
C
X
Y
Wespace note that the values of the function f here may lie in any normed vector Z. The most important special cases occur when Z is JR, C, JR n , n . In these cases the verification of Propositions 1-3 obviously reduce to ortheecase ofverbatim their proofrepetitions for Z Rof theButproof for Zof thelR thecorresponding proofs of Propositions 1 and 2 are propositions for a one-dimensional 17.1), and(Sect.Proposition corollary of Propositionintegral 1 and (see FubiniSect.' s theorem 11. 4 ). 3 is a simple =
17.5.2
=
Improper Multiple Integrals Depending on a Parameter
If theis understood set X JRn asor the function fimproper ( x, y) in the integral ( 17. 72) is unbounded, itexhaustion the limit of integrals over sets of a suitable of X. In studying multiple improper integrals depending on a parameter, as a rule, oneone-dimensional is interested incase.particular exhaustions liwith ke those that we studied in the In complete accord the one-dimensional case, we remove the .: : neighborhood of the singularities/ integrals overintegrals the remaining parts Xc: of X and then find the limit offindtheIfthethis values of the over Xc: as -+ +0. passageintegral is uniform with respect to the parameter y E Y, we say thatlimiting the improper ( 17. 72 ) converges uniformly on Y. Example The integral F().. ) JJ e - .\ ( x 2 + y 2 ) dx d y C
c:
1.
=
JR2
7 That is, the points in every neighborhood of which the function f is unbounded. If the set X is also unbounded, we remove a neighborhood of infinity from it.
4 78
1 7 Integrals Depending on a Parameter
results from the limiting passage JJR2J e- A (x2 +y2 ) dxdy := e!i�o 2 +Jy2J� 1 je2 e- A(x2 +Y2 ) dxdy x and, as one canit converges easily verifyuniformly using polar coordinates, it converges for A > 0. Furthermore, on the set EAo = { A E IR J A 2: Ao > 0} , since for A E EAo > 0<
and last integral tends to 0 as e -+ 0 (the original integral F(A) converges at A this = A o > 0) . Example Suppose, as always, that B(a, r) { x E !Rn J ix - ai < r} is the ball of radius r with center at a E !Rn , and let y E !Rn . Consider the integral Yl dx := lim J l x - y l dx . F(y) = J (1l x -xi) e-HO (1 - l x l )<> l O< B ( O, l ) B ( O, l - e ) Passing to polar coordinates i n !Rn , we verify that this integral converges only for < to1. theIf theparameter value y
2.
a
a
C
1 7 . 5 . 3 Improper Integrals with a Variable Singularity
Example 3. As is known, the potential of a unit charge located at the point x E JR3 is expressed by the formula U(x, y) = l x �y l , where y is a variable point of JR3 . If the charge is now distributed in a bounded region X JR3 C
479
17.5 Multiple Integrals Depending on a Parameter
with a distributed bounded density J.Lway ( x ) ( equal to zero outside X), the potential of a charge in this can be written (by virtue of the additivity of potential) as U (y ) I U ( x, y) J.L ( x ) dx I �;x2 �� . (17.75) X of the parameter in this last integral is played by the variable pointThey role E JR3 . If the point y lies in the exterior of the set X, the integral (17. 75) is a proper integral; but if y E X, then l x - Yl 0 as X x y, and y becomes a singularity of the integral. As y varies, this singularity thus moves. Since U (y) c--t+O lim Uc (y) , where I l xJ.L (-x)y l dx ' =
=
R3
-+
3
-+
=
X\B ( y , c )
it is naturalconverges to consider, as before, thatset theY ifintegral (17.75) with a variable singularity uniformly on the U ) U ( y ) on Y as c +0. We have assumed that IJ.L (x) l :::; M E lR on X,c (Yand therefore 2 I XnBI(y ,c) J.Ll x(x-) dxYl l -< MB (Iy ,c) � l x - Yl 2rrMc . Thi s estimate shows that y E JR 3 , that I U ( y ) - Uc ( y ) I 271" Mc 2 for every 3 converges uniformly on theU set isY continuous JR . is, theIn particular, integral (17.75) with re if we verify that the function c (Y) spect to y, we wi l l then be able to deduce from general considerations that the potential U ( y ) is continuous. But the continuity of Uc ( Y ) does not follow formally froma parameter, Propositionsince1 onin thethe continuity of thean improper integral de pending on present case domain of integration X \ B ( y, c) changes when y changes. For that reason, we need to examine the question of the that continuity We remark for IY -ofYUco I (:::;y )c,more closely. J.L (x) dx + J.L (x ) dx · Uc ( Y ) I I x y l X\B ,c nB o, c l x - y l X\B ( y0 , 2 c ) l ( ( y )) ( y 2 ) The first of these two integrals is continuous with respect to y assuming IY - Yo I < c, being a proper integral with a fixed domain of integration. The absolute value of the second does not exceed Mdx 8rrMc2 . x-y I B ( yo, 2 c ) 1 -1 ::4
-+
=
:::;
=
=
=
480
17 Integrals Depending on a Parameter
Hence the inequality IUo: ( Y) - Uo: ( Yo) l c + 1 6 Mc2 holds for all values of y sufficiently close to y0 , which establishes that Uo:(Y) is continuous at the pointThusYo EweJR.3.have shown that the potential U(y) is a continuous function in the whole space JR.3. These examples provide the basis for adopting the following definition. Definition 1. Suppose the integral ( 1 7.72) is an improper integral that con of singularities the set X obtained verges forfrom each Xy theE Y.c-neighborhood by Let X" be theof portion removing the set of of the inte gral, 8 and let Fo: (Y) = J f(x , y ) dx . We shall say that the integral (17.72) converges uniformly on the set Y if Fo:(Y) F (y) on Y as +0. Theandfollowing useful proposition an immediate inition considerations similar toisthose illustratedconsequence in Exampleof3. this def Proposition 4 . If the function f in the integral (17. 72) admits the estimate l f (x , y) l :::; l x �l " , where M E JR. , x E X JR.n , y E Y JR.n , and n, then the integral converges uniformly on Y . Example 4 . In particular, we conclude on the basis of Proposition 4 that the integral ( y ) = I fJ (x )( x i - y i ) dX ' Ix - y13 obtained by formal differentiation of the potential ( 1 7. 75) with respect to the variable yi (i = 1 , 2, 3) converges uniformly on Y = JR.3 , since I J.L(��(x�;;/l I :::; l x -yAsl 2 . in Example 3, it follows from this that the function V;(y) is continuous on JR.3.Let us now verify that the function U(y) - the potential ( 17.75) - really g� and that g� (y) = V;(y). doesTohave a partial derivativesuffices do this it obviously to verify that <
7r
x.
::::+
C
t r. v,
c --+
C
X
M
Ia V; (yl , y2 , y3) dyi = U(yl , y2 , y3 ) � �' =a . b
8
See the footnote on page 477.
a <
17.5 Multiple Integrals Depending on a Parameter
481
But in fact, b b i f p, (x)(xi - yi ) i ! \I;(Y) dY = j dY I x - y 1 3 dx = ba( 1 ) i b (xi - yi ) i = J p, (x) dx J l x - Yl 3 dy = j p, (x) dx j [)yi l x - Yl dy = = ( / p,l x(x)- dxY l ) l b = U(y ) j b . The only nontrivial pointin order in thistocomputation is theofreversal of the order ofit suffices integration. In general, reverse the order improper integrals, multiple This integral that converges absolutely with respect tothetheinterchange wholeto sethaveisofajustified. variables. condition holds in the present case, so that Of course, it could also be justified directly due to the simplicity of the function involved. Thus,inweJRhave shown that the potential U(y) generated by a charge dis 3 with a bounded density is continuously differentiable in the tributed entireThespace. and reasoning used in inExamples 3 and 4 enable us to studyLetthetechnique::; following more general situation a very similar way. F(y) = j K(y - cp(x)) 7J; (x, y) dx , ( 1 7. 76) wheretheX domain is a bounded measurable domaincp Xin -+!Rn !Rm , the parameter y ranges over is a smooth mapping Y !Rm , with n satisfying rankcp'(x) = n, and 0, that is, cp defines an n cp '(x) l 2: l dimensional parametrized surface, or, more precisely, an n-path in !Rm . Here C(!Rm \ 0, IR), that is, the function K(z) is continuous everywhere in K!RmEexcept at continuous z = 0, near which it may be unbounded; and 7j; X x Y -+ lR isintegral a bounded function. We shall assume that for each y E Y the ( 1 7. 76) (which in general is an improper integral) exists. In the integral ( 1 7. 75) that we considered above, in particular, we had n = cp(x) = x , 7/J (x, y) = p, (x) , K(z) = l z l - 1 . It is not difficult to verify that underof thetheseintegral restrictions on the function convergence ( 1 7. 76) means that for cp,everyDefinition0 one1 ofcanuniform choose E 0 such that a
t
a
X
a
X
a
X
y' =a
y' = a ·
X
X
C
::;;
m,
:
c
>
:
m ,
o:
>
>
482
17 Integrals Depending on a Parameter
I J K (y IY (x) I <E -
(x) ) 'I/J(x , y) dx
i
< a ,
( 17.77)
whereThethefollowi integral is taken overholdthe forset9the{xintegral E X I IY - (x) l E } . ng propositions ( 1 7. 76) . Proposition 5. If the integral ( 1 7. 76) converges uniformly on Y under the hypotheses described above on the functions '1/!, and K, then F E C(Y,IR). Proposition 6 . If it is known in addition that the function '1/J in the integral ( 1 7. 76) is independent of the parameter y (that is, '1/!(x, y) = '1/!(x)) and K E C ( 1 )(JRm \ 0, IR), then if the integral ! 8K IJ i ( y -
<
X
y
converges uniformly on the set y E Y, one can say that the function F has a continuous partial derivative g:, , and
aF j aK IJyi (y) = IJyi ( y - (x) ) 'l/!(x) dx .
X
( 1 7. 78)
Thein proofs of these propositions, as stated, are completely analogous to thoseWe Examples 3 and 4, and so we shall not take the time to give them . noteexhaustion) only that implies the convergence of convergence. an improper Inintegral (under3 andan arbitrary its absolute Examples 4 the hypothesis of absolute convergence was used in the estimates and in reversing an illustration possible uses Propositionsthe 5order and 6,ofletintegration. us considerAsanother example offromthepotential theory.of Suppose a chargev(x).is distributed onofa such smooth compact surface SExample IR3 with surface density The potential a charge distribution isintegral called a single-layer potential and is obviously represented by the surface (x) da(x) c
5.
U(y) =
J
v
( 17. 79) lx - yl . Suppose is a bounded function. Then for y 'f. S this integral is proper, andButthe iffunction U(y) is infinitely differentiable outside S . y E S, the integral has an integrable singularity at the point y. v
s
The singularity is integrable because the surface S is smooth and differs by 9
Here we are assuming that the set X itself is bounded in IE. n . Otherwise one must supplement inequality ( 17. 77) with the analogous inequality in which the integral is taken over the set {x E X l lxl > 1/e:}.
17.5 Multiple Integrals Depending on a Parameter
483
little from aofpiece of1 /thea isplane �2 nearin the point ify E S; 2.andUsing we know that a singularity type integrable the plane Proposition r 5, we can turn this general consideration into a formal proof. If we represent S locally in a neighborhood Vy of the point y E S in the form x
lit
=
C
=
=
Vy
V,
=
17.5.4 *Convolution, the Fundamental Solution, and Generalized Functions in the Multidimensional Case
�n Definition 2. The convolution u * v of real- or complex-valued functions u and v defined in �n is defined by the relation ( 1 7.80) (u * v) (x) J u(y)v(x - y) dy . Example Comparing formulas (17. 75 ) and ( 1 7.80) , we can conclude, for example, that the potential U of a charge distributed in �3 with density J.l(x) is the convolution (J.l * E) of the function and the potential E of a unitRelation charge located at the origin of �3. ( 1 7.80) is a direct generalization of the definition of convolution given infor17.4.theForcasethatn reason, all the properties of the convolution considered in�n 17.4 1 and their proofs remain valid if � is replaced by .An approximate identity in �n is defined j ust as in � with � replaced by �n The and U(O) understood to becontinuity a neighborhood of the point 0 E � n in � n . concept of uniform of a function f G C on a set E G, and with it the basic Proposition 5 of Sect. 17.4 on convergence of the convolution dimensional case.f * L1a to f, also carry over in all its details to the multi a. Convolution in
:=
6.
JFt n
J.l
=
c
:
---+
484
17 Integrals Depending on a Parameter
We note only that in Example 3 and in the proof of Corollary 1 of Sect. must minor be replaced in the definition of the functions . Only changesbyarel x l needed in the approximate identity.dngi(x)venandin Exampleof4periodic of Sect. functions 1 7.4 for the proof of the Weierstrass theorem on approxi mation by trignometric polynomials. In this case it is a question of approximating a function f (x1 , . . . , xn ) that is continuous and pe riodicThewithassertion periodsamounts T1 , T2 , . . . , Tn respectively in the variables x 1 , x 2 , . . . , xn . to the statement thatwiforth theeveryrespective > 0 one can exhibit a trigonometric polynomial in n vari a bles periods T1 , T2 , . . . , Tn that approximates f on lRn within We confine ourselves to these remarks. An independent verification of the properties ( 1 7.80) for n E N, which were proved for the case n = 1 ofintheSect.convolution 1 7.4, will be an easy but useful exercise for the reader, helping to promote an adequate understanding of what was said in Sect. 17.4. b. Generalized Functions of Several Variables We now take up cer tain multi-dimensional aspects ofintheSect.concepts connected with generalized functions, which were introduced 1 7.4. As before, let cC 00 l (G) and c6oo \G) denote respectively the sets of in finitely differentiable in thesupport domaininGG. IfJRnG and the set of infinitely differentiable functionsfunctions of compact = JRn , we shall use the respective abbreviations C ( oo ) and c6oo l . Let := ( , n ) be a multi index and ( {) ) m1 . . . . . ( {) ) mn 'P . 'P ( m ) ·. {)xn - [)xl In c6oo l (G) we introduce convergence ooof functions. As in Definition 7 of Sect. we consider that 'Pk c.p in c6 l (G) as k oo if the supports of all the1 7.4, functions of the sequence { 'Pk } are contained in one compact subset m m ( ) offunctions G and c.p� ) c.p on G for every multi-index as k oo, that is, the uniformly, and songdodefinition. all of their partial derivatives. Given converge this, we adopt the followi Definition 3. The vector space c6oo l (G) with this convergence is denoted V( G) ( and simply V if G = JRn ) and is called the space of fundamental or test functions. Continuous linear functionals on V( G) are called generalized functions or distributions. They form the vector space of generalized functions, denoted V'(G) ( or V' when G = lRn ) . Convergence in V' (G), as in the one-dimensional case, is defined as weak ( pointwise ) convergence of functionals ( see Definition 6 of Sect. 17.4) . The definition of a regular generalized function carries over verbatim to the multi-dimensional case. 17.4 x c.p (x)
E
E.
m
::::l
-+
C
m1 , . . .
m
-+
m
-+
17.5 Multiple Integrals Depending on a Parameter
485
The (denoted definitiono (ofxo),theoro-function andbutthenoto-function shiftedo (tox-xthe))point xremain more often, always happily, 0 also 0 E G the same. Now let us consider some examples. Example Set lxl2 ' 1 L1t (x) := (2a..firi)n e 4;;"2t where > 0, t > 0, x E JR.n . We shall show that these functions, regarded as regular distributions in JR.n , converge tothatthetheo-function on functions JR.n as t --+ + 0. For the proof it suffices to verify fami l y of L1t is an approxi m ate identity in JR.n as t --+ + 0. Usingand a change of variable, reduction of theintegral, multipleweintegral integral, the value of the Euler-Poisson find to an iterated 7.
a
Next, for any fixed value of > 0 we have J L1t (x) dx = ( �)n J e - 1 � 1 2 d.; --+ 1 , r ) ( ' 2 aVt as tFinally, --+ +0. taking account of the fact that L1 (x) is nonnegative, we conclude t that these functions indeed constitute an approximate identity in JR.n . generalization of theino-function (corresponding, for example, to aOsExample unit(corresponding chargeAlocated at the origin JR.n ) is the following generalized function to a distribution of charge over a piecewise-smooth surface S with a distribution of unit surface density). The effect of Os on the function cp E V is defined by the relation ( os, cp) := j cp(x) da . s Like the distribution o, the distribution Os is not a regular generalized function. of a distribution as inMultiplication the one-dimensional case. by a function in V is defined in JR.n just Example If E V, then J-lOs is a generalized function acting according to the rule ( 1 7.81) ( J-LOs, cp) = J cp(x)J-L(x) da . s r
B(O,r)
8.
9.
J-1
B o
486
1 7 Integrals Depending on a Parameter
If the function J-t(x) were defined only on the surface S, Eq. ( 1 7.81 ) could beanalogy, regarded as the definition of theintroduced generalizedin this function Js.calledBy anatural the generalized function way is single layer on the surface S with density J-t. Differentiation ofprinciple generalizedas infunctions in the multi-dimensional casea fewis defined by the same the one-dimensional case, but has peculiarities. If F E V' (G) and G !Rn , the generalized function g;: is defined by the relation J-t
C
It follows that ( 17.82) where m ( m 1 , . . . , mk ) is a multi-index and l m l Ln m; . is naturalof theto verify the relation �:f:x �:bx But that follows from the Itequality right-hand sides in the relations =
=
i
8
=
8
i= l
'.
which follows from the classical equality : lf : 't , which holds for every function cp E Example Now consider an operator D L where m = are numer ( m 1 , . . . , m n ) is a multi-index, D ( � ) ·. . .· ( � ) coefficients,operator. and the sum extends over a finite set of multi-indices. This isicalaThe differential transpose or adjoint of D is the operator usually denoted t D or D* and defined by the relation 1 0.
2 8 , xi
D.
m
=
8 1
m,
=
( DF, cp) ( F, t Dcp) , cp E V F E V'.
which musttheholdexplicit for allformulaand now write
rn
=:
2 EJ i x '
m
am D m ,
EJ n
mn
, am
Starting from Eq. ( 1 7.82) , we can
for the adjoint of the differential operator D.
17.5 Multiple Integrals Depending on a Parameter
487
In particular, if all the values of l m l are even, the operator D is self It is clearof dithatfferentiation the operation of differentiation in V' (JR.n ) preserves all the properties in V' (JR.) . However, let us consider the following important example, which is specific to the multi-dimensional case. Let S be a smooth (n - 1 )-dimensional submanifold of JR.n , that is,isExample S is a smooth hypersurface. Assume that the function f defined on JR.n \ S infinitely differentiable and that all its partial derivatives have a limit at every point x E S under one-sided approach to x from either (local) side of S. the The surfacedifference between these two atlimits will bex corresponding the jump J {Nrto ofa par the partial derivative under consideration the point ticular direction of passageis across the The surfacejumpS atcanx. thus The besignregarded of the jump changes if that direction reversed. as a function defined on an oriented surface if, for example, we make the con vention that the direction of passage is given by an orienting normal to the surface. Theby function {Nr is defined, continuous, and locally bounded outside S, and the assumptions just made f is locally ultimately bounded upon S itself. Since S is a submanifold of JR.n , no mat approach to the surface ter how wediscontinuities complete theondefinition of {Nr on S, we obtain a function with possible only S, and hence locally integrable in JR.n . But integrable functions without that diffworrying er on a about set of themeasure zero have equal inte S, we may assume grals, and therefore, values on that {Nr generates some regular generalized function { {Nr } according to the rule adjoint, that is, t D = D. 11.
We shallimportant now showformula that iffholdsis regarded as aofgeneralized function, then the following in the sense differentiation of generalized functions: 8f 8f ( 1 7.83) 8xi = { 8xi } + (If) s cos O:i bs whereofthethelastfunction term isf understood in the sense of Eq. ( 1 7.81 ) , (If ) s is the jump at x E S correpsonding to either of the two possible unit normal directions onto the xiof-axisthe (that is, = (costoo:S1 , at. . . x,, cosandak )cos) . o:i is the projection of Proof. Formula ( 1 7.83) generalizes Eq. ( 1 7.64) , which we use to derive it. '
n
n
n
488
17 Integrals Depending on a Parameter
For definiteness we consider the case i = 1 . Then
J :;I dx + J · · · J (Jf) cp dx2 . . . dxn . Here thethrough jump Iftheofsurface f is taken at thepointpointin thex direction (xi , x2 , , xn) E S as one-axis. passesThe at that of(I f)the positi ve value Hence, of the function in computing the product is taken XatI the same point. this last integral can be written as a surface integral of first kind j (Jf) cp cos dO" , where isStheat angle between the direction of the positive XI-axis and the normal to x , direct so that in passing through x in the direction of that normal0. Ittheremains functiononlyf hasto remark preciselythattheifjump I f. This means only that cos we choose thecosineotherwould direction I for the normal, the sign of the jump and the sign of the reverse simultaneously; hence the product (If) cos I does not change. bothD Remark As can be seen from this proof, formula ( 17.83) holds once the jump (I f) of f is defined at each point x E S, and a locally integrable partial derivative jb exists outside of S in !Rn , perhaps as an improper integral generating a regular generalized function { jb } . Remark 2. At points x E S at which the direction of the x i -axis is not transversal totheS, jump that is,f itin isthetangent to S, difficulties maybeariseseeninfrom the definition of given direction. But i t can I ( 17.83) that its last term is obtained from the integral =
JR n
=
s
ai
a
:z:2 • • . :z: n
.
.
•
ai
�
1.
a
8
! ! ···
:z:2 . , . :z: n
(If) cp dx 2 · · · dx n .
The projections of the set E on x2 , . . . ,x n -hyperplane has (n - I) dimensional measure zero andthetherefore has no effect on the value of the integral. Hence we can regard form ( 17.83) as having meaning and being valid always if (If) cos ai is given the value 0 when cos ai 0. 8
=
17.5 Multiple Integrals Depending on a Parameter
489
Remark Similar considerations make it possible to neglect sets of area zero; therefore one can regard formula ( 17.83) as proved for piecewise-smooth surfaces. As aourcannextbe example wedirectly shall show howthe thedifferential classical relation Gauss-Ostrogradskii formul obtained from ( 1 7.83) , and ininformed a form thethatreader is maximally free of the extra analytic requirements that we of previously. Example in JR.n field bounded bycontinuous a piecewise-smooth surface S. LetLetA =G (Abe1 a, . finite ben a vector that is in G and . . , An)domain . such that the function divA = L ��: is defined in G and integrable on G, possibly inregard the improper sense. If we the field A as Szeroof theoutside G, then the jump of this field at each poi n t x of the boundary domain G when leaving G is -A(x). Assuming that n is a unit outward normal vector to S, applying formula ( 17.83) to each component A i of the field A and summing these equalities, we arrive at the relation ( 17.84) divA = {divA} - (A · n) 6s , inpointwhichx EA·S.n is the inner product of the vectors A and n at the corresponding Relation ( 17.84) is equality of generalized functions. Let us apply it to the 00 l function E c6beenequal to 1 onmoreG (the existence and construction ofevery such afunction functionr.p1/J Ehas discussed than once previously). Since for V ( 17.85 ) (divA, r.p) = - J (A · 'Vr.p) dx (which follows immediately fromAtheanddefinition of the1/Jderivative of a have gen eralized function), for the field the function we obviously (divA, 'lj;) = 0. But, when we take account of Eq. ( 17.84 ) this gives the relation 0 = ( {div A},'lj; ) - ( (A · n)6s,1/J ) , which in classical notation 0 = J divA dx - J (A · n) dcr ( 17.86 ) is the same as the Gauss-Ostrogradskii formula. now consider several important examples connected with differenta tionLetof usgeneralized functions. 3.
1 2.
i= l
]R n
G
S
490
17 Integrals Depending on a Parameter
We consider3 the vector field A = fxP- defined in JR3 \ 0 and show that in the space V' (JR ) of generalized functions we have the equality div � �3 = 4m5 . ( 1 7.87) We remark first that for x =1- 0 we have div fxP- = 0 in the classical sense. Now, the definition of divAdivifxPn the= form ( 1 7.85) , the definition using of ansuccessively improper integral, the equality 0 for x =1- 0, the Gauss-Ostrogradskii port, we obtain formula ( 1 7.86) , and the fact that tp has compact sup Example
1 3.
\ div � �3 , tp) = - I ( 1 �3 · \7tp(x)) dx = JR3 = lim - I ( 1 � 3 · \7tp(x)) dx = tp ( x3 ) ) d = = lim d" c I V I lxl c: --+ + 0
c: < l x l < l /c:
X
c: --+ +0
For the operator A : V' (G) V' ( g ) , as before, we define a fundamental solution to be a generalized function E V' (G) for which A (E) = c5. Example We verify that the regular generalized function E(x)2 = - 2j 3 in V' (JR2 ) is a fundamental solution of the Laplacian Ll = ( � ) + ( � ) + ( a�3 ) · Indeed, Ll = div grad, and gradE(x) = f for x =1- 0, and therefore the equality divExample gradE 1= , c5onefollows from relation ( 1 7.87) . As i n can verify that for any n E N, n 2, we have the 3 following relation in JRn : ---+
14.
c
8 1
8 2
4 1r x l
4 7r x l 3
:2':
( 1 7.87' )
where an = f(r�:�) is the area of the unit sphere in JR" . Hence we can conclude upon taking account of the relation Ll = div grad that Llln l X I = 27rc5 in JR2 and 1 Ll1 1 n- 2 = - (n - 2)an c5 in JRn , n > 2 . X
491
17.5 Multiple Integrals Depending on a Parameter
Let us verify that the function H;, e - s; , E(x, t) = (2a nt) n where x E JRn , t E JR, and H is the Heaviside function (that is, we set E(x, t) = 0 when t 0) satisfies the equation ( :t - a2 L1 ) E = J . Here .1 is the Laplacian with respect to x in JRn , and J = J ( x, t) is the iS-function in JR� x lRt = JRn+l . When t > 0, we have E E C ( oo ) (JR n+l ) and by direct differentiation we verify that ( :t - a2 L1) E = O when t > O . Takifunction ng this factt.p EintoV(JRaccount along with the result of Example 7, we obtain n+l for any Example
1 5.
<
)
+oo
J0 dt J E(x, t) ( �� + a2 .1t.p) dx = +oo = - �� J dt J E(x, t) ( �� + a 2 .1t.p ) dx = = -
� ,i 0 V
�
�
2'.".'
l�
o
c:
JRn
o
Tdt1 ( ';;:
R (x, £ )1"(x, D) dx +
R (x, £ ) I" ( X , 0) dx +
) d]
- hiE l" x
1 R (x, £) (l" (x, £) - l" (x, D)) dx]
�
lim+0 J E(x, ) t.p (x, 0) dx = t.p(O, 0) = (J, t.p) . Example Let us show that the function 1 E(x, t) = H ( at lxl ) , 2a where equationa > 0, x E JR�, t E JR�[), 2and H [)is2 the Heaviside function, satisfies the ( fJt 2 - a2 fJx2 ) E = J ' in which J = J(x, t) is the iS-function in the space V'(JR� x JRD = V'(JR2 ). =
1 6.
c: -+
JRn
-
E
-
492
17 Integrals Depending on a Parameter
Let 'P E V(�2 ). Using the abbreviation Da := .g;, - a2 l;r , we find ( D a E ,
�
-
a
-
u
1 7.
1 8.
u.
17.5 Multiple Integrals Depending on a Parameter
493
offunction the equation �� - .du = f, for example, under the assumption that the f is continuous and bounded, which guarantees the existence of the convolution *E. We note that these assumptions are made only for example, and are far ffrom obligatory. Thus, from the point of view of generalized functions one could pose the question of the solution of the equation �� .du = f taking as f(x, t) the generalized function cp(x) · o(t), where cp E V(JRn ) andThe o E V'(JR). formal substitution of such a function f under the integral sign leads to the relation � ) ed� . u(x, t) = J [2acp(.Jii; t] n Applying thethatrulethisfor function differentiating an integral depending on��a parameter one can verify is a solution of the equation a.du = 0 for t > 0. We note that u(x, t) cp(x) as t +0. This follows from the result of Example 7, where it was established that the family of functions encountered here is an approximate identity. Example recalling the fundamental solution of the Laplace oper ator obtainedFinally, in Example 14, we find the solution u (x ) = J !(� ) d� ! x -� ! 2 4a
rr
JR n
-+
-+
t
-
1 9.
JRn
lx - ��
ofis thethe same Poissonas theequation .du = - 47r f, which up to notation and relabeling potential ( 1 7.75) for a charge distributed with density J , whiIfchthewe function consideredf isearlier. takenintoas v(x)os, whereleads S is toa piecewise smooth surface in JR3 , formal substitution the integral the function u (x ) = v(l�x) -da(� � ) ' � s
J
as wedistributed know, is aoversingle-layer the potential ofwhich, a charge the surfacepotential; S JR3 more with precisely, surface density v(x). C
17.5.5 Problems and Exercises 1. a) Reasoning as in Example 3, where the continuity of the three-dimensional potential ( 17.75) was established, show that the single-layer potential ( 17.79) is continuous. b) Verify the full proof of Propositions 4 and 5.
494
17 Integrals Depending on a Parameter
2. a ) Show that for every set M C ll�n and every E > 0 one can construct a function f of class c (ool (R n , R) satisfying the following three conditions simultaneously: 'Vx E R n (0 :::; f(x) :::; 1 ) ; 'Vx E M (f(x) = 1 ) ; supp f C M, , where M, is the e:-blowup ( that is, the €-neighborhood ) of the set M. b ) Prove that for every closed set M in R n there exists a nonnegative function f E C (oo) (R n , R ) such that ( !(x) = 0) {o} (x E M). 3 . a ) Solve Problems 6 and 7 of Sect. 17.4 in the context of a space R n of arbitrary dimension. b ) Show that the generalized function 8s ( single layer ) is not regular.
4. Using convolution, prove the following versions of the Weierstrass approximation theorem. a) Any continuous function f : I -+ R on a compact n-dimensional interval I C R n can be uniformly approximated by an algebraic polynomial in n variables. b ) The preceding assertion remains valid even if I is replaced by an arbitrary compact set K C R and we assume that f E C(K, C) . c ) For every open set G C R n and every function f E C ( rn l ( G, R ) there exists a sequence { Pk } of algebraic polynomials in n variables such that P� a) :::; f ( a) on each compact set K C G as k -+ oo for every multi-index a = (at , . . . , a n ) such that l a l S m. d ) If G is a bounded open subset of R n and f E c (oo) (G, R ) , there exists a sequence {Pk } of algebraic polynomials in n variables such that P� a ) :::; f ( a) for every a = ( a t , . . . , a n ) as k -+ oo . e ) Every periodic function f E C ( R n , R ) with periods Tt , T2 , . . . , Tn in the variables x 1 , . . . , x n , can be uniformly approximated in R n by trigonometric poly nomials in n variables having the same periods Tt , T2 , . . . , Tn in the corresponding variables. 5. This problem contains further information on the averaging action of convolu tion. a) Previously we obtained the integral Minkowski inequality
(I
l a(x) + b(x) I P dx
) 1/p ( I <
X
l a i P (x) dx
) 1/p ( I +
X
IW (x) dx
) 1/p
for p 2: 1 on the basis of this numerical Minkowski inequality. The integral inequality in turn enables us to predict the following generalized integral Minkowski inequality:
(I I I
p ) 1/p ( l :::; I I
f(x, y) dy dx
I J I P (x, y) dx
) 1/p
dy .
Prove this inequality, assuming that p 2: 1 , that X and Y are measurable subsets example, intervals in R"' and R n respectively ) , and that the right-hand side of the inequality is finite. ( for
17.5 Multiple Integrals Depending on a Parameter
495
b) By applying the generalized Minkowski inequality to the convolution f * g, show that the relation ll f * g ii P :'::: ll f ll 1 · II9II P holds for p 2': 1 , where, as always, 1 1P dx) . lluii P =
(J,: l u i P (x)
c�oo ) (JRn , JR) with 0 ::::: <.p(x) ::::: 1 on lRn and I <.p(x) dx 1 . Assume that <.p,(x) := � 'P ( � ) and f, := f 'P• for E > 0. Show that if f E Rp(JR n ) (that is, if the integral I IJI P (x) dx exists) , then f, E c ( oo ) (JR n , JR) and ll f, ll p ::::: IIJII P " c) Let 'P E
=
JRn
*
JRn
We note that the function f, is often called the average of the function f with kernel 'P• · d) Preserving the preceding notation, verify that the relation
ll f, - f lip,[ :'::: sup• li Th ! - f lip,[ , Ihi<
(
)1 P
holds on every interval I C lR n , where llull p,I = [ i uiP(x) dx / and Th j (x) = f(x - h). e) Show that if f E Rp(JR n ), then II Th f - f lip, I -+ 0 as h -+ 0. f) Prove that ll f, II P :'::: IIJII P and ll f, - f li P -+ 0 as E -+ +0 for every function f E Rp(JR n ), p 2': 1 . g ) Let Rp(G) b e the normed vector space o f functions that are absolutely inte grable on the open set G C lR n with the norm I llp, G · Show that the functions of class C ( oo ) (G) n Rp(G) form an everywhere-dense subset of Rp(G) and the same is true for the set C� oo ) (G) n Rp(G). h) The following proposition can be compared with the case p = oo in the preceding problem: Every continuous function on G can be uniformly approximated on G by functions of class c ( oo ) (G). i) If f is a T-periodic locally absolutely integrable function on JR, then, setting 1 a T IJI P (x) dx) / P , we shall denote the vector space with this norm by ll f ll p .T = R� . Prove that ll f, - f ll p, T -+ 0 as E -+ +0 . j) Using the fact that the convolution of two functions, one of which is periodic, is itself periodic, show that the smooth periodic functions of class c(oo ) are everywhere dense in R� .
(l
6. a) Preserving the notation of Example 1 1 and using formula (17.83) , verify that if f E C ( l ) (JE.n \ S) , then fJj 82 ! = 82 ! f) i fJx fJxJ fJxi fJxJ + fJxJ f)s cos aiOs) + fJx i ) 8 cos Oj Os .
{
}
(J
( (J
(J
(J �) (
b) Show that the sum £: fxlr) s cos ai equals the jump s of the normal ,= 1 derivative of the function f at the corresponding point x E S, this jump being independent of the direction of the normal and equal to the sum §t + I£; ) (x) of the normal derivatives of f at the point x from the two sides of the surface S.
17 Integrals Depending on a Parameter
496
c) Verify the relation
(
) (
(Jf)
5 is where :n is the normal derivative, that is, :n F,
f
under the assumption that G is a finite domain in JRn bounded by a piecewise smooth surface S; and
f
f
7. a) The function 1 � 1 is the potential of the electric field intensity A created in JR3 by a unit charge located at the origin. We also know that
div
( �) -
�
3
=
47r8 ,
div
( �f3 ) -
�
=
47rq8 ,
div grad
C!l )
=
-�
4m5 .
Starting from this, explain why it was necessary to assume that the function U(x) = J ���:�� must satisfy the equation L1 U = -47rp,. Verify that it does indeed JR3 satisfy the Poisson equation written here. b) A physical corollary of the Gauss-Ostrogradskii formula, known in electro magnetic field theory as Gauss ' theorem, is that the flux across a closed surface S of the intensity of the electric field created by charges distributed in JR3 equals Q/co (see p. 279, where Q is the total charge in the region bounded by the surface S. Prove this theorem of Gauss. 8. Verify the following equalities, understood in the sense of the theory of general ized functions. a) L1E = 8, if for x E JR2 , 2� In E(x) - 2 7r nr(/ 2 nn)- 2) -( n - 2) for X E lRn , n > 2 .
{
(
lxl lxl
17.5 Multiple Integrals Depending on a Parameter
497
x e-ik l x b ) ( Ll + k 2 ) E = 8, .1f E(x ) = - ei4 rrk ll x ll or 1f. E(x ) = - 47iTXT"l and x E JR.3 . 2 2 c) D a E = 8, where D a = � a 2 [ ( -/;;r ) + + ( a �n ) ] , and E = H ( at - l x l) H 2 4 7r a t 8s at = 2 7r( t)a 8(a 2 t 2 - l x l 2 ) for x E IR.3 ' 2 tr a y'a 2 2 l l 2 for x E IR. or E = ..J!!!:d, -
·
t -x
·
·
-
t E R Here H( t ) is the Heaviside function, Sat = {x E IR.3 I I x l = at} is a sphere, and a > 0. d) Using the preceding results, present the solution of the equation Au = f for the corresponding differential operator A in the form of the convolution f * E and verify, for example, assuming the function f continuous, that the integrals depending on a parameter that you have obtained indeed satisfy the equation Au = f. Differentiation of an integral over a liquid volume. Space is filled with a moving substance (a liquid) . Let v = v( t, x) and p = p ( t, x) be respectively the velocity of displacement and the density of the substance at time t at the point x. We observe the motion of a portion of the substance filling the domain flo at the initial moment of time. a) Express the mass of the substance filling the domain flt obtained from flo at time t and write the law of conservation of mass. b) By differentiating the integral F(t) = I j( t , x ) dw with variable domain of 9.
n,
flt (the volume of liquid) , show that F' (t) = I %f dw + I f(v, ) da, where flt , an� , dw, da, v, ( ' ) are respectively the domain, its boundary, the integration
n,
n,
a n,
n
element of volume, the element of area, the unit outward normal, the flow velocity at time t at corresponding points, and the inner product . c) Show that F' ( t) in problem b) can be represented in the form F' ( t) = I ( %f + div ( Jv) ) dw.
n,
d) Comparing the results of problems a) , b) , and c) , obtain the equation of continuity � + div (pv) = 0. (In this connection, see also Subsect. 14.4.2.) e) Let l flt I be the volume of the domain flt . Show that diZr 1 = I div v dw . n,
f) Show that the velocity field v of the flow of an incompressible liquid is divergence-free ( div v = 0) and that this condition is the mathematical expression of the incompressibility (conservation of volume) of any portion of the evolving medium. g) The phase velocity field (p, q) of a Hamiltonian system of classical mechanics satisfies the Hamilton equations p = �� , q = �� , where H = H(p, q) is the Hamiltonian of the system. Following Liouville, show that a Hamiltonian flow pre serves the phase volume. Verify also that the Hamiltonian H (energy) is constant along the streamlines (trajectories) . -
1 8 Fourier Series and t he Fourier Transform
18. 1 Basic General Concepts Connected with Fourier Series
18. 1 . 1 Orthogonal Systems of Functions
Expansion of a vector in a vector space During this course of analysis wea.spaces haveinmentioned several times thatarithmetic certain classes of functions formexample, vector relation to the standard operations. Such, for are the basicreal-,classes of analysis, which consist of smooth, continuous, or integrable complex-, or vector-valued functions on a domain X JR. n . From the point of view of algebra the equality C
where j, h , . . . , fn are functions of the given class and O:i are coefficients from or C, simply means that the vector f is a linear combination of the vectors h , . . . , fn of the vector space under consideration. a rule, it ofis the necessary tions"In analysis, -- series ofasfunctions form to consider "infinite linear combina JR.
(18.1)
The definition of the sum of inthetheseriesvectorrequires that some topology (init particular, a metric) be defined space in question, making possiblen to judge whether the difference f - Sn tends to zero or not, where Sn = 2:.::: O: k fk · k=l The mainisdevice usedsomein classical analysis toor introduce a metric on a vector space to define norm of a vector inner product of vectors in that space.nowSect. 10. 1 was devoted to a discussion of these concepts. We are going toshall consider only ,spaces endowed with an inner product (which, as before, we denote ) ) . In such spaces one can speak of ( orthogonal orthogonal systemsEuclidean of vectors,spaceandfamiliar orthogonalfrombases, just asgeometry. in the casevectors, of three-dimensional analytic
500
18 Fourier Series and the Fourier Transform
Definition 1 . The vectors x and y in a vector space endowed with an inner product ( , ) are orthogonal ( with respect to that inner product ) if ( x, y ) = 0. Definition 2 . The system of vectors { x k ; k E K} is orthogonal if the vectors in it corresponding to different values of the index k are pairwise orthogonal. Definition 3 . The system of vectors {e k ; k E K} is orthonormalized ( or orthonormal) if ( e i , e J ) = bi,J for every pair of indices i, j E K, where bi ,J is 1 if i = j the Kronecker symbol, that is, bi,J = { 0 if i =1- j . Definition 4. A finite system of vectors x 1 , . . . , X n is linearly independent if the equality a 1 x 1 + et2 X2 + + Ctn Xn = 0 is possible only when et 1 = a2 = = Ct n = 0 ( in the first equality 0 is the zero vector and in the second it is the Anzeroarbitrary of the coefficient fieldvectors ). system of of a vector space is a system of linearly independent vectors if every finite subsystem of it is linearly independent. The main questionsystem thatofwilllinearly interestindependent us now is thevectors. question of expanding a vector in a given Having in mind later applications to spaces of functions (which may be infinite-dimensional as well ) we must reckon with the fact that such an ex pansion may, inenters particular, leadstudy to a series offundamental the type ( 18.1). That is precisely where analysis into the of the and essentially alge braicAsquestion we have posed. is known fromhaveanalytic geometry, expansions over in orthogonal andin orar thonormal systems many technical advantages expansions bitrary linearly independent systems. ( The coefficients of the expansion are easy to compute; i t is easy to compute the and innersoproduct of two vectors from theirIt coefficients in an orthonormal basis, on. ) is for systems. that reasonIn function that we shall bethese mainlywiinterested in expansions in or thonormal spaces l l be expansions in orthogonal systems of functions or Fourie? series, to the study of which this chapter is devoted. '
'
'
· · ·
· · ·
1 J.-B.J. Fourier ( 1768-1830) - French mathematician. His most important work Theorie analytique de la chaleur (1822) contained the heat equation derived by Fourier and the method of separation of variables (the Fourier method) of solving it (see p. 516) . The key element in the Fourier method is the expansion of a function in a trigonometric (Fourier) series. Many outstanding mathematicians later undertook the study of the possibility of such a representation. This, in particular, led to the creation of the theory of functions of a real variable and set theory, and it helped to promote the development of the very concept of a function.
18. 1 Basic General Concepts Connected with Fourier Series
501
Extending Example of Sect. 10.1, we introduce an inner product ( 18. 2 ) (!, g) := l (f · g)(x) dx on thelocally vectorsquare-integrable space 'R2 (X, C)(asconsisting of improper functions integrals). on the set X JR.n that are proper or Since I,f · 9unambiguously. 1 :S ! (1/1 2 + l g l 2 ) , the integral in ( 18. 2 ) converges and hence defines ) (!areg discussing real-valued functions, relation ( 18.2) in the real space If we n2 (x, JR.) reduces to the equality ( 18.3) (!, g) := l u · g)(x) dx . Relyi ng anon inner properties of thelistedintegral, one10.1can areeasilysatisfied verify inthatthisallcase, the axi o ms for product in Sect. provided zero. we identify two functions that differ onlytexton portion a set of ofn-dimensional measure Throughout the following, in the 2 ) and inner products of functions will be understood in the sense of Eqs.the( 18.section, b. Examples of Orthogonal Systems of Functions 12
X
C
X
( 18.3) .
We recall that for integers m and n if m # n , x x n . m i -i e e ( 18.4) dx = I 2 if m = n ; rr , r if m # n , I cosmx cosnxdx = r2rrrr , , ifif mm == nn #= OO ;, ( 18.5) ( 18.6) I cosmxsinnxdx = 0 ; . sin mx sin nx dx = ' if m # n , ( 18 . 7) I if m = n # O . rr , r E Z} is an orthogonal system of Thesein relations show'R that rr],{ eiCnx ; nrelative vectors the space to the inner product ( 18. 2 ) [-rr, ) 2( and the trigonometric system { 1 , cos nx, sin nx; n E N} is orthogonal in 'R2 ( [-rr, rr] , JR.) . If we regard the trigonometric system as a set of vectors incomplex 'R2 ([- rr, rr], C ) , that is, if we allow linear combinations of them to have coefficients, then by Euler ' s formulas einx = cos nx + i sin nx, Example
1.
7r
- ?r
7r
- ?r
7r
- ?r
- ?r
502
18 Fourier Series and the Fourier Transform
cos nx -- 2 ( ei n x + e -i nx ) ' sin nx - 2 i ( ei nx - e -inx ) ' we see that these two systems canequivalent. be expressedForlinearly in terms ofexponential each other,system that is,{ ethey inx ; nareE Z}algebraically that reason the issystem also called · the trigonometric system or more precisely the trigonometric in complex notation. Relationswhile (18.4)-(18. 7) show{ that these systems are orthogonal, but not normalized, the systems k einx ; n E Z} and { 1 1 cos nx, 1 sin nx; n E N} 2 7f 7f 7f are Iforthonormal. the closed interval [ of] vari is replaced bycananobtain arbitrary closed interval [-l, l] { JR, then by a change a ble one the analogous sys tems ei f nx ; n E Z} and { 1, cos fnx, sin fnx; n E N} , which are orthogonal in the spacessystems R2 ( [-l, l] , C ) and R2 ([-l, l] , JR) and also the corresponding orthonormal 1 7f 1 . 7f { 1 ei -" nx n E '7/ } and { 1 vrz V'ii V'ii ' cos T nx, vrz sm T nx; n E N } ' Example Let Ix be an interval in JRm and Iy an interval in JR n , and let {fi (x)} be an orthogonal system of functions in R.2 (Ix , lR) and {gj (y)} an orthogonal from Fubini ' s theorem, thesystem systemof offunctions functionsin R{u2i(Ij (x,y , JR).y) Then,/j (x)gas jfollows (y) is orthogonal in R2 (lx X fy , JR). Example We remark that for a f. (3 l J Sill. ax Sill. (3X dX = 21 ( sin(aa --(3(3)l sin(aa ++/3(3)l ) = _- cos al cos (3l · (3 tan aal2 -- /3a2tan (3l . Hence, if a and (3 are such that ta� = ta�f3l , the original integral equals zero. Consequently, if 6 6 · · · �n · · · is a sequence of roots of the equation tan �l c�, where c is an arbitrary constant, then the system of functions ( �n x) ; n E N} is orthogonal on the interval [0, l] . In particular, for c = 0, {wesinobtain the familiar system { sin ( fnx) ; n E N}. Example 4. Consider the equation ( d�2 + q(x) ) u(x) = Au(x) , where E C ( oo ) ( [a, b] , JR) and A is a numerical coefficient. Let us assume that 2 ) ( the functions u1 , u2 , . . . are of class C ( [a, b], JR) and vanish at the endpoints .l.
!
;;;; rrc , v v
;;;; v
- 1r , 1r
C
I
j
ILl
2.
3.
-
-
0
=
q
<
<
<
ad
<
18. 1 Basic General Concepts Connected with Fourier Series
503
ofwiththeparticular closed interval [a, b] and that each of them satisfies the given equation values A 1 , A2 , . . . of the coefficient A. We shall show that if Ai =J Aj , then the functions u i and Uj are orthogonal on [a, b]. Indeed, integrating by parts, we find that J [ ( d�2 + q (x ) ) ui (x )] uJ (x ) dx = J ui (x ) [ ( d�2 + q (x ) ) uj (x )] dx . According to the equation, we obtain from this the relation b
b
a
a
and, since Ai =J Aj , we now0conclude that[a,(ub]i , Uj= )[0,= 0.] we again find that the on [a, b] and In particular, if ( system { sin nx ; n EqN}x ) is orthogonal on [0, ] Further examples, including examples of orthogonal systemsat theof impor tance in mathematical physics, will be found in the problems end of this section. Orthogonalization It is well-known that in a finite-dimensional Eu clidean space,waystarting with a linearly independent system of vectors, there is aofcanonical of constructing an orthogonal and even orthonormal system 2 -Schmidt3 orthog vectors equi v alent to the given system, using the Gram onalization process. By thesystem same method one'lj;can, 'lj; obviously orthonormalize any linearly independent of vectors , . . . in any vector space 1 2 having an inner product. We recall, that the orthogonalization process leading to the orthonormal system
1r .
1r
c.
'1/J
-
nL1 ('1/Jn ,
�---------
--------
-
2 J.P. Gram (1850-1916) - Danish mathematician who continued the research of P.L. Chebyshev and exhibited the connection between orthogonal series expan sions and the problem of best least-squares approximation ( see Fourier series below ) . It was in these investigations that the orthogonalization process and the famous Gram matrix arose ( see p. 185 and the system ( 18. 18) on p. 510) . 3 E. Schmidt (1876-1959) - German mathematician who studied the geometry of Hilbert space in connection with integral equations and described it in the language of Euclidean geometry.
504
18 Fourier Series and the Fourier Transform
The process of orthogonalizing the linearly independent system leads to the system of orthogonal polyno mials known as the Legendre polynomials. We note that the name Legendre polynomials is often given not to the orthonormal system, but to a system ofcanpolynomials proportional to these polynomials. The proportionality factor be chosen from various considerations, for example, requiring the leading coefficient to be 1 orof the requiring theis polynomial to these have therequirements, value 1 at xbut= in1. The orthogonality system unaffected by general orthonormality is lost. the standard Legendre polynomials, which We have already encountered are defined by Rodrigues ' formula Example
5.
{ 1, x, x 2 , . . . , } in R2 ( [ - 1, 1], �)
= 1. Let us write out the first few Legendre For these polynomials polynomials, normalizedPnby(1)requiring the leading coefficient to be 1: p3 ( X ) = X - -3 X . Po(x) 1 , 5 The orthonormalized Legendre polynomials have the form -
=
3
P�n (x) = y� �-2- Pn (x) ,
whereOne can= 0,verify 1, 2, . . . . by direct computation that these polynomials are orthogo nal on the closed interval [-1, 1]. Taking Rodrigues ' formula as the definition ofalsthe{P polynomial Pn ( x), let us verify that the system of Legendre polynomi 1]. To do this, it suffices n (x)} is orthogonal on the closed intervaln[-1, toof degree verify that Pn (x) is orthogonal to 1, x, . . . , x - l , since all polynomials Pk k are linear combinations of these. Integrating by parts for k we indeed find that n
< n
< n,
Awillcertain picture of thelast origin of orthogonal systemsandofinfunctions in analat be given i n the subsection of this section the problems ysis the end ofconnected the section.wiAtth present we shallofreturn to thein terms fundamental general problems the expansion a vector of vectors of a given system of vectors in a vector space with inner product.
18. 1 Basic General Concepts Connected with Fourier Series
505
d. Continuity of the Inner Product and the Pythagorean Theorem
We shallsums have(series to work not only with finite sums of vectors but also with infinite ) . In this connection we note that the inner product is a function, enabling us oftoseries. extend the ordinary algebraic properties ofcontinuous theLetinner product to the case X be a vector space with an inner product ( , ) and the norm it induces 00 ll x ll := y'(x,X} ( see Sect. 10. 1 ) . Convergence of a series iI:= l X i = x of vectors X i E X to the vector x E X will be understood in the sense of convergence in this norm. Lemma 1. ( Continuity of the inner product ) . Let ( , ) : X ---* C be an inner product in the complex vector space X. Then 1--+
(x, y) (x, y ) is continuous jointly in the two variables; 00 b) if x = I: x i , then (x, y) = I: (x i , y); i= l i= l 00 ) if e 1 , e 2 , . . . , is an orthonormal system of vectors in X and x = I: x i ei i= l 00 00 i i and y = I: y e i , then (x, y) = I: x fi . i= l i= l a) the function 00
c
inequality (see Sect. Assertion a) follows from2 the Cauchy-Bunyakovskii l (x - xo, Y - Yo) l :::; ll x - xo ll 2 · IIY - Yo ll 2 · Assertion b) follows from a) , since
Proof. 10. 1 ) :
and i=I:00n+l Xi ---* 0 as n ---* oo. Assertion ) follows by repeated application of b) , taking account of the relation (x, y) = (y, x). D The following result is an immediate consequence of the lemma. Theorem 1. ( Pythagoras4 ) . c
a) If {x i } is a system of mutually orthogonal vectors and x =
ll x ll 2 = I:i ll xi ll 2 •
I:i xi, then
4 Pythagoras of Samos (conjectured to be 580-500 BCE) - famous ancient Greek
mathematician and idealist philosopher, founder of the Pythagorean school, which, in particular, made the discovery that the side and diagonal of a square are incommensurable, a discovery that disturbed the ancients. The classical Pythagorean theorem itself was known in a number of countries long before Pythagoras (possibly without proof, to be sure) .
506
18 Fourier Series and the Fourier Transform
b) If { ei } is an orthonormalized system of vectors and x = L:i xi e i , ll x ll 2 = L:i I xi 1 2
then
•
1 8 . 1 . 2 Fourier Coefficients and Fourier Series
a. Definition of the Fourier Coefficients and the Fourier Series
Let
orthonormal system( ,and) . { li } an orthogonal system of vectors in i } be Xan with a{ espace inner product Suppose that x = L:i xi zi . The coefficients xi in this expansion of the vector x can be found directly: X i = (l(x,i , llii )) . If li = e i , the expression becomes even simpler: We remark that the formulas for x i make sense and are completely deter mined if the vector x itself and the orthogonal system { l i } (or { e i } ) are given.i The to compute x equality x = L:i xi zi (or x = L:i xi e i ) is no longer needed from these formulas. Definition 5 . The numbers { (�·t� } are the Fourier coefficients of the vector x E X in the orthogonal system { l i }. If the system { e i } is orthonormal, the Fourier coefficients have the form {(x, e i )}. Fromx EtheX geometric point ofof view the ithinFourier coefficient (x, e i ) of the vector is the projection that vector the direction of the unit vector e i . In the familiar case of three-dimensional Euclidean ispace E3 with a given orthonormal frame eof1 , ethe2 , e3vector the Fourier coefficients x = (x, e i ), i = 1 , 2, 3, are the coordinates x in the basis e , 1 e 2 , e 3 appearing in the expansion x = x 1 e 1 + x 2 e 2 + x3 e 3 . If we were given only the two vectors e 1 and e 2 instead of allnotthreebe e 1 , e 2 , e 3 , the expansion x = x 1 e 1 +x 2 e 2 in this system would certainly valid for all vectors x E E3 . Nevertheless, the Fourier coefficients xi 2= (x, e i ), 1 i = 1 , 2, would be defined in this case and the vector Xe = x e 1 of+ xthee 2vectors would bee and the orthogonal projection of the vector x onto the plane L e 2 . Among all the vectors in that plane, the vector Xe is distinguished by1 E being closest to x in the sense that ll x - Yll llx - Xe ll for any vector L. This is the remarkable extremal property of the Fourier coefficients, to which we shall return below in the general situation. y
�
18. 1 Basic General Concepts Connected with Fourier Series
507
Ifis anX orthogonal is a vectorsystem spaceof with innervectors product , ) andfor ( then l , . . . , l , . . . nonzero in X , each2 vectorn x E X one can form the series (18. 8 ) X k�=l ((lx,k , llkk)) lk . This series is the Fourier series of x in the orthogonal system {lk }. IfIn thethe system {lank }orthonormal is finite, the system Fourier {series reduces toseries its finiteof asum. the Fourier vector case of e } k x E X has a particularly simple expression: 00 (18. 8 ') x rv L ( x, e k ) e k . k=l Example Let X = 'R. 2 ( [ -1r, 7r] , � ) . Consider the orthogonal system {1,coskx,sinkx; k E N} ofseriesExample 1. To the function f E 'R-2 ( [ - 1r, 1r], �) there corresponds a Fourier f rv ao;n + kf=l ak (f) cos kx + bk (f) sin kx in this system. The tocoefficient � is included in the zeroth term so as to give a unified appearance the following formulas, which follow from the definition of the Fourier coefficients: Definition 6.
h,
"'
�
-
6.
.,..
(18. 9 ) ak (f) = � � f(x) coskxdx , k = 0, 1, 2, . . . , k = 1, 2, . . . . (18.10) bk (f) = ;1 I f(x) smkxdx . = ( -1) k + l f , Let us set f(x) = x. Then a 0, k = 0, 1, 2, . . . , and b = k k k = 1, 2, . . .. Hence in this case we obtain 00 k+ l k2 sin kx . f(x) = X rv L( -1) k=l Example Let us consider the orthogonal system { e i k x ; k E Z} of Example 1 in5 and the space 'R.2 ( [ - 7r , 1r] , C). Let f E 'R.2 ( [ - 7r , 7r] , C ) . According to Definition relations (18. 4by), the Fourier coefficients {eikx } are expressed the following formula: {ck (f)} of f in the system kx (18.11) Ck (f) = 21�,. I f(x)e -i kx dx ( ((f(x),e kei x , eiikx )) ) .,..
- 71"
7.
.,..
- 71"
_
_
·
508
18 Fourier Series and the Fourier Transform
Comparing Eqs. ( 18.9 ) , ( 18. 10 ) , and ( 18. 1 1 ) and taking account of Euler ' s formula eicp = cos cp + i sin cp, we obtain the following relations between the Fourier coefficients of a given function in the trigonometric systems written in real and complex forms: if k 2: 0 , if k < O .
( 18. 12 )
In order that the case k = 0 not be an exception in formulas ( 18.9 ) and a 0 to denote not the Fourier coefficient itself, but rather its double, as was done above. ( 1 8 . 1 2 ) , it is customary to use
b. Basic General Properties of Fourier Coefficients and Series The following geometric observation is key in this section.
Lemma ( Orthogonal complement ) . Let {l k } be a finite or countable system of nonzero pairwise orthogonal vectors in X , and suppose the Fourier series of x E X in the system { l k } converges to X! E X . Then in the representation x = X! + h the vector h is orthogonal to x 1 ; moreover, h is orthogonal to the entire linear subspace generated by the system of vectors { l k } , and also to its closure in X.
Proof. Taking account of the properties of the inner product , we see that it suffices to verify that ( h , lm) = 0 for every lm E {l k } · We are given that
(x, l k ) l k · � h = X - X! = X - " (lk , lk ) k
-
Hence
Geometrically this lemma is transparent , and we have already essentially pointed it out when we considered a system of two orthogonal vectors in three-dimensional Euclidean space in Subsect . 18. 1 . 2a. On the basis of this lemma we can draw a number of important general conclusions on the properties of Fourier coefficients and Fourier series. Bessel's inequality
Taking account of the orthogonality of the vectors X! and h in the de composition x = X! + h , we find by the Pythagorean theorem that ll x ll 2 = ll xd l 2 + ll h ll 2 2: ll x d l 2 ( the hypotenuse is never smaller than the leg ) . This relation, written in terms of Fourier coefficients, is called Bessel ' s inequality.
18.1 Basic General Concepts Connected with Fourier Series
Let us write it out. By the Pythagorean theorem
509
(18. 13)
Hence
"'"""' l(x,(l llk )l) 2 - I I X 1 2 • (18. 14) k kk This is Bessel orthonormal system's inequality. of vectorsIt{ ehask }: a particularly simply appearance for an ( 18 . 15) L l(x, e kW l x l 2 . k In terms of the Fourier coefficients O:k themselves Bessel ' s inequality ( 18. 14) can be written as I: l o: k l 2 l l k l 2 :::; l x l 2 , which in the case of an orthonormal system reduces kto I:k l o: k l 2 :::; l x l 2 . We allowing have included thevectors absolutespaces valueX.sigInn this in thecaseFourier coefficient, since wemayare complex the Fourier coefficient assume complex values. Bessel ' s inequality we made use of the assump We note that in deriving tion that the vector x1 exists and that Eq. (18. 13) holds. But if the system { l k } is finite, there is no doubt that the vector Xt does exist (that is, that the Fouri er seriesof {lconverges in X). Hence inequality (18. 14) holds for every finite subsystem },k and then it must hold for the whole system as well. Example For the trigonometric system (see formulas (18.9) and ( 18. 10)) 's inequality has the form Bessel l ao � W + f l a k ( J W + l bk (f) l 2 � J l f l 2 (x) dx · ( 18. 16) k=l ikx ; k E Z} (see formula ( 1 8. 1 1 ) ) Bessel ' s inequality can {e For the system be written in a particularly elegant form: L...t
<
:S:
8.
:S:
71"
- 11"
(18. 1 7) Convergence of Fourier series in a complete space
Suppose I:k xk ek = l:k (x, ek ) ek is the Fourier series of the vector x E X ' inI: thexk 2orthonormal system {ek }. By Bessel theorems inequality (18.15) the series lk l converges. By the Pythagorean
18 Fourier Series and the Fourier Transform
510
By the Cauchy convergence criterion for a series, the right-hand side of this equality becomes less than any e > 0 for all sufficiently large values of m and n > m. Hence we then have
ll xm e m + · · + xn e n ll < ,fi · Consequently, the Fourier series E x k e k satisfies the hypotheses of the k Cauchy convergence criterion for series and therefore converges provided the original space X is complete in the metric induced by the norm ll x ll = y' (x, x). ·
To simplify the writing we have carried out the reasoning for a Fourier series in an orthonormal system. But everything can be repeated for a Fourier series in any orthogonal system. The extremal property of the Fourier coefficients We shall show that if the Fourier series E x k e k =
E ((ekx, ,eekk)) e k of the k k vector x E X in the orthonormal systsem { e k } converges to a vector x 1 E X, then the vector X! is precisely the one that gives the best approximation of x 00 among all vectors = E o: k e k of the space L spanned by {e k }, that is, for k= l every E L, ll x - x d l � ll x - Yll , and equality holds only for = x 1 • Indeed, by the orthogonal complement lemma and the Pythagorean the y
y
y
orem,
ll x - Yll 2 = ll (x - X!) + (x l - Y ) ll 2 = llh + (xl - Y ) ll 2 = = llhll 2 + ll x 1 - Yll 2 � ll hll 2 = ll x - x 1 ll 2 · Example 9. Digressing slightly from our main purpose, which is the study of expansions in orthogonal systems, let us assume that we have an arbitrary system of linearly independent vectors x1 , . • . , X n in X and are seeking the n best approximation of a given vector x E X by linear combinations E O: k X k k =l o f vectors o f the system. Since we can use the orthogonalization process to construct an orthonormal system e 1 , . . . , e n that generates the same space L that is generated by the vectors X 1 , . . . , X n , we can conclude from the extremal property of the Fourier coefficients that there exists a unique vector X ! E L such that ll x - x d l = inf ll x - Y ll · Since the vector h = x - X ! is orthogonal to yE L the space L, from the equality x 1 + h = x we obtain the system of equations � (x ) ( · · x,) , + . ( · .. .. .. ( 18. 18) . (x l , Xn )o: l + + (x n , Xn )O: n - (x, Xn )
{�
� :: � �� �:)�� · · ·
:�
·
.
18.1 Basic General Concepts Connected with Fourier Series
511
n O: of the unknown = l k Xk k system x 1 , . . • , X n· The existence
for the coefficients a: 1 , . . . , O: n of the expansion X !
=
I:
vector X! in terms of the vectors of the and uniqueness of the solution of this system follow from the existence and uniqueness of the vector x 1 • In particular, it follows from this by Cramer ' s theorem that the determinant of this system is nonzero. In other words, we have shown as a by-product that the Gram determinant of a system of linearly independent vectors is nonzero. This approximation problem and the system of equations ( 18. 18) corre sponding to it arise, as we have already noted, for example, in processing experimental data by Gauss ' least-squares method. (See also Problem 1 . )
c.
Complete Orthogonal Systems and Parseval's Equality
Definition 7. The system {x0; a: E A} of vectors of a normed space X is complete with respect to the set E C X (or complete in E) if every vector x E E can be approximated with arbitrary accuracy in the sense of the norm of X by finite linear combinations of vectors of the system.
If we denote by L {xa} the linear span in X of the vectors of the system (that is, the set of all finite linear combinations of vectors of the system) , Definition 7 can be restated as follows: The system {X a } is complete with respect to the set E C X if E is con tained in the closure L{ X a} of the linear span of the vectors of the system.
Example 1 0. If X = E3 and e� , e 2 , e 3 is a basis in E3 , then the system { e 1 , e 2 , e 2 } is complete in X . The system { e 1 , e 2 } is not complete in X , but it is complete relative to the set L{ e 1 , e 2 } or any subset E of it.
Example 1 1 . Let us regard the sequence of functions 1, x, x 2 , • • . as a system of vectors {x k ; k = 0, 1 , 2, . . . } in the space R2 ( [a, b] , JR.) or R2 ( [a, b] , C) . If C[a, b] is a subspace of the continuous functions, then this system is complete with respect to the set C[a, b]. Proof Indeed, for any function f E C[a, b] and for every number e > 0, the Weierstrass approximation theorem implies that there exists an algebraic polynomial P(x) such that max l f(x) - P(x) l < e. But then .
x E [a , b]
II ! - P ll :=
b
J I f - P l 2 (x) dx < evb - a a
and hence one can approximate the function r in the sense of the norm of the space R2 ( [a, bl ) with arbitrary accuracy. D
512
18 Fourier Series and the Fourier Transform
We note that, in contrast to the situation in Example 9, in the present case not every continuous function on the closed interval [a, b] is a finite linear combination of the functions of this system; rather, such a function can only be approximated by such linear combinations. Thus C[a, b] C L{ x n } in the sense of the norm of the space R2 [a, b].
Example 1 2. If we remove one function, for example the function 1 , from the system { 1 , cos kx, sin kx; k E N}, the remaining system { cos kx, sin kx; k E N} is no longer complete in R2 ( [-1r, 1r] , C ) or R2 ( [ - 1r, 1r], 1R.) . Proof. Indeed, by Lemma 3 the best approximation of the function f(x) among all the finite linear combinations
=
1
n Tn (x) = 2_)a k cos kx + bk sin kx ) k=l
of any length n is given by the trigonometric polynomial Tn (x)in which a k and b k are the Fourier coefficients of the function 1 with respect to the orthogonal system { cos kx, sin kx; k E N}. But by relations ( 18.5), such a polynomial of best approximation must be zero. Hence we always have
and it is impossible to approximate 1 more closely than /2ii by linear com binations of functions of this system. D Theorem ( Completeness conditions for an orthogonal system ) . Let X be a vector space with inner product ( , ) , and h , l 2 , . . . , ln , . . . a finite or count able system of nonzero pairwise orthogonal vectors in X . Then the following conditions are equivalent: a) the system { l k } is complete with respect to the set!' E c X; b) for every vector x E E c X the following (Fourier series) expansion holds: (x, l k ) k ; (18. 19) X=" L...., k (lk , lk )
c ) for every vector x E E C X Parseval 's6 equality holds:
(18. 20) 5 The set E may, in particular, consist of a single vector that is of interest for one reason or another. 6 M.A. Parseval (1755-1836) - French mathematician who discovered this relation for the trigonometric system in 1799.
18.1 Basic General Concepts Connected with Fourier Series
513
Equations ( 18. 19) and ( 18.20) have a particularly simple form in the case of an orthonormal system { e k } . In that case ( 18. 19' ) and
ll x ll = L l (x, e k ) l 2 · k
( 18.20' )
Thus the important Parseval equality (18.20) or ( 18.20') is the Pythagorean theorem written in terms of the Fourier coefficients. Let us now prove this theorem.
Proof. a ) :::} b) by virtue of the extremal property of Fourier coefficients; b) :::} c) by the Pythagorean theorem; c) :::} a ) since by the lemma on the orthogonal complement (see Subsub sect . b) above) the Pythagorean theorem implies
Remark. We note that Parseval ' s equality implies the following simple nec essary condition for completeness of an orthogonal system with respect to a set E C X: E does not contain a nonzero vector orthogonal to all the vectors in the system. As a useful supplement to this theorem and the remark just made, we prove the following general proposition. Proposition. Let X be a vector space with an inner· product and x 1 , x 2 , . . . a system of linearly independent vectors in X . In order for· the system { x k } to be complete in X , a ) a necessary condition is that there be n o nonzero vector i n X orthogonal to all the vectors in the system ; b) if X is a complete (Hilbert) space, it suffices that X contain no nonzero vector orthogonal to all the vectors in the system. Proof. a) If the vector h is orthogonal to all the vectors in the system {x k }, we conclude by the Pythagorean theorem that no linear combination of vectors in the system can differ from h by less than llhll · Hence, if the system is complete, then llhll = 0. b) By the orthogonalization process we can obtain an orthonormal system { e k } whose linear span L{ e k } is the same as the linear span L{ x k } of the original system. We now take an arbitrary vector x E X. Since the space X is complete, the Fourier series of x in the system { e k } converges to a vector Xe E X. By the
514
18 Fourier Series and the Fourier Transform
lemma on the orthogonal complement, the vector h = x - Xe is orthogonal to the space L{ek } = L{xk } . By hypothesis h = 0, so that x = Xe , and the Fourier series converges to the vector x itself. Thus the vector x can be approximated arbitrarily closely by finite linear combinations of vectors of the system { e k } and hence also by finite linear combinations of the vectors of the system {xk } . 0 The hypothesis of completeness in part b ) of this proposition is essential, as the following example shows.
Example 1 3. Consider the space l 2 ( see Sect. 10.1) of real sequences a = 00 1 2 (a , a , . . . ) for which j=I: (ai) 2 oo . We define the inner product of the l vectors a = (a 1 , a 2 , . . . ) and b = (b 1 , b 2 , . . . ) in l 2 in the standard way: (a, b) := 00 2:: aibi. j= l Now consider the orthonormal system e k (0, . . . , 0, 1, 0, 0, . . . ) k k = 1, 2, . . . . The vector e 0 = (1, 0, 0, . . . ) does not belong to this system. We now add to the system {e k ; k E N} the vector e = (1, 1/ 2 , 1/ 2 2 , 1/ 2 3 , . . . ) and consider the linear span L{ e, e 1 , e 2 , . . . , } of these vectors. We can regard this linear span as a vector space X ( a subspace of l 2 ) with the inner product from l 2 . We note that the vector e0 = (1, 0, 0, . . . ) obviously cannot be ob tained as a finite linear combination of vectors in the system e, e 1 , e 2 , . . . , and therefore it does not belong to X . At the same time, it can be ap proximated as closely as desired in l 2 by such linear combinations, since e - I:=n -dt e k = (1, 0, . . . , 0, 2 n\ 1 , 2 n\2 , · · · ) · k l Hence we have established simultaneously that X is not closed in l 2 ( and therefore X , in contrast to l 2 , is not a complete metric space ) and that the closure of X in l 2 coincides with l 2 , since the system e0, e 1 , e 2 , . . . generates the entire space l 2 . We now observe that in X = L{ e, e 1 , e 2 , . . . } there is no nonzero vector orthogonal to all the vectors e 1 , e 2 , . . . . n Indeed, let x E X , that is, x = ae + 2:: a k e k , and let (x, e k ) 0, k = k= 1, 2, . . . . Then (x, e n+l ) = 2 n� ' = 0, that is, al = 0. But then O k = (x, e k ) = 0, k = 1, . . . , n. Hence we have constructed the required example: the orthogonal system e 1 , e 2 , . . . is not complete in X , since it is not complete in the closure of X, which coincides with l 2 . This example is of course typically infinite-dimensional. Fig. 18.1 repre <
�
=
sents an attempt to illustrate what is going on.
18. 1 Basic General Concepts Connected with Fourier Series
515
Fig. 1 8 . 1 .
We note that in the infinite-dimensional case (which is so characteristic of analysis) the possibility of approximating a vector arbitrarily closely by linear combinations of vectors of a system and the possibility of expanding the vector in a series of vectors of the system are in general different properties of the system. A discussion of this problem and the concluding Example 14 will clarify the particular role of orthogonal systems and Fourier series for which these properties hold or do not hold simultaneously (as the theorem proved above shows) . Definition 8 . The system x 1 , x 2 , . . . , X n , . . . of vectors of a normed vector space X is a basis of X if every finite subsystem of it consists of linearly inde pendent vectors and every vector x E X can be represented as x = 2::: O: k X k l
k
where o: k are coefficients from the scalar field of X and the convergence (when the sum is infinite) is understood in the sense of the norm on X . How i s the completeness of a system of vectors related t o the property of being a basis? In a finite-dimensional space X completeness of a system of vectors in X, as follows from considerations of compactness and continuity, is obviously equivalent to being a basis in X . In the infinite-dimensional case that is in general not so.
Example 14. Consider the set C ( [ - 1, 1 ] , 1R ) of real-valued functions that are continuous on [ - 1, 1] as a vector space over the field lR with the standard inner product defined by (18.3). We denote this space by C2 ( [ - 1, 1 ] , 1R ) and consider the system of linearly independent vectors 1, x, x 2 , . . . in it . This system is complete in C2 ( [ - 1, 1], 1R ) (see Example 11), but is not a basis. ()()
2::: o: k x k converges in C2 ( [ - 1, 1] , 1R ) , k=O that is, in the mean-square sense on [ - 1, 1], then, regarded as a power series, it converges pointwise on the open interval ] - 1, 1 [. Proof. \Ve first show that if the series
516
18 Fourier Series and the Fourier Transform
Indeed, by the necessary condition for convergence of a series, we have
ll a k x k II --+ 0 as k --+ oo. But
1 2 k ll a k x ll = J ( a k x k ) 2 dx = a� 2k � 1 . -1 Hence l a k l J2k + 1 for all sufficiently large values of k. In that case 00 the power series I: a k x k definitely converges on the interval ] - 1 , 1 [. k =O We now denote the sum of this power series on ] - 1, 1 [ by We remark that on every closed interval [a, b] c] - 1, 1 [ the power series converges uni <
cp .
formly to cp I [a , b] " Consequently it also converges in the sense of mean-square deviation. It now follows that if a continuous function f is the sum of this series in the sense of convergence in C2 ( [ - 1, 1], then f and cp are equal on ] - 1, 1[. But the function cp is infinitely differentiable. Hence if we take any function in C2 ( [ - 1, 1], lR) that is not infinitely differentiable on ] - 1, 1 [ it cannot be expanded in a series in the system { x k ; k = 0, 1, . . . } . D
IR),
Thus, if we take, for example, the function x = l x l and the sequence of numbers {en = * ; E N } , we can construct a sequence {Pn (x); E N} of finite linear combinations Pn (x) = a0 + a 1 x + · · · + a n x n of elements of the system {x k ; k E N} such that II ! - Pn ll < * ' that is, Pn f as If necessary, one could assume that in each such linear combination Pn (x) the coefficients can be assumed to have been chosen in the unique best-possible 00 way ( see Example 9). Nevertheless, the expansion f = I: a k x k will not arise
n
--+
n n --+ oo.
k =O
since in passing from Pn ( x) to Pn+l ( x), not only the coefficient an+l changes, but also possibly the coefficients ao , . , a n . If the system is orthogonal, this does not happen ( no , . . . , an do not change ) because of the extremal property of Fourier coefficients. For example, one could pass from the system of monomials {x k } to the or thogonal system of Legendre polynomials and expand f(x) = l x l in a Fourier series in that system.
..
1 8 . 1 . 3 *An Important Source of Orthogonal Systems of Functions in Analysis
We now give an idea as to how various orthogonal systems of functions and Fourier series in those systems arise in specific problems.
Example 1 5. The Fourier method. Let us regard the closed interval [ 0, l ] as the equilibrium position of a homogeneous elastic string fastened at the endpoints of this interval, but
18. 1 Basic General Concepts Connected with Fourier Series
517
otherwise free and capable of making small transverse oscillations about this equilibrium position. Let u(x, t) be a function that describes these oscilla tions, that is, at each fixed instant of time t = t0 the graph of the function u(x, t0 ) over the closed interval 0 :::; x :::; l gives the shape of the string at time t0 . This in particular, means that u(O, t) = u(l, t) = 0 at every instant t, since the ends of the string are clamped. It is known (see for example Sect . 14.4) that the function u(x, t) satisfies the equation EPu = a2 82 u , ( 18. 2 1)
8t 2
8x 2
where the positive coefficient a depends on the density and elastic constant of the string. The equation ( 18.21) alone is of course insufficient to determine the func tion u(x, t). From experiment we know that the motion u(x, t) is uniquely determined if, for example, we prescribe the position u(x, 0) = cp(x) of the string at some time t = 0 ( which we shall call the initial instant) and the velocity �� (x, 0) = 1./J(x) of the points of the string at that time. Thus, if we stretch the string into the shape cp(x) and let it go, then 'lj;(x) = 0 . Hence the problem of free oscillations of the string7 that is fixed at the ends of the closed interval [0, l] has been reduced to finding a solution u(x, t) of Eq. ( 18.21) together with the boundary conditions
u(O, t) = u(l, t) = 0
( 18. 22 )
and the initial conditions
u(x, 0) = cp(x) ,
(18.23)
To solve such problems there exists a very natural procedure called the method of separation of variables or the Fourier method in mathematics. It consists of the following. The solution u(x, t) is sought in the form of a series 00 I: Xn (x)Tn (t) whose terms X(x)T(t) are solutions of an equation of special
n= l
form (with variables separated) and satisfy the boundary conditions. In the present case, as we see, this is equivalent to expanding the oscillations u(x, t) into a sum of simple harmonic oscillations (more precisely a sum of standing waves) . Indeed, if the function X(x)T(t) satisfies Eq. (18.2 1 ) , then X(x)T" (t) = a2 X"(x)T(t), that is, T" ( t) X"(x) ( 18. 2 4) ' 2
a T(t)
X(x) In Eq. (18. 2 4 ) the independent variables x and t are on opposite sides of
the equation (they have been separated) , and therefore both sides actually 7 We note that the foundations of the mathematical investigation of the oscillations of a string were laid by Brook Taylor.
518
18 Fourier Series and the Fourier Transform
represent the same constant .\. If we also take into account the boundary conditions X(O)T(t) = X(l)T(t) = 0 that the solution of stationary type must satisfy, we see that finding such a solution reduces to solving simultaneously the two equations
T"(t) = .\ a 2 T(t) , X" (x) = .\X(x) under the condition that X (O) = X (l) = 0.
(18. 25) (18.26)
It is easy to write the general solution of each of these equations individ ually:
T(t) = A cos V>..at + B sin V>..at , X(x) = C cos J>..x + D sin J>..x .
(18. 2 7) (18.28)
If we attempt to satisfy the conditions X (O) = X (l) = 0, we find that for .\ =/= 0 we must have C = 0, and, rejecting the trivial solution D = 0, we find that sin ../5.l = 0, from which we find ../). = ±mr /l , n E N. Thus it turns out that the number .\ in Eqs. ( 18.25) and (18.26) can be chosen only among a certain special series of numbers (the so-called eigenval ues of the problem) , An = (mr/l) 2 , where n E N. Substituting these values of .\ into the expressions ( 18.27) and (18. 2 8) , we obtain a series of special solutions 1r
(
)
na t + B sm . na u11 (x, t) = sm n y x An cos n l n nlt , .
(18. 2 9)
satisfying the boundary conditions u n (O, t) = u n (l , t) = 0 (and describing a standing wave of the form
(18.30)
18. 1 Basic General Concepts Connected with Fourier Series
and
519
( 18.31) '1/J ( x) = n"L.= l n 7 Bn sinn y x · Thus the problem has been reduced to finding the coefficients An and Bn , which up to now have been free, or, what is the same, to expanding the Fourier [0,series in the system { sinnfx; n E N } , which functions is orthogonalandon '1/Jtheininterval l] . to remark that the functions { sin nfx; n E N } , which arose fromIt Eq.is useful ( 18.26) can be regarded as eigenvectors of the linear operator A = � corresponding to the eigenvalues A n = nf, which in turn arose from the condition that the operator A acts on the space of functions in c< 2 ) [0, l] that vanish at the endpoints of the closed interval [0, l] . Hence Eqs. (18.30) and ( 18.31) can be interpreted as expansions in eigenvectors of this linear operator. linearofoperators connected with particularin analysis. problems are one of the mainThe sources orthogonal systems of functions nowsystems recall another fact known from algebra, which reveals the reason whyWe such are orthogonal. Let Z equal be a vector space with inner product ( , ) , and let E be a subspace (possibly to Z itself) that is dense in Z. A linear operator A : E --+ Z iseigenvectors symmetric if ( Ax, y ) = ( x, Ay ) for every pair of vectors x, y E E. Then: of a symmetric operator corresponding to different eigenvalues 00
cp
are orthogonal.
Proof.
Indeed, if Au = au and Av = (3v , and a =/= (3, then a (u, v) = (Au, v) = (u, Av) = (3 (u, v) ,
from which it follows that (u, v) = 0. 0 It is now useful to look at Example 3 from this point of view. There we were essentially considering the eigenfunctions of2the operator A = ( � onclosedthe interval space of[a,functions in C ( ) [a, b] that vanish at the q(x)) operating endpoints of the b] . Through integration by parts one can verify thatinnerthisproduct operator( 18.4)is symmetric on this space (with respect to the standard ) , so that the result of Example 4 is a particular manifestation of this algebraic fact. In particular, when q(x) 0 the operator A becomes d�2 , which for [a, b] = [0, l] occurred in the last example (Example 15) . We note and also that(seeinrelations this example theandquestion reduced to expanding the functions (18.30) ( 18.31)) in a series of eigenfunc '1/J A = d� Here of course the question arises whether tions of the operator 2• itequiisvtheoretically possible to formtosuchthe anquestion expansion, this question alent, as we now understand, of theandcompleteness of theis +
=
cp
18 Fourier Series and the Fourier Transform
520
system of eigenfunctions for the operator in question in the given space of functions. The completeness of the trigonometric system (and certain other particu lar systems of orthogonal functions) in R2 [-1r, 1r] seems to have been stated explicitly for the first time by Lyapunov.8 The completeness of the trigono metric system in paraticular was implicitly present in the work of Dirich let devoted to studying the convergence of trigonometric series. Parseval ' s equality, which is equivalent to completeness for the trigonometric system, as already noted, was discovered by Parseval at the turn of the nineteenth cen tury. In its general form, the question of completeness of orthogonal systems and their application in the problems of mathematical physics were one of the main subjects of the research of Steklov,9 who introduced the very con cept of completeness ( closedness) of an orthogonal system into mathematics. In studying completeness problems, by the way, he made active use of the method of integral averaging (smoothing) of a function (see Sects. 17.4 and 17.5) , which for that reason is often called the Steklov averaging method. 1 8 . 1 .4 Problems and Exercises
The method of least squares. The dependence y = j ( x 1 , . . . , xn ) of the quantity on the quantities x 1 , . . . , Xn is studied experimentally. As a result of m (� n ) experiments, a table was obtained 1.
y
X1
X2
Xn
I Y
. .�L . � � . . . . . . . . _t_t�. I b l -
-
each of whose rows contains a set ( ai , a�, . . . , a�) of values of the parameters and the value b' of the quantity y corresponding to them, measured by some device with a certain precision. From these experimental data we would n like to obtain an empirical formula of the form y = 2::: a;x; convenient for com i= l putation. The coefficients a 1 , 0: 2 , . . . , an of the required linear function are to be X 1 , X 2 , . . . , Xn
(
)
chosen so as to minimize the quantity f bk - f= a;a7 , which is the meank= l k= l square deviation of the data obtained using the empirical formula from the results obtained in the experiments. 8
2
A . M. Lyapunov (1857-1918) - Russian mathematician and specialist in mechan ics, a brilliant representative of the Chebyshev school, creator of the theory of stability of motion. He successfully studied various areas of mathematics and mechanics. 9 V.A. Steklov ( 1864-1926) - Russian/Soviet mathematician, a representative of the Petersburg mathematical school founded by Chebyshev and founder of the school of mathematical physics in the USSR. The Mathematical Institute of the Russian Academy of Sciences bears his name.
18.1 Basic General Concepts Connected with Fourier Series
521
Interpret this problem as the problem of best approximation of the vector (b 1 , . . . , brn ) be linear combinations of the vectors (a} , . . . , ai ) , i = 1 , . . . , n and show that the question reduces to solving a system of linear equations of the same type as Eq. {18. 18) . 2. a) Let O[a, b] be the vector space of functions that are continuous on the closed interval [a, b] with the metric of uniform convergence and 02 [a, b] the same vector space but with the metric of mean-square deviation on that closed interval (that is, b
J I f - g l 2 (x) dx) . Show that if functions converge in O[a, b] , they also converge in 02 [a, b] , but not conversely, and that the space 02 [a, b] is not complete, in contrast to O[a, b] . b) Explain why the system of functions { 1 , x, x 2 , . . . , } is linearly independent and complete in 02 [a, b] ' but is not a basis in that space. c) Explain why the Legendre polynomials are a complete orthogonal system and also a basis in 02 [- 1 , 1] . d) Find the first four terms of the Fourier expansion of the function sin JrX on the interval [ - 1 , 1] in the system of Legendre polynomials. e) Show that the square of the norm I I Pn ll in 02 [- 1 , 1] of the nth Legendre polynomial is d(f, g) =
2 2n + 1
(
�
2) · · · 2n (n + 1) = (-1t 2�n
j
)
(x 2 - 1 t dx .
-1 f) Prove that among all polynomials of given degree n with leading coefficient 1 , the Legendre polynomial Pn (x) is the one closest to zero on the interval [- 1 , 1] . g) Explain why the equality
where {Po , P1 , . . . } is the system of Legendre polynomials, necessarily holds for every function f E 02 ( [- 1 , 1] ,
3. a) Show that if the system {x 1 , x 2 , . . . , } of vectors is complete in the space X and X is an everywhere-dense subset of Y, then {x 1 , x 2 , . . . , } is also complete in Y. b) Prove that the vector space O[a, b] of functions that are continuous on the closed interval [a, b] is everywhere dense in the space R 2 [a, b] . (It was asserted in Problem 5g of Sect. 17.5 that this is true even for infinitely differentiable functions of compact support on [a, b] .) c) Using the Weierstrass approximation theorem, prove that the trigonometric system { 1 , cos kx, sin kx; k E N} is complete in R 2 [-1r, 1r] . d ) Show that the systems { 1 , x , x 2 , . . . } and { 1 , cos kx, sin kx; k E N} are both complete in R 2 [ -Jr, 1r] , but the first is not a basis in this space and the second is.
522
18 Fourier Series and the Fourier Transform e) Explain why Parseval's equality 1 :;
rr
J
- rr
holds, where the numbers
ao 2
l -l + � 2 2 I f I 2 (x) dx = L_., l a k I + l b k l 2 k= l oo
a k and bk are defined by (18.9) and (18. 10) .
f) Using the result of Example 8, now show that I:: � = 00
rr 2 6
•
n= l 4. Orhogonality with a weight function. a) Let po, . . . , Pn be continuous functions that are positive in the domain D . Verify that the formula Pl ,
J P (x)f(k)(x)g(k)(x)dx
k=O D k defines an inner product in C ( n ) (D, C) . (!, g) = t
b) Show that when functions that differ only on sets of measure zero are iden tified, the inner product
(!, g) =
J p( x) f( x)g( x) dx ,
D
involving a positive continuous function p can be introduced in the space R(D, C) . The function p here is called a weight function, and if (!, g, ) = 0, we say that the functions f and g are orthogonal with weight p. c) Let 'P : D -+ G be a diffeomorphism of the domain D C R n onto the domain G C JR n , and let { u k (y); k E N} be a system of functions in G that is orthogonal with respect to the standard inner product ( 18.2 ) or (18.3 ) . Construct a system of functions that are orthogonal in D with weight p( x) = I det i.p1 ( x) I and also a system of functions that are orthogonal in D in the sense of the standard inner product. d) Show that the system of functions {e rn , n (x, y) = e i(rnx+n y ) ; m, n E N} is orthogonal on the square I = {(x, y) E R 2 l l x l :::; 1r 1\ IYI :S: 1r } . e) Construct a system of functions orthogonal on the two-dimensional torus T2 c R2 defined by the parametric equations given in Example 4 of Sect. 12. 1 . The inner product o f functions f and g o n the torus i s understood as the surface integral J fg da. T2
5 . a) It is known from algebra (and we have also proved it in the course of discussing constrained extremal problems) that every symmetric operator A : E n -+ E n on n-dimensional Euclidean space E n has nonzero eigenvectors. In the infinite dimensional case this is generally not so. Show that the linear operator f ( x) >-+ x f ( x) of multiplication by the independent variable is symmetric in C2 ( [a, b], R ) , but has no nonzero eigenvectors.
18. 1 Basic General Concepts Connected with Fourier Series
523
b) A Sturm 10 -Liouville problem that often arises in the equations of mathemati cal physics is to find a nonzero solution of an equation u"(x)+ [q(x)+Ap(x)]u(x) = 0 on the interval [a, b] satisfying certain boundary conditions, for example u(a) = u(b) = 0. Here it is assumed that the functions p(x) and q(x) are known and continuous on the interval [a, b] in question and that p(x) > 0 on [a, b]. We have encountered such a problem in Example 15, where it was necessary to solve Eq. (18.26) under the condition X (O) = X(l) = 0. In this case we had q(x) = 0, p(x) = 1 , and [a, b] = [0, l] . We have verified that a Sturm-Liouville problem may in general turn out to be solvable only for certain special values of the parameter A, which are therefore called the eigenvalues of the corresponding Sturm -Liouville problem. Show that if the functions f and g are solutions of a Sturm-Liouville problem corresponding to eigenvalues AJ f. A9, then the equality d: (g' f - !'g) = (AJ A9 )pf g holds on [a, b] and the functions f and g are orthogonal on [a, b] with weight p. c) It is known (see Sect. 14.4) that the small oscillations of an inhomogeneous string fastened at the ends of the closed interval [a, b] are described by the equation (pu�)� = pu�� ' where u = u(x, t) is the function that gives the shape of the string at each time t, p = p(x) is the linear density, and p = p(x) is the elastic constant at the point x E [a, b] . The clamping conditions mean that u(a, t) = u(b, t) = 0. Show that if we seek the solution of this equation in the form X(x)T(t), the question reduces to a system T" = AT, (pX')' = ApX, in which A is the same number in both equations. Thus a Sturm-Liouville problem arises for the function X(x) on the closed interval [a, b], which is solvable only for particular values of the parameter A (the eigenvalues) . (Assuming that p ( x ) > 0 on [a, b] and that p E C ( l l [a, b] we can obviously bring the equation (pX' )' = ApX into a form in which the first derivative is missing by the change of variable x = f !ch . ) a
d) Verify that the operator S(u) = (p(x)u'(x))' - q(x)u(x) on the space of functions in c< 2 l [a, b] that satisfy the condition u(a) = u(b) = 0 is symmetric on this space. (That is, (Su, v) = (u, Sv), where ( , ) is the standard inner produt of real-valued functions.) Verify also that the eigenfunctions of the operator S corre sponding to different eigenvalues are orthogonal. e) Show that the solutions X 1 and X2 of the equation (pX')' = ApX corre sponding to different values A 1 and A 2 of the parameter A and vanishing at the endpoints of the closed interval [a, b] are orthogonal on [a, b] with weight p(x) . 6. The Legendre polynomials as eigenfunctions. a) Using the expression of the Legendre polynomials Pn ( x ) given in Example 5 and the equality (x 2 - 1 ) n = (x - 1t(x + 1) n , show that Pn (1) = 1 .
1 0 J.Ch.F. Sturm (1803-1855) French mathematician (and, as it happens, an honorary foreign member of the Petersburg Academy of Sciences) ; his main work was in the solution of boundary-value problems for the equations of mathematical physics. -
18
524
Fourier Series and the Fourier Transform
b) By differentiating the identity (x 2 - l ) d';, (x 2 - l ) n = 2nx(x 2 - l) n , show that Pn ( x) satisfies the equation (x 2 - 1) P:; (x) + 2x P� (x) - n(n + l)Pn (x) = 0 . ·
·
c) Verify that the operator
d + 2x d d [ ( x 2 - 1)d] - 1)dx 2 dx dx dx is symmetric in the space c< 2 l [-1, 1] c R 2 [-1, 1]. Then, starting from the relation A : = ( x2
2
=
A(Pn ) = n(n + l)Pn , explain why the Legendre polynomials are orthogonal. d) Using the COmpleteness of the system { 1, X, x 2 , . . . } in C ( 2) [ -1, 1], show that the dimension of the eigenspace of the operator A corresponding to the eigenvalue n(n + 1) cannot be larger than 1. e ) Prove that the operator A = d';, [ ( x 2 - 1) d';, ] cannot have eigenfunctions in the space c< 2) [ -1, 1] except those in the system {Po ( X ) , pl ( X ) , } of Legendre poly nomials, nor any eigenvalues different from the numbers {n(n + 1); n = 0, 1, 2, . . . } . 7. Spherical functions. a ) I n solving various problems i n IR3 (for example, problems of potential theory connected with Laplace's equation Llu = 0) the solutions are sought in the form of a series of solutions of a special form. Such solutions are taken to be homogeneous polynomials Sn (x, y, z ) of degree n satisfying the equation Llu = 0 . Such polynomials are called harmonic polynomials. In spherical coordinates ( r,
0 0
·
Pn (cos B) , Pn ,rn (cos B) cos m
18. 1 Basic General Concepts Connected with Fourier Series
525
a) Show that Ho (x) = 1 , H1 (x) = 2x, H2 (x) = 4x 2 - 2. b) Prove that Hn (x) is a polynomial of degree n. The system of functions { Ho ( x), H1 ( x), . . . } is called the system of Hermite polynomials. c) Verify that the function Hn (x) satisfies the equation H;: (x) - 2xH� (x) + 2nHn (x) = 0. 2 d) The functions '1/J n (x) = e -x Hn (x) are called the Hermite functions. Show that '1/J� (x) + (2n + 1 - x 2 )'1/J n (x) = 0, and that '1/J n (x) --+ 0 as x --+ oo. + oo
e) Verify that J '1/Jn 'I/Jm dx = 0 if m i- n. oo
2 f) Show that the Hermite polynomials are orthogonal on lR with weight e -x . 9. The Chebyshev-Laguerre 1 1 polynomials {L n (x) ; n = 0, 1 , 2 , . . . } can be defined ). by the formula L n ( x ) : = e x d ( xdn e-"' xn Verify that a) L n (x) is a polynomial of degree n; b) the function L n (x) satisfies the equation xL � (x) + ( 1 - x)L� (x) + nL n (x) = 0 ; -
c) the system {L n ; n = 0, 1 , 2, . . . } of Chebyshev-Laguerre polynomials is or thogonal with weight e - x on the half-line (0, +oo[. 10. The Chebyshev polynomials {To (x) = 1, Tn (x) = 2 1 - n cos n(arccos x); n E N} for l xl < 1 can be defined by the formula dn (-2) n n! � ,., ( ) = 1! - x 2� n ( 1 - X 2 ) n - ! . .L n X y (2n) ! dx Show that: a) Tn (x) is a polynomial of degree n; b) Tn (x) satisfies the equation ( 1 - x 2 )T;: (x) - xT� (x) + n 2 Tn (x)
=
0;
c) the system {Tn ; n = 0, 1 , 2, . . . } of Chebyshev polynomials is orthogonal with on the interval ] - 1 , 1 [. weight p(x) = yr!---:; 1 - x2 1 1 . a) In probability theory and theory of functions one encounters the following system of Rademacher1 2 functions : { '1/J n (x) =
1 1 E.N. Laguerre ( 1834-1886) - French mathematician. 1 2 H.A. Rademacher (1892-1969) - German mathematician (American after 1936) . 1 3 A. Haar (1885-1933) - Hungarian mathematician.
526
{ -� ,
18 Fourier Series and the Fourier Transform
Xn , k (x) =
1'
if ��+r < x < ��:jJ ,
if ��'fl < X < 2 �i1 ,
at all other points of [0, 1 ) . Verify that Haar system is orthogonal on the closed interval [0, 1 ) .
1 2 . a) Show that every n-dimensional vector space with an inner product is iso metrically isomorphic to the arithmetic Euclidean space JRn of the same dimension. b) We recall that a metric space is called separable if it contains a countable everywhere-dense subset. Prove that if a vector space with inner product is sepa rable as a metric space with the metric induced by the inner product, then it has a countable orthonormal basis. c) Let X be a separable Hilbert space ( that is, X is a separable and complete metric space with the metric induced by the inner product in X). Taking an or thonormal basis { e; ; i E N} in X, we construct the mapping X 3 x f-t ( c 1 , c2 , . . . ) , where c; = (x, e; ) are the Fourier coefficients of the expansion of the vector in the basis {e; } . Show that this mapping is a bijective, linear, and isometric mapping of X onto the space l 2 considered in Example 14. d) Using Fig. 18. 1 , exhibit the basic idea of the construction of Example 14, and explain why it comes about precisely because the space in question is infinite dimensional. e) Explain how to construct an analogous example in the space of functions C [a , b] C 'R.2 [a, b] .
1 8 . 2 Trigonometric Fourier Series 1 8. 2 . 1 Basic Types of Convergence of Classical Fourier Series
a. Trigonometric Series and Trigonometric Fourier Series A classical trigonometric series is a series of the form 1 4
�0 + L= ak cos kx + bk sin kx , 00
k l
( 18.32)
obtained on the basis of the trigonometric system { 1 , cos kx, sin kx; k E N}. The coefficients {a 0 , a k , b k ; k E N} here are real or complex numbers. The partial sums of the trigonometric series ( 18.32) are the trigonometric polyno mials n 0 Tn (x) = + L a k cos kx + bk sin kx (18.33)
n.
�
k= l
corresponding to degree 1 4 Writing the constant term in the form ao/2, which is convenient for Fourier series, is not obligatory here.
527
18.2 Trigonometric Fourier Series
If the series ( 18.32) converges pointwise on JR, its sum f(x) is obviously a function of period 21r on JR. It is completely determined by its restriction to any closed interval of length 27r. Conversely, given a function of period 21r on lR (oscillations, a signal, and the like) that we wish to expand into a sum of certain canonical periodic func tions, the first claimants for such canonical status are the simplest functions of period 27r, namely { 1 , cos kx, sin kx; k E N} , which are simple harmonic oscillations of corresponding degree n, with frequencies that are multiples of the lowest frequency. Suppose we have succeeded in representing a continuous function as the sum ao 00 . ( 18.34) f(x) = 2 + "' L a k cos kx + b k sm kx = k l of a trigonometric series that converges uniformly to it . Then the coefficients of the expansion ( 18.34) can be easily and uniquely found. Multiplying Eq. ( 18.34) successively by each function of the system { 1 , cos ks, sin kx; k E N} , using the fact that termwise integration is possible in the resulting uniformly convergent series, and taking account of the relations 7r
7r
7r
J 1 2 dx = 27r ,
- 7r
J cos mx cos nx dx = J sin mx sin nx dx 7r
7r
- 7r
- 7r
=
0 for m =I n , m, n E N ,
J cos2 nx dx = J sin2 nx dx = 7r ,
we find the coefficients ak bk
= ak (f) =
bk (f)
7r
� J f(x) cos kx dx , 1 J f(x) sm. kx dx , - 7r
7r
:;
- 7r
nEN,
k = 0, 1 , . . .
'
(18.35)
k = 1 , 2, . . . .
( 18.36)
of the expansion ( 18.34) of the function f in a trigonometric series. We have arrived at the same coefficients that we would have had if we had regarded ( 18.34) as the expansion of the vector f E R2 [ - 1r , 1r] in the orthogonal system { 1, cos kx, sin kx; k E N}. This is not surprising, since the
528
18 Fourier Series and the Fourier Transform
uniform convergence of the series ( 18.34) of course implies convergence in the mean on the closed interval [-1r, 1r] , and then the coefficients of (18.34) must be the Fourier coefficients of the function f in the given orthogonal system (see Sect . 18. 1 ) . Definition 1 . If the integrals (18.35) and (18.36) have meaning for a func tion f , then the trigonometric series
f "'
� + kf:= l ak (f) cos kx + bk (f) sin kx
ao f)
( 18.37)
assigned to f is called the trigonometric Fourier series of f. Since there will be no Fourier series in this section except trigonometric Fourier series, we shall occasionally allow ourselves to drop the word "trigono metric" and speak of just "the Fourier series of f." In the main we shall be dealing with functions of class 'R ( [-7r, 1r] , C) , or, slightly more generally, with functions whose squared absolute values are integrable (possibly in the improper sense) on the open interval J - 1r, 1r[. We retain our previous notation 'R2 [-1r, 1r] to denote the vector space of such functions with the standard inner product
(!, g) =
7r
j fg dx .
( 18.38)
Bessel ' s inequality
l ao �W + f: l a k UW + l bk UW ::::; � j l f l 2 (x) dx , k= l 7r
(18.39)
- 7r
which holds for every function f E 'R2 ([ - 7!" , 1r] , C) , shows that by no means every trigonometric series ( 18.3 2 ) can be the Fourier series of some function f E 'R2 [-1r, 1r] . Example 1 . The trigonometric series
f: sin kx k= l
v'k '
IR,
as we already know (see Example 7 of Sect . 16. 2 ) converges on but is not 2 the Fourier series of any function f E 'R2 [ -7r, 1r] , since the series f: ( �)
k=l
diverges. Thus, arbitrary trigonometric series ( 18.32) will not be studied here, only Fourier series (18.37) of functions in 'R2 [-1r, 1r] , and functions that are abso lutely integrable on J - 1r, 1r [.
18.2 Trigonometric Fourier Series
529
b. Mean Convergence of a Trigonometric Fourier Series Let
Sn (x)
=
; + t= ak (f) cos kx + bk (f) sin kx
ao f )
( 18.40)
k l be the nth partial sum of the Fourier series of the function f E R 2 [ - 71" , 1r] . The deviation of Sn from f can be measured both in the natural metric of the space R2 [-1r, 1r] induced by the inner product (18.38) , that is, in the sense of the mean-square deviation I I / - Sn ll
=
11"
J If
- 11"
-
Sn l 2 (x) dx
(18.41)
of Sn from f on the interval [-1r, 1r] , and in the sense of pointwise convergence on that interval. The first of these two kinds of convergence was studied for an arbitrary series in Sect. 18. 1 . Making the results obtained there specific in the context of a trigonometric Fourier series involves first of all noting that the trigonometric system { 1 , cos kx, sin kx; k E N} is complete in R 2 [-1r, 1r] . (This has already been noted in Sect . 18. 1 and will be proved independently in Subsect . 18.2.4 of the present section. ) Hence, the fundamental theorem of Sect. 18. 1 enables us t o say i n the present case that the following theorem is true. Theorem 1. (Mean convergence of a trigonometric Fourier series) . The Fourier series ( 18.37) of any function f E R 2 ( [-1r, 1r] , C) converges to the function in the mean ( 18.41 ) , that is,
ao ( f )
. f (x) ;: -- + � L...t a k (f) cos kx + bk (f) sm kx , 2 2 k= l and Parseval 's equality holds: ( 18.42) We shall often use the more compact complex notation for trigonometric polynomials and trigonometric series, based on the Euler formulas ei x cos x + i sin x, cos x = � ( ei x + e - i x ) , sin x = � ( e i x - e - i x ) . Using them, we can write the partial sum (18.40) of the Fourier series as =
Sn (x) =
n
L
k= -n
Ck e i kx ,
( 18.40' )
530
18 Fourier Series and the Fourier Transform
and the series ( 18.37) itself as
_
where
Ck -
{
(18.37') � (a k - ib k ) , 1 2 ao ,
if k
Ck
0,
if k = 0 ,
� (a -k + ib_ k ) , if k
that is,
>
1f
f(x)e - i kx dx , = ck (f) = 2_ 27r J
<
( 18.43)
0, kEZ,
(18.44)
and hence the numbers Ck are simply the Fourier coefficients of f in the system {ei kx ; k E Z} . We call attention to the fact that summation of the Fourier series ( 18.37') is understood in the sense of the convergence of the sums ( 18.40') . In complex notation Theorem 1 means that for every function f E
R2 ( [ - n, n] , C )
and
00
2� 11 ! 11 2 = f00 i ck UW -
·
( 18.45)
c. Pointwise Convergence of a Trigonometric Fourier Series Theorem 1 gives a complete solution to the problem of mean convergence of a Fourier series ( 18.37) , that is, convergence in the norm of the space R2 [-n, n]. The remainder of this section will be mainly devoted to studying the conditions for and the nature of pointwise convegence of a trigonometric series. We shall consider only the simplest aspects of this problem. The study of pointwise convergence of a trigonometric series, as a rule, is such a delicate matter that , despite the traditional central position occupied by Fourier series after the work of Euler, Fourier, and Riemann, there is still no intrinsic descrip tion of the class of functions that can be represented by trigonometric series converging to them at every point (the Riemann problem). Until recently it was not even known whether the Fourier series of a continuous function must converge to it almost everywhere (it was known that convergence need not occur at every point ) . Previously A.N. Kolmogorov 1 5 had even given an 1 5 A.N. Kolmogorov ( 1903-1987) - outstanding Soviet scholar, who worked in probability theory, mathematical statistics, theory of functions, functional analysis, topology, logic, differential equations, and applied mathematics.
18.2 Trigonometric Fourier Series
531
example of a function f E L[ - n, n] whose Fourier series diverged everywhere ( where L[ -n, n] is the space of Lebesgue-integrable functions on the interval [-n, n] , obtainable as the metric completion of the space R[ - n, n]) , and D.E. Men ' shov 1 6 constructed a trigonometric series ( 18.32) with coefficients not all zero that nevertheless converged to zero almost everywhere ( Men 'shov 's null-series ) . The problem posed by Luzin 1 7 ( Luzin's problem ) of deter mining whether the Fourier series of every function f E £ 2 [ - n, n] converges almost everywhere ( where £ 2 [-n, n] is the metric completion of R2 [ - n, n]) was answered in the affirmative only in 1966 by L. Carleson. 1 8 It follows in particular from Carleson ' s result that the Fourier series of every function f E R2 [-n, n] ( for example a continuous function ) must converge at almost all points of the closed interval [ -n, n] .
N. N .
18.2.2 Investigation of Pointwise Convergence of a Trigonometric Fourier Series a. Integral Representation of the Partial Sum of a Fourier Series Let us now turn our attention to the partial sum ( 18.40) of the Fourier series ( 18.37) and, using the complex notation ( 18.40') for the expression ( 18.44) for the Fourier coefficients, we make the following transformations:
7r
�J
2
- 1r
f(t)
( kL=n- n eik (x -t) ) dt .
( 18.46)
But u
Dn ( )
:=
n
ei (n+ l )u - e - i n u ei. k u = ---,-.---L ei u - 1 k= - n
ei (n+ ! )u - e - i (n+ ! )u ei ! u e - i ! u
and, as can be seen from the very definition, D n ( u ) Hence sin ( n + � ) u Dn ( U ) = . 21 u ' sm
_
=
( 2n + 1) if e i u
( 1 8.47) =
1.
( 18.48)
1 6 D.E. Men'shov (1892-1988) - Soviet mathematician, one of the greatest special ists in the theory of functions of a real variable. 1 7 N.N. Luzin (1883-1950) - Russian / Soviet mathematician, one of the greatest specialists in the theory of functions of a real variable, founder of the large Moscow mathematical school ( "Lusitania" ) . 1 8 L. Carleson ( b. 1928) - outstanding Swedish mathematician whose main works are in various areas of modern analysis.
532
18 Fourier Series and the Fourier Transform
where the ratio is regarded as 2n + 1 when the denominator of the fraction becomes zero. Continuing the computation ( 18.46) , we now have
Sn ( x )
=
� J f (t) Dn (X - t) t . 71"
2
( 18.49)
d
- 71"
We have thus represented Sn ( x ) as the convolution of the function f with the function ( 18.48) , which is called the Dirichlet kernel. As can be seen from the original definition ( 18.47) of the function Dn (u), the Dirichlet kernel is of period 2rr and even, and, in addition
� J Dn (u) du � J Dn (u) du = 1 . 71"
2
=
- 71"
71"
( 18.50)
0
Assuming the function f is of period 2rr on R or is extended from [ -rr, rr] to R so as to have period 2rr, and making a change of variable in ( 18.49) , we obtain 71" 71" sin ( + .!2 ) t 1 1 (x dt . (18.51) t) Dn (t) dt = - f (x - t) Sn ( x ) = f . 1 2 1r 2 1r sm 2t
!
n
!
- 7r
- 7r
In carrying out the change of variable here, we used the fact that the integral of a periodic function is the same over every interval whose length equals a period. Taking account of the fact that Dn (t) is an even function, we can rewrite Eq. (18.51) as
Sn ( x )
= =
� J ( / (x - t) 1 (f(x - t) 2 J 71"
2
0
1r
71"
+ f(x + t)) Dn (t) d t + f(x + t))
sin
0
=
n
( + .!2 ) t t. . 1 sm 2t d
( 18.5 2)
The Riemann-Lebesgue Lemma and the Localization Principle The representation ( 18.52) for the partial sum of a trigonometric :Fourier series, together with an observation of Riemann stated below, forms the basis for studying the pointwise convergence of a trigonometric Fourier series. Lemma 1 . ( Riemann-Lebesgue ) . If a locally integmble function f ]wb w 2 [---+ R is absolutely integmble (perhaps in the improper sense) on an open interval ]w 1 , w2 [, then W2
J f(x)ei>..x x ---+ 0 as A ---+ oo ,
WI
d
AER.
( 18.53)
18.2 Trigonometric Fourier Series
533
Proof. If ]w 1 , w2 [ is a finite interval and f(x) = 1 , then Eq. ( 18.53) can be verified by direct integration and passage to the limit. We shall reduce the general case to this simplest one. Fixing an arbitrary E: > 0, we first choose an interval [a, b] c]w 1 , w2 [ such that for every >. E � W2
IJ
f(x)e i>.x dx -
J f(x)ei>.x x l b
< E:
d
a
.
( 18.54)
In view of the estimates
W2
IJ w1
f(x)ei >.x dx a
:::; J W1
J f(x)ei>.x x l b
d
a
l f(x)e i>.x l dx +
:::; w2
J b
l f(x)ei>.x l dx =
a
w2
J l fl (x) dx J lfl (x) dx +
W1
b
and the absolute integrability of f on ]w 1 , w2 [, there does of course exist such a closed interval [a, b] . Since f E R([a, b] , � ) (more precisely !I [a,b] E R( [a, bl ) ) , there exists a n lower Darboux sum 2.:: mj LlXj , where mj = inf f(x) , such that
j= l
x E [Xj - 1 ,Xj )
b
0<
J f(x) dx - j=Z: mj LlXj < E . n
l
a
Now introducing the piecewise constant function g(x) we find that g(x) :::; f(x) on [a, b] and
[xj - ! , Xj ] , j = 1 ,
. . . , n,
I / f(x)ei>.x dx - J g(x)ei>.x dx l :::; :::; J lf(x) - g(x) l l ei>.x l x J {J(x) - g(x)) b
mj for x E
b
0 :::;
a b
a
d
a
=
b
( 18.55)
dx < E .
a
but b
J a
g(x)ei >.x dx
= =
8n J mj ei>.x dx Xj
J-
Xj - 1
=
;_ � (mJ· ei >.x ) l Xxj3 - 1 L.....,; I A j= l
--+
0 as >. --+
oo ,
>. E � ( 18.56) .
534
18 Fourier Series and the Fourier Transform
Comparing relations ( 18.53)-( 18.56) , we find what was asserted. D
Remark
1.
Separating the real and imaginary parts in ( 18.53) , we find that
W2
j f ( x) cos Ax dx --7 0
Wl
and
--7 oo,
W2
j f(x) sin Ax dx --7 0
Wl
(18.57)
as A A E R If the function f in the preceding integrals had been complex, then, separating the real and imaginary parts in them, we would have found that relations (18.57) , and consequently (18.53) , would actually be valid for complex-valued functions f : ] w 1 , w 2 [ -7 C.
Remark 2. If it is known that f E R2 [ -7r, 7r ] , then by Bessel ' s inequality ( 18.39) we can conclude immediately that 71"
J f(x) nx x --7 0 cos
d
71"
J f(x) nx x --7 0
and
- 71"
- 71"
n --7 oo, n
sin
d
as E N. Theoretically, we could have gotten by with just this discrete version of the Riemann-Lebesgue lemma for the elementary investi gations of the classical Fourier series that will be carried out here. Returning now to the integral representation ( 18.52) of the partial sum of a Fourier series, we remark that if the function f satisfies the hypotheses of the Riemann-Lebesgue lemma, then, since sin �t 2: sin � 8 > 0 for 0 < 8 :::; t :::; 71" , we can use ( 18.57) to write
Sn (x) =
1 2 7r
8
j (f (x - t) + f (x + t) ) sin (n + 2 ) .
1
sm 2 t
0
! t
d t +o ( 1 ) as
n --7 oo . (18.58)
The important conclusion one can deduce from ( 18.58) is that the con vergence or divergence of a Fourier series at a point is completely determined by the behavior of the function in an arbitrarily small neighborhood of the point . We state this principle as the following proposition.
Theorem 2. (Localization principle ) . Let f and g be real- or complex-valued locally integrable functions on ] - 71" , 7r [ and absolutely integrable on the whole interval (possibly in the improper sense). If the functions f and g are equal in any (arbitrarily small} neighborhood of the point xo E] - 71", 71" [ , then their Fourier series +oo
f(x) "' L Ck ( f ) eikx ' - oo
+oo
g(x) "' L ck ( g ) eikx - oo
18.2 Trigonometric Fourier Series
535
either both converge or both diverge at xo . When they converge, their limits are equal. 1 9 Remark 3. As can be seen from the reasoning used in deriving Eqs. ( 18.52) and ( 18.58) , the point x0 in the localization principle may also be an endpoint of the closed interval [ - 1r , 1r ] , but then (and this is essential!) in order for the periodic extensions of the functions f and g from the closed interval [ - 7r , 1r ] to lR to be equal on a neighborhood of x0 it is necessary (and sufficient) that the original functions be equal on a neighborhood of both endpoints of the closed interval [ - 1r , 1r ] . c. Sufficient Conditions for a Fourier Series to Converge at a Point Definition 2 . A function f : U C defined on a deleted neighborhood of a point x E lR satisfies the Dini conditions at x if a) both one-sided limits
--+
lim0 f(x - t) , f(x_ ) = t --++
exist at x; b) the integral
J +O
lim0 f(x + t) f(x + ) = t --++
(f(x - t ) - f(x _ )) : (f(x + t ) - f(x+ )) dt
converges absolutely. 2 0 Example 2. If f is a continuous function in U(x) satisfying the Holder con dition l f(x + t ) - f(x) l ::; M I W , 0 < a ::; 1 , then, since the estimate f(x + t ) - f(x) < __J!_ - I W-"' t now holds, the function f satisfies the Dini conditions at x. It is also clear that if a continuous function f defined in a deleted neighbor hood U(x) of x has one-sided limits f(x_ ) and f(x + ) and satisfies one-sided Holder conditions l f(x + t ) - f(x+ ) l :S M t a , l f(x - t ) - f(x - ) 1 :S M t a ,
I
I
where t > 0, 0 < a ::; 1 , and M is a positive constant , then Dini conditions for the same reason as above.
19 Although the limit need not be J(xo)
20
What is meant is that the integral
f0
f will satisfy the
g(xo ) . converges absolutely for some value E > 0. =
18 Fourier Series and the Fourier Transform
536
Definition 3. We shall call a real- or complex-valued function f piecewise continuous on the closed interval [a, b] if there is a finite set of points a = x0 < x 1 < · · · < Xn = b in this interval such that f is defined on and has one-sided limits on approach to each interval ] x3 _ 1 , x3 [, j = 1 , its endpoints.
. . . , n,
Definition 4 . A function having a piecewise-continuous derivative on a closed interval is piecewise continuously differentiable on that interval.
Example 3. If a function is piecewise continuously differentiable on a closed interval, then it satisfies the Holder conditions with exponent a = 1 at every point of the interval, as follows from Lagrange ' s finite-increment (mean-value) theorem. Hence, by Example 1 , such a function satisfies Dini ' s conditions at every point of the interval. At the endpoints of the interval, of course only the corresponding one-sided pair of Dini ' s conditions needs to be verified. Example 4 . The function f( x ) point x E even at zero.
IR,
=
sgn x satisfies Dini ' s conditions at every
Theorem 3. (Sufficient conditions for convergence of a Fourier series at a point ) . Let f : lR -+ C be a function of period 2rr that is absolutely integrable on the closed interval [ - rr , rr ] . If f satisfies the Dini conditions at a point xE then its Fourier series converges at that point, and
IR,
� c (f) ei x = f ( x_ ) ; f(x -
oo
k
k
+
)
(18.59)
Proof. By relations ( 18.52) and ( 18.50)
Sn ( x )
_
f (x ) + f (x+ ) = 2 7r _- -1 { J(x - t) - f( x ) ) + { J(x + t) - f ( x + ) ) sm. ( + 1 ) t dt . 1 2 rr 2 sin 2t _
J 0
-
n
-
Since 2 sin ! t "' t as t -+ + 0 , by the Dini conditions and the Riemann Lebesgue lemma we see that this last integral tends to zero as 0
n -+ oo.
Remark 4. In connection with the theorem just proved and the localization principle, we note that changing the value of the function at a point has no influence on the Fourier coefficients or the Fourier series or the partial sums of the Fourier series. Therefore the convergence and the sum of such a series at a point is determined not by the particular value of the function at the point, but by the integral mean of its values in an arbitrarily small neighborhood of the point. It is this fact that is reflected in Theorem 3.
537
18.2 Trigonometric Fourier Series
Example
5.
In Example 6 of Sect . 18. 1 we found the Fourier series x
"'
00 ( - 1 ) k + l sin kx 2
2::::
( 1 8.60)
k
k=1
for the function f(x) = x on the closed interval [-1r, 1r] . Extending the func tion f(x) periodically from the interval ] - 1r, 1r [ to the whole real line, we may assume that the series (18.60) is the Fourier series of this extended function. Then, on the basis of Theorem 3 we find that
oo ( - 1 ) k + l 2:::: 2 k sin kx = k =1
{X,
if l x l < 1r ,
0 ' if l x l = 1r .
In particular, for x = � it follows from this relation that
� (-l) n = � . L..n = O 2n + 1 4
Example 6. Let a E lR and l a l < 1 . Consider the 21r-periodic function f(x) defined on the closed interval [-1r, 1r] by the formula f (x) = cos a x . B y formulas (18.3 5 ) and (18.36) we find its Fourier coefficients
( -1) n sin 1r a J cos a x cos nx dx = ; 7r 1
bn (f) =
7r
2a
- 7r
7r
� J cos ax sin nx dx = 0 . - 7r
By Theorem 3 the following equality holds at each point x 2 sin 7ra 1 � (-1) n cos nx cos a x = a + L..- a 2 - n 2 2 a 2 n=1 1r
(
)
E [-1r , 1r] :
When x = 1r, this relation implies 2 a 00 1 1 cot 1r a - - = - ""' L..- a 2 - n 2 1r n=1 7r a
--=----=-
I
I
(18.61)
1 < a0 < 1 ' then a 2-n2 < n2 -1 a g ' and hence the series on the If l a l right-hand side of Eq. (18.61) converges uniformly with respect to a on every closed interval l a l :s; a0 < 1 . Hence it is legitimate to integrate it termwise, that is,
538
18 Fourier Series and the Fourier Transform 7r
J o
00 2::: ( cot 7rcY - 2_ ) dx = :
7r
J � - n2 1
n= o
?TO'
and
a
a a ,
d
00 sm ?TX 2::: : ln 1 - x2 , l n -- 1TX 1T 2 n= 1
yielding
.
(
_
)
oo sin ?TX IJ x2 when l x l < 1 . ( 18.62) = 1 -2 n ?TX n= 1 We have thus proved relation ( 18.62) , which we mentioned earlier when deriving the complement formula for Euler ' s function T(x) (Sect. 1 7.3) .
and finally,
)
(
d . Fejer's Theorem. Let us now consider the sequence of functions
O"n (x) := So(x) +n· +· · 1+ Sn (x)
,
that are the arithmetic means of the corresponding partial sums S0(x), . . . , Sn (x) of the trigonometric Fourier series ( 18.37) of a function f : JR. � C of period 21r. From the integral representation (18.51) of the partial sum of the Fourier series we have 7r
J(x - t)Fn (t) dt , O"n (x) = _]:_ 21T
J
- 7r
where
1 ( Do (t) + · · · + Dn (t) ) . Fn (t) = n+1 Recalling the explicit form (18.48) of the Dirichlet kernel and taking ac count of the relation n sin2 ( n+2 1 ) t 1 ) -1 n I: sin k + -21 ) t = -21 sin -t L ( cos kt - cos(k + 1)t ) = . -1 t , 2 sm 2 k� k�
(
we find
(
sin2 n+2 l t . Fn (t) = (n + 1 ) sin2 21 t The function Fn is called the Fejer kernel, more precisely the nth Fejer kernel. 2 1 21 L. Fe}§ r ( 1880-1956) - well-known Hungarian mathematician.
18.2 Trigonometric Fourier Series
539
account of the original definition ( 1 8.47) of the Dirichlet kernel Dwhose ,n Taking one can conclude that the Fejer kernel is a smooth function of period 2n value equals ( n + 1) where the denominator of this last fraction equals zero. The properties of the Fejer and Dirichlet kernels are similar in many ways, but in contrast to the Dirichlet kernel, the Fejer kernel is also nonnegative, so that we have the following lemma. Lemma 2. The sequence of functions
L1n (x) = { 2�F0n (x), , ifif l xx l 7r , l l > 7r :S:
is an approximate identity on JR.
L1n (x)
Proof. The nonnnegativity of is clear. Equality ( 18.50) enables us to conclude that CXJ
7r
- CXJ
- 7[
7r
J L1n (x) dx = J .dn (x) dx = � J Fn (x) dx = 2
15
- 7[
Finally, for every > 0 7r
7r
J L1n (x) dx = J L1n (x) dx =JL1n (x) dx ::::; 2n ( n + 1 ) J sm. �x2xl -t 0 as n -t + oo. D Theorem 4. (Fejer) . Let f : IR -t C be a function of period 2n that zs absolutely integrable on the closed interval [ n] . If - li
0 :S:
- CXJ
a)
+ CXJ
li
1
li
li
-n,
f is uniformly continuous on the set E c IR, then
O"n (x) f(x) on E as n -t oo ;
b) f E C(IR, C) , then
::4
O"n (x) f(x) on 1R as n -t oo ; c) f is continuous at the point x E IR, then O"n (x) -t f(x) as n -t oo . ::4
540
18 Fourier Series and the Fourier Transform
Proof. Statements b) and c) are special cases of a) . Statement a) itself is a special case of the general Proposition 5 of Sect. 17.4 on the convergence of a convolution, since
an (x)
=
� 2rr
71"
J f(x - t):Fn (t) t d
- 71"
=
( ! * Ll n )(x) . D
Corollary 1 . (Weierstrass ' theorem on approximation by trigonometric polynomials) . If a function f : [-rr, rr] C is continuous on the closed interval [-rr, rr] and f ( -rr) = f(rr) , then this function can be approximated uniformly on the closed interval [-rr, rr] with arbitrary precision by trigono metric polynomials. Proof. Extending f as a function of period 2 rr, we obtain a continuous 2rr periodic function on JR, to which the trigonometric polynomials an (x) con verge uniformly by Fejer ' s theorem. D Corollary 2 . If f is continuous at x, its Fourier series either diverges at x or converges to f (x) . Proof Only the case of convergence requires formal verification. If the se quence Sn ( x ) has a limit as then the sequence an ( x ) = So (x) ��i Sn (x) has that same limit. But by Fejer ' s theorem an (x ) f ( x ) as and hence Sn (x ) f ( x ) also whenever the limit Sn (x) exists as D Remark 5. We note that the Fourier series of a continuous function really can diverge at some points.
�
�
n � oo,
�
n �n oo.� oo,
1 8 . 2 . 3 Smoothness of a Function and the Rate of Decrease of the Fourier Coefficients
a. An Estimate of the Fourier Coefficients of a Smooth Function We begin with a simple, yet important and useful lemma. Lemma 3 . (Differentiation of a Fourier series) . If a continuous function f E C ( [-rr, rr] , C) assuming equal values at the endpoints of the closed inter val [ -rr, rr] is piecewise continuously differentiable on [ -rr, rr] , then the Fourier series of its derivative
00
-
oo
can be obtained by differentiating formally the Fourier series
00 -
of the function itself, that is, c k (f ' )
=
oo
i kck (f ) ,
kEZ.
(18.63)
18.2 Trigonometric Fourier Series
541
Proof. Starting from the definition of the Fourier coefficients (18.44) , we find through integration by parts that 7r
7r
- 7[
- 7[
ck (f') = 27r Jf'(x)e - 1' kx dx = 217r f(x) e - 1' kx 1 7r- 1r + 2ik7r Jf(x)e - 1' k x dx = ikck (f), 1
since
f(1r) e - i k1f - f(- 7r ) e i k7r = 0.
D
Proposition 1. (Connection between smoothness of a function and the rate of decrease of its Fourier coefficients) . Let f E C ( m - ll ( [ - 7r , 7r ] , IC ) and f (j ) ( -1r) = fUl ( 7r ), j = 0, 1, . . . , m - 1 . If the function f has a piecewise continuous derivative f (rn ) of order m on the closed interval [ - 1r, 1r] , then
(18.64) and as moreover,
k --+ oo , k E 7l ;
(18.65)
00
L "'(� < oo. - oo
Proof. Relation ( 18.64 ) follows from an m-fold application of Eq. ( 18.63) : Setting "'/k
= l c k (f ( m l) l and using Bessel ' s inequality
we obtain ( 18.65 ) from ( 18.64 ) . D
Remark 6. In the proposition just proved, as in Lemma 3, instead of asssum ing the conditions f ( j l ( - 1r) = fUl ( 7r ), we could have assumed that f is a function of period 21r on the entire line.
Remark 7. If a trigonometric Fourier series were written in the form ( 18.37 ) , rather than in the complex form ( 18.37' ) , it would be necessary to replace the simple relations ( 18.64 ) by noticeably more cumbersome equalities, whose meaning, however, would be the same: under these hypotheses a Fourier series can be differentiated termwise whichever from it is written in, ( 18.37) or ( 18.37') . As for the estimates of the Fourier coefficients, a k (f) and b k (f) of ( 18.37 ) , since a k (f) = c k (f) + c_ k ( J) and b k (f) = i(c k (f) - c_ k (f)), (see formulas ( 18.43 ) ) it follows from ( 18.65 ) that if a function f satisfies the hypotheses of the proposition, then
542
18 Fourier Series and the Fourier Transform
(18.64') 00
where I: a: � < = '"Yk + '"Y-kk ·l
oo and kI:00= l (3� < oo, and we can assume that a:k = fJk =
b. Smoothness of a Function and the Rate of Convergence of its Fourier Series
-+
Theorem 5. If the function f : [ - n , n] C is such that a) f E C ( m - l ) [ - n , n] , m E N, b) f Ul ( - n ) = f Ul ( n ) , j = 0, 1 , . . . , m - 1 , c) f has a piecewise continuous mth derivative f ( m ) on [ - n , n ] , m :2: 1 , then the Fourier series of f converges absolutely and uniformly o n [ - n , n] to f, and the deviation of the nth partial sum Sn (x) of the Fourier series from f ( x) has the following estimate on the entire interval:
where
{En } is a sequence of positive numbers tending to zero.
Proof. We write the partial sum ( 1 8.40) of the Fourier series in the compact notation ( 18.40'):
n -n
According to the assumptions on the function f and Proposition 1 we have
l c k (f) l '"Yk / l k l m , and L '"Yk / l kml m < oo : since 0 :::; '"Yk / l k l m :::; � ('"'(� + 1/k2m ) and m :2: 1 , we have L '"Yk / l k l < oo. Hence the sequence Sn (x) converges =
uniformly on [ - n , n] ( by the Weierstrass M-test for series and the Cauchy criterion for sequences ) . By Theorem 3 the limit S(x) of Sn (x) equals f(x), since the function f satisfies the Dini conditions at each point of the closed interval [ - n , n] ( see Example 3) . And, since f ( - n ) = f (n) , the function f can be extended to lR as a periodic function with the Dini conditions holding at each point x E JR. Now, using relation (18.63) , we can proceed to obtain an estimate:
l f(x) - Sn (x) l = I S(x) - Sn (x) l = 00
± k=n+ l
I ± kf==n+ l ck ( f ) ikx l :::; e
00
± k =n+ l
18.2 Trigonometric Fourier Series
543
The first factor on the right-hand side of the Cauchy-Bunyakovskii inoo equality here tends to zero as -+ since L:; 'Y� <
n oo,
oo.
Next (see Fig. 18.2) 00
'"' 1/ L..... k =n + l
00 12m k -< J xd2mx = 2m1- 1 . _ 2m n -1 .
n
We thus obtain the assertion of Theorem 5. D
n n+ l
Fig. 1 8 . 2 .
X
In connection with these results we now make a number of useful remarks.
Remark 8. One can now easily obtain again the Weierstrass approximation theorem stated in Corollary 1 from Theorem 5 (and Theorem 3, of which essential use was made in the proof of Theorem 5), independently of Fejer ' s theorem. Proof It suffices to prove this result for real-valued functions. Using the uni form continuity of f on [-rr, rr] , we approximate f on this closed interval uniformly within c/2 by a piecewise-linear continuous function cp(x) assum ing the same values as f at the endpoints, that is, cp( -rr) = cp(rr) = f (rr) (Fig. 18.3) . By Theorem 5 the Fourier series of cp converges to cp uniformly on the closed interval [-rr, rr] . Taking a partial sum of this series that dify
Fig. 18.3.
X
544
18 Fourier Series and the Fourier Transform
fers from 'P(x) by less than c/2, we obtain a trigonometric polynomial that approximates the original function f within F: on the whole interval [ -1r, 1r] . D
Remark 9. Let us assume that we have succeeded in representing a function f having a jump singularity as the sum f = 'P '1/J of a certain smooth function '1/J and a simple function 'P having the same singularity as f (Fig. 18.4 a, c, b) . Then the Fourier series of f is the sum of the Fourier series of '1/J, which converges rapidly and uniformly by Theorem 5, and the Fourier series of the function 'P· The latter can be regarded as known, if we take the standard function 'P (shown in the figure as 'P( x) = -1r - x for -1r < x < 0 and 'P(x) = 1r - x for 0 < x < 1r) .
+
c.
Fig. 18.4.
This observation can be used both in theoretical and computational prob lems connected with series (it is Krylov ' s 22 method of separating singularities and improving the convergence of series) and in the theory of trigonometric Fourier series itself (see for example the Gibbs2 3 phenomenon, described be low in Problem 1 1 ) .
Remark 1 0. (Integration of a Fourier series) . By Theorem 5 we can state and prove the following complement to Lemma 3 on differentiation of a Fourier series. Proposition 2 . If the function f : [-1r, 1r] --+ C is piecewise continuous, 00 then after integration the correspondence f (x) "' E ck ( f ) ei k x becomes the oo equality -
X
j f ( t ) dt = c U) x + 2.:::oo , ckiV) ( ei kx - 1 ) , 0
o
00
-
22 A.N. Krylov ( 1863-1945) - Russian/Soviet specialist in mechanics and mathe matics, who made a large contribution to computational mathematics, especially in methods of computing the elements of ship::; . 2 3 J.W. Gibbs (1839-1903) - American physicist and mathematician, one of the founders of thermodynamics and statistical mechanics.
18.2 Trigonometric Fourier Series
545
where the prime indicates that the term with index k = 0 is omitted from the n sum; the summation is the limit of the symmetric partial sums 'L: , and the -n
series converges uniformly on the closed interval [ -1r, 1r] . Proof. Consider the auxiliary function F (x )
X
= j f(t) dt - co (J)x 0
on the interval [-1r, 1r] . Obviously F E C[-1r, 1r] . Also F( -1r)
F(1r) - F(-1r) =
7r
= F(1r) , since
J f(t)dt - 2Jrco(J) = 0 ,
as follows from the definition of c0 ( ! ) . Since the derivative F' (x) = f(x) 00 c0 (f) of the function F is piecewise continuous, its Fourier series I:: c k ( F )e i k x -
oo
converges uniformly to F on the interval [-1r, 1r] by Theorem 5. By Lemma 3 we have c k (F) = c k ;[') for k -1- 0. But c k (F') = c k (F) if k -1- 0. Now writing 00 the equality F(x) = I:: c k (F)ei k x in terms of the function f and noting that
F(O)
oo
= 0, we obtain the assertion of the proposition. -
0
18.2.4 Completeness of the Trigonometric System
a. The Completeness Theorem In conclusion we return once again from
pointwise convergence of the Fourier series to its mean convergence ( 1 8.41 ) . More precisely, using the facts we have accumulated on the nature of point wise convergence of the Fourier series, we now give a proof of the completeness of the trigonometric system { 1 , cos kx, sin kx; k E N} in R2 ( [-7r, 1r] , IR) inde pendent of the proof already given in the problems. In doing so, as in Subsect. 18.2. 1 , we take R2 ( [-1r, 1r] , IR) or R2 ( [-1r, 1r] ,
546
18 Fourier Series and the Fourier Transform
Every function can be approximated arbitrarily closely in mean a ) by functions of compact support in ] - 7!", K[ that are Riemann integrable over the closed interval [ - 7!", K]; b) by piecewise-constant functions of compact support on the closed inter val [ - K, K]; c ) by piecewise-linear continuous functions of compact support on the closed interval [-7!", K]; d) by trigonometric polynomials.
Theorem 6. (Completeness of the trigonometric system) .
f
E R2 [ -7!", 7!"]
Proof. Since it obviously suffices to prove the theorem for real-valued func tions, we confine ourselves to this case. a) It follows from the definition of the improper integral that
�
�-8
J� f2 (x) x = 8-r +O �j8 f2 (x) x lim
d
Hence, for every
E
d .
-+
> 0 there exists b > 0 such that the function
fo ( x )
=
{
l x l < 7l" - b , if 7l" - b :::; l x l :::; 7l" ,
f (x) , if 0
E,
[ - 7!", 7!"] by less than since � � - �+8 2 2 j (f - fo) (x) dx = j f (x ) dx + j f2 (x) dx . �-8 -� -� b) It suffices to verify that every function of the form fo can be approxi mated in R 2 ( [ - 7!", K], JR ) by piecewise-constant functions of compact support in [ - 7!", K]. But the function fo is Riemann-integrable on [ - K +b, 7!" - b]. Hence it is bounded there by a constant M and moreover there exists a partition - 7!" + b = x0 < x1 < · · · < Xn = 7!" -n b of this closed interval such that the corresponding lower Darboux sum 2::::: mu1x i of the function fo differs from i= l the integral of fo over [ - 7!" + J, 7!" - J ] by less than > 0.
will differ in mean from f on
Now setting
g(x) = we obtain
{ mi , 0,
E
x E]x i _ 1 , x i [ , i = 1 , . . . , n , at all other points of [ - 7!", K] , if
18.2 Trigonometric Fourier Series
547
11" 11" 2 11"- 0 ju, - g) (x) dx ::; jl f, -g)l l fo +gl (x ) dx ::; 2M j ( f, - g) (x ) dx ::; 2Mc , -11" - 11" - 11" +0 and hence fo really can be approximated arbitrarily closely in the mean on [-n , n ] by piecewise-constant functions on the interval that vanish in a
neighborhood of the endpoints of the interval. c) It now suffices to learn how to approximate the functions in b) in mean. Let be such a function. All of its points of discontinuity X I , . . . , X n lie in the open interval ] There are only finitely many of them, so that for every E > 0 one can choose 5 > 0 so small that the 5-neighborhoods of the points XI , . . . , X n are disjoint and contained strictly inside the interval ] , [ , and 25nM < E, where M = sup Now replacing the function l x l�" on - 5, i = 1 , . . . , n, by the linear interpolation between the values 5) that assumes at the endpoints of this interval, we 5) and obtain a piecewise linear continuous function that is of compact support in -1r, 1r] . By construction I M on [ -1r, 1r] , so that
g
- n , n[.
-n n l g (x) l. gg(x -[xi xi +g(x5] , + g i i g0 [ l g0 ( x) ::; 11" 2 11" j (g - g0 ) (x) dx ::; 2M j l g - gol(x) dx = -11" x; o - 11" n + = 2M L j l g - gol(x) dx ::; 2M . (2M . 25) . n < 4Mc , t = I x; - o
and the possibility of the approximation is now proved. d) It remains only to show that one can approximate any function of class ] by a trigonometric polynomial. But for every E > 0 c) in mean on and every function of type Theorem · 5 enables us to find a trigonometric polynomial Tn that approximates uniformly within E on the closed interval and the possibility of an arbitrarily Tn ) 2 Hence J <
[-n, n 0 g , g0 [-n, n]. -11"11" (g0 - dx 2nc2 , precise approximation in mean by trigonometric polynomials on [ 1r] for any function of class c) is now established. By the triangle inequality in Rd - n , 1r] we now conclude that all of The orem 6 on the completeness of these classes in R 2 [-1r , n ] is also proved. D b. The Inner Product and Parseval's Equality Now that the complete ness of the trigonometric system in R2 ([- n , n] , C) has been proved, we can use Theorem 1 to assert that ( 18.66 ) ak ( f) cos kx + bk ( f) sin kx f = ao�f) + f k =I for every function f E R 2 ([-n, n] ,C), or, in complex notation, - 7r ,
548
18 Fourier Series and the Fourier Transform 00 -
(18.67) oo
where the convergence is understood as convergence in the norm of R2 [ -1r, 1r] , that is, as mean convergence, and the limiting passage in (18.67) is the limit n of sums of the form Sn ( x ) = E as -n If we rewrite Eqs. (18.66) and ( 18.67) as
ck ( f) ei kx n --+ oo.
(18.66') (18.67') then the right-hand sides contain series in the orthonormal systems
{
vk-ei kx;
1
1 1 r:;; cos kx, r:;; sin kx; k E N } 7r v 7r v 7r
2 , �
y
k E Z} . Hence by the general rule for computing the inner and { product of vectors from their coordinates in an orthonormal basis, we can assert that the equality
� ( !, g) = ao ( f �ao (g) + kf=l ak ( f) ak (g) + bk ( f) bk (g) holds for functions f and g in R 2 ( [ -7r , 1r ] , C ) , or, in other notation, 00 1 , = (f g) 'L00 ck ( f) ck (g) , 7r 2
( 18.68)
( 18.69)
-
where, as always,
7r
(f,g) J f (x ) g (x ) dx . In particular, if f = g, we obtain the classical Parseval equality from (18.68) and (18.69) in two equivalent forms: 2 (18.70) � 11 ! 11 2 = l ao �)l + kf=l l ak ( f)l 2 + l bk (J W ' ( 18.71) � 11 ! 11 2 = f l ck ( f)l 2 =
2
- (X)
·
18.2 Trigonometric Fourier Series
549
We have already noted that from the geometric point of view Parseval ' s equality can be regaded as an infinite-dimensional version of the Pythagorean theorem. Parseval ' s relation provides the basis for the following useful proposition. Proposition 3. (Uniqueness of Fourier series) . Let f and g be two functions in 'R. 2 [-1r, 1r] . Then a ) if the trigonometric series
converges in mean to f on the interval [-1r, 1r] it is the Fourier series of f ; b) if the functions f and g have the same Fourier series, they are equal almost everywhere on [1r , ?r] , that is, f = g in 'R.2 [-7r, 7r] . Proof. Assertion a) is actually a special case of the general fact that the expansion of a vector in an orthogonal system is unique. The inner product , as we know (see Lemma lb) shows immediately that the coefficients of such an expansion are the Fourier coefficients and no others. Assertion b) can be obtained from Parseval ' s equality taking account of the completeness of the trigonometric system in R. 2 ( [-7r, ?r] , C ) , which was just proved. Since the difference (! - g) has a zero Fourier series, it follows from Par seval ' s equality that II / - g ll n2 = 0. Hence the functions f and g are equal at all points of continuity, that is, almost everywhere. D Remark
11.
f:=O t<�!( a) ( x - a ) n we noted previn 00
When studying Taylor series
ously that different functions of class C ( ) (IR, lR) can have the same Taylor series (at some points a E lR). This contrast with the uniqueness theorem just proved for the Fourier series should not be taken too seriously, since every uniqueness theorem is a relative one in the sense that it involves a particular space and a particular type of convergence. For example, in the space of analytic functions (that is, functions that can 00 be represented as power series 2:: an ( z - z0 ) n converging to them pointwise) ,
n=O
two different functions have distinct Taylor series about every point . If, in turn, in studying trigonometric series we abandon the space 'R.2 [ -?r, 1r] and study pointwise convergence of a trigonometric series, then, as already noted (p. 531) one can construct a trigonometric series not all of whose coefficients are zero, which nevertheless converges to zero almost everywhere. According to Proposition 3 such a null-series of course does not converge to zero in the mean-square sense.
550
18 Fourier Series and the Fourier Transform
In conclusion, we illustrate the use of the properties of trigonometric Fourier series obtained here by studying the following derivation, due to Hur witz, 2 4 of the classical isoperimetric inequality in the two-dimensional case. In order to avoid cumbersome expressions and accidental technical difficulties, we shall use complex notation. Example 7. Between the volume V of a domain in the Euclidean space En , n ?: 2 , and the ( n - 1 )-dimensional surface area F of the hypersurface that bounds it , the following relation holds:
( 18. 72 ) called the isoperimetric inequality. Here Vn is the volume of the n-dimensional unit ball in E n . Equality in the isoperimetric inequality ( 18. 72) holds only for the ball. The name "isoperimetric" comes from the classical problem of finding the closed plane curve of a given length L that encloses the largest area S. In this case inequality (18. 72) means that
(18. 73) It is this inequality that we shall now prove, assuming that the curve in question is smooth and is defined parametrically as x = cp(s) , y = '1/J(s) , where s is arc length along the curve and cp and '1/J belong to c< l ) [0, L] . The condition that the curve be closed means that cp(O) = cp(L) and '1/J(O) = '1/J(L) . We now pass from the parameter s t o the parameter t = 27r f - 1r, which ranges from -7f to 1r, and we shall assume that our curve is defined paramet rically as y = y(t) , x = x(t) , - 7f � t � 7f , ( 18.74) with y( -7r) = y(7r) . (18.75) We write (18.74) as a single complex-valued function
z
= z(t) ,
-7f
�
t
�
where z(t) = x(t) + i y(t) and by (18.75) z ( -1r) We remark that
l z ' (tW
7f ,
= z (1r) .
(18.74 ' )
= (x ' (t)) 2 + (y' (t)) 2 = (�;f ,
and hence under our choice of parameter
( 18. 76) 2 4 A. Hurwitz (1859-1919) - German mathematician, a student of F. Klein.
551
18.2 Trigonometric Fourier Series
Next , taking into account the relations zz' = ( x - iy ) ( x' iy' ) = ( xx' yy ' ) i ( xy' - x'y ) , and using Eqs. ( 18.75) , we write the formula for the area of the region bounded by the closed curve (18.74) :
+
+
+
S=
� j( IT
xy ' - yx ' ) ( t ) dt
= ;i
j IT
z ' ( t ) z ( t ) dt .
(18.77)
We now write the Fourier series expansion of the function (18.74') : (X) z ( t ) = L c k ei k t . (X) Then (X) z ' ( t ) "' L ik ck ei k t . (X) Equalities (18.76) and (18.77) mean in particular that -
-
�2 ll z' ll 2 = 2� ] l z' ( t ) l 2 dt = 4�22 , - IT
and
2_ 2n ( z ' , z )
= 2_ 2n
IT
j
z ' ( t ) z ( t ) dt
= !:_ S . 1r
In terms of Fourier coefficients, as follows from Eqs. (18.69) and (18.71 ) , these relations assume the form (X) L 2 = 4n 2 L l kck l 2 , (X) -
Thus,
(X) L2 - 4n S = 4n 2 L ( 2 - ) l l 2 . (X) The right-hand side of this equality is obviously nonnegative and vanishes only if 0 for all E Z except = 0 and 1. Thus, inequality (18.73) is proved, and at the same time we have found the equation z (t) - 7r ::::: t ::::: 7r ' of the curve for which it becomes equality. This is the complex form of the parametric equation of a circle with center at in the complex plane and radius -
ck =
l c1l ·
k
k = co + c1 e'" t ,
k k ck k= c0
552
18
Fourier Series and the Fourier Transform
1 8 . 2 . 5 Problems and Exercises 1.
a) Show that
00 """' sm nx 1r - x for 0 < x < 21r , L.. 2 n =l n and find the sum of the series at all other points of JR. Using the preceding expansion and the rules for operating with trigonometric Fourier series, now show that the following equalities are true.
b) � L..J
.
_
_
si n 2 k x - for 0 < X < 7r . k=l 2 k = 4 2 s i n( 2kk -ll ) x 4 for O < X < 7r . c) � L..J 2k=l 00 ( d ) I; � sin nx = � for lxl < n =l e) x 2 = i + 4 f= < -n't cos nx for l x l < n =l 4 � cos( 2 k - l ) x rr f) x - 2 - -; L..J ( 2 k - l 2 for 0 � x � ) k=l 2!:
=
'E.
2!:
)n- 1
1r .
_
1r . 1r .
h) Sketch the graph of the sums of the trigonometric series here over the entire real line JR. Using the results obtained, find the sums of the following numerical series: oo 00 00
(-1) n
1
(-1t
' n2 . =O 2n + 1 ' nL =l n 2 nL: =l nL
Show that: a) if f : [ - 1r , 1r ] -+ IC is an odd (resp. even) function, then its Fourier coefficients have the following property: a k (f) = 0 ( resp. bk (f) 0) for k = 0 , 1, 2, . . . ; b) if f : lR -+ IC has period 27r/m then its Fourier coefficients c k (f) can be nonzero only when k is a multiple of m; c) if f : [-1r, 1r] -+ lR is real-valued, then c k (f) c _ k ( J ) for all k E N; d) la k (f) l � 2 sup l f(x) l , l bk (f) l � 2 sup l f(x) l , lck (f) l � sup l f(x) l . 2.
=
=
l x l
l x l
l x l
3. a) Show that each of the systems {cos kx; k = 0, 1, . . . } and {sin kx; k E N} is complete in the space R 2 [a, a + 1r] for any value of a E JR. b) Expand the function f(x) = x in the interval [0, 1r] with respect to each of these two systems. c) Draw the graphs of the sums of the series just found over the entire real line. d) Exhibit the trigonometric Fourier series of the function f(x) = l x l on the closed interval [-1r, 1r] and determine whether it converges uniformly to this function on the entire closed interval [ -7r, 1r] .
I8.2 Trigonometric Fourier Series 4.
553
The Fourier series f C k (f)ei k t of a function f can be regarded as a special case
of a power series f Ck Z k - (X)
(
=
E Ck Z k + f0 Ck Z k ) , in which z is restricted to the
- (X)
unit circle in the complex plane, that is, z = ei t . Show that if the Fourier coefficients Ck (f) of the function f : [ 1r] -+ I and k�+oo lim ic k (f) j 1 / k = c + < I , then k� - oo a) the function f can be regarded as the image of the unit circle under a function represented in the annulus c= 1 < jzj < c:j: 1 by the series f: c k z k ; - oo b) for z = x + iy and In __!_ c _ < y < In __!_ c + the series f: Ck (f)ei k x converges absolutely (and, in particular, its sum is independent of the order of summation of the terms). c) in any strip of the complex plane defined by the conditions a ::; Im z ::; b, where In c� < a < b < In c� , the series f c k (f)ei k x converges absolutely and uniformly; . d) using the expansion e z = I + TI + �!2 + and Euler's formula e' x = cos x + i sin x, show that I + cos x + . . . + cos nx + . . . = e cos x cos (sin x) , n! 1! sin x + . . . sin nx . . . = cos x -sin(sin x) ; + e + 1! n!
-?T
-
-
.
. t he expans10ns cos z = e ) usmg verify that
+ I)x f ) - I t cos(2n (2n + I ) ! + I)x I) - I t sin(2n (2n + I ) ! f:( - I t cos 2nx (2n) ! f:( - I t sin(2n!)2nx
n=O
n=O
n =O
n=O
z2 21
00
00 ·
-I-
,
·
·
z• + 4T -
· ·
·
. z=z and sm
-
z3 3T
zs + 5T
- , · ·
·
sin( cos x) cosh(sin x) , sin( cos x) sinh(sin x) , cos( cos x) cosh(sin x) , cos( cos x) sinh(sin x) .
Verify that a) the systems { I , cos k 2; x, sin k � x ; k E N } and { ei k 2T x ; k E Z } are orthog onal in the space R2 ( [a, a + T] ,
18 Fourier Series and the Fourier Transform
554
c) if C k (f) and
Ck (g) are the Fourier coefficients of T-periodic functions f and a+ oo 1 iT (x) dx = ck ( j ) ck ( g) ; f(x) L g T oo a d) the Fourier coefficients c k (h) normalized by the factor t of the "convolution " T h(x) = � I f(x - t)g(t) dt
g, then
0
of T-periodic smooth functions f and g and the coefficients c k (f) and functions themselves are related by C k (h) c k (f)c k (g), k E Z. 6. Prove that if a is incommensurable with 7!', then N " i kt .. e dt · a) N-t lim ..!.. L e i k ( x+n a ) ...!. J N 2 7r oo n =I b) for every continuous 27r-periodic function f : JR. --+ IC 1 1 N lim f(x + na ) = j (t) dt . N� oo N L 27!' n= l
=
=
-
ck (g) of the
'
7T
" I
- 7r
7. Prove the following propositions. a) If the function f JR. --+ IC is absolutely integrable on JR., then
:
1 1 f(x)eiAx dx l l l f (x + �) - j (x) l dx . -
S
(X)
- CX)
b) If the functions f : JR. --+ IC and g : JR. --+ IC are absolutely integrable on JR. and g is bounded in absolute value on JR., then I j (x + t)g(t) e''A t dt =: . --+ 00
oo .
c) If f : JR. --+ IC is a 27r-periodic function that is absolutely integrable over a period, then the remainder Sn (x) - j (x) of its trigonometric Fourier series can be represented as 7r
Sn (x) - j(x) = � I( L1? f)(x , t)Dn (t) dt , 0
where D n is the nth Dirichlet kernel, and ( L1 2 j )(x, t) = f(x + t) - 2 j (x) + f(x - t). d ) For every 8 E]O, 7r[ the formula for the remainder just obtained can be brought into the form 0
sin nt (L1 2 f)(x, t) dt o (1) , Sn (x) - f(x) = - I -+ t 1
7l'
0
where o ( 1 ) tends to zero as n --+ oo , and uniformly on each closed interval [a, b] on which f is bounded.
18.2 Trigonometric Fourier Series
555
e) If the function f : [-1r, 1r] -+ C satisfies the Holder condition i f(xi ) - J(x 2 ) l :S Mlx 1 - x 2 l"' on [-1r, 1r] (where M and a are positive numbers) and in addition f( -1r) = j(1r) , then the Fourier series of f converges to it uniformly on the entire interval. 8. a) Prove that if f : R -+ R is a 21r-periodic function having a piecewise contin uous derivative f ( m ) of order m (m E N) , then f can be represented as
f(x) where Bm (u)
=
�0 + � I Bm (t - x)f ( m ) (t) dt , .,.
_ .,.
L
00 cos(ku+ 2� ) , k '"'
m E N. n= l b) Using the Fourier series expansion obtained in Problem 1 for the function " ;"' on the interval [0 , 21r] , prove that B (u) is a polynomial of degree 1 and Bm (u) 1 is a polynomial of degree m on the interval [0 , 27r] . These polynomials are called the Bernoulli polynomials. 2 11" c) Verify that J Bm (u) du = 0 for every m E N. 9.
a) Let Xm
=
=
0
;;_;;_ , m = 0, 1, . . . , 2n. Verify that
2n 2 cos kxm cos lxm = 8kl , L 2n + 1 m=O 2n 2 '""' sin kxm sin lxm = 8kl , 2n + 1 m� =O 2n sin kxm cos lxm = 0 , L m =O where k and l are nonnegative integers, 8kl = 0 for k #- l, and 8kl = 1 for k = l. b) Let f : R -+ R be a 27r-periodic function that is absolutely integrable over a period. Let us partition the closed interval [0, 27r] into 2n + 1 equal parts by the points Xm = ;;_;;_ , m = 0, 1 , . . . , 2n. Let us compute the integrals ---
a k (f)
=
2 11"
� I f ( x) cos kx dx , 0
bk (f)
=
2 11"
� I f ( x) sin kx dx 0
approximately using the rectangular method corresponding to this partition of the interval (0, 21r] . We obtain the quantities 2n 2 cos kx m , L 2n + 1 m =O f(xm) 2n 2 '""' f(xm) sin kxm , 2n + 1 m� =O
--
556
18 Fourier Series and the Fourier Transform
which we place in the nth partial sum Sn (f, x) of the Fourier series of f instead of the respective coefficients a k (f) and b k (f). Prove that when this is done the result is a trigonometric polynomial Sn (f, x) of order n that interpolates the function f at the nodes Xm , m = 0, 1 , . . . , 2n, that is, at these points f(x m ) = Sn (x, m ) . 1 0 . a) Suppose the function f : [a, b] --+ R is continuous and piecewise differ entiable, and suppose that the square of its derivative f' is integrable over the interval ]a, b[. Using Parseval's equality, prove the following: a) if [a, b] = [0, 1rj , then either of the conditions f(O) implies Steklov 's inequality
=
j ( 1r )
=
0 or I f(x) dx = 0 7r
0
I f2 (x) dx ::; l (f') 2 (x) dx , 7r
7r
0
0
in which equality is possible only for f(x)
=
a cos x;
b) if [a, b] = [ - 1r , 1r] and the conditions f( - 1r ) hold, then Wirtinger 's inequality holds: 7r
=
j ( 1r ) and I f(x) dx = 0 both 7r
7r
I l (x) dx ::; I (J' ) 2 (x) dx , where equality is possible only if f(x)
=
a cos x + b sin x.
1 1 . The Gibbs phenomenon is the behavioral property of the partial sums of a trigonometric Fourier series described below, first observed by Wilbraham (1848) and later (1898) rediscovered by Gibbs ( Mathematical Encyclopedia, Vol. 1, Moscow, 1977) . a) Show that
ao sin(2k - 1)x for x i < 7r . sgn x = -7r4 L I k= l 2 k - 1 b) Verify that the function Sn (x) = � £: s i n ��k_-/ ) x has a maximum at x = 27\"n k= l and that as n --+ oo 7r 2 ...f-, sin(2k - 1 ) 271"n -1r 2 sin x Sn = - � --+ - -dx � 1 . 179 . 2n 1r (2k - 1) n 1r x 2n k= l 0 Thus the oscillation of Sn ( x) near x = 0 as n --+ oo exceeds the jump of the function sgn x itself at that point by approximately 18% (the jump of Sn (x) "due to inertia" ) . c ) Describe the limit of the graphs of the functions Sn (x) i n problem b) . Now suppose that Sn (f, x) is the nth partial sum of the trigonometric Fourier series of a function f and suppose that Sn (f, x) --+ f(x) in a deleted neighborhood
( ) -
7r
·
11\"
18.2 Trigonometric Fourier Series
557
0 < ix - .;1 < J of the point .; as n --+ oo and that f has one-sided limits f(.; _ ) and f(.;+ ) at .;. For definiteness we shall assume that f (.; - ) :::; f(.;+ ) · We say that Gibbs ' phenomenon occurs for the sums Sn (!, x) at the point .; if lim Sn (f, x) < f(.; - ) ::; f(.; + ) < lim S (f, x) . n-->oo n n-->oo d) Using Remark 9 show that Gibbs' phenomenon occurs at the point .; for every function of the form tp(x) + c sgn (x - .;) , where c oj 0, 1.;1 < 1r, and tp E CC l l [-7r, 7r] . 12.
Multiple trigonometric Fourier series. a) Verify that the system of functions
e i k x , where k = (k 1 , . . . , k n ) , X = (x 1 , . . . , X n ) , kx = k 1 X 1 + + k n X n , and ( 2 1r �n ; 2 k 1 , . . . , kn E /Z, is orthonormal on any n-dimensional cube I = { x E lR n I aj :'::: Xj :'::: · ·
ai + 27r, j = 1 , 2, . . . , n}.
b) To a function f that is integrable over I we assign the sum f
"'
·
f: Ck (f) e i k x ,
- oo which is called the Fourier series of f in the system { 2 " �n 1 2 ei k x }, if C k (f) = - i k x dx. The numbers C k (f) are called the (Fourier coefficients of f 2( " �n ; 2 II f (x)e · the system { 27r 1n / 2 ei kx } . tn ( ) In the multidimensional case the Fourier series is often summed via the partial sums SN (x) = L Ck (f) e i k x , i k i :S N where lkl :'::: N means that N = (N1 , . . . , Nk ) and l ki l :'::: Nj , j = 1 , . . . , n. Show that for every function f(x) = f(x 1 , . . . , x n ) that is 21r-periodic in each variable
where D Nj (u) is the Njth one-dimensional Dirichlet kernel. c) Show that the Fejer sum
of a function f(x) represented as
f(x l , . . . , X n ) that is 27r-periodic in each variable can be
n where P N (u) = IT :FNj (ui ) and :FNj is the Njth one-dimensional Fejer kernel. j=l d) Now extend Fejer's theorem to the n-dimensional case. e) Show that if a function f that is 21r-periodic in each variable is absolutely integrable over a period I, possibly in the improper sense, then I if(x + u) I f(x) l dx --+ 0 as u --+ 0 and I i f - aN i (x) dx --+ 0 as N --+ oo . -
I
558
18
Fourier Series and the Fourier Transform
f) Prove that two functions f and g that are absolutely integrable over the cube
I can have equal Fourier series (that is, C k (f) = C k g) for every multi-index k) only if f(x) = g(x) almost everywhere on I. This is a strengthening of Proposition 3 on
uniqueness of Fourier series. g) Verify that the original orthonormal system { (27r�n / 2 e i k x } is complete in R2 ( I) , so that the Fourier series of every function f E R2 ( I ) converges to f in the mean on I . h) Let f be a function in c C =l (JR n ) of period 21r in each variable. Verify that ck (f ( "' ) ) = i l "' l k "' c k (f) , where a = (a 1 , . . . , a n ), k = (k 1 , . . . , kn ), l a l = l a 1 l + + k';: n , and aj are nonnegative integers. I a n I , k"' = kr' i) Let f be a function of class c C mn l (JR n ) of period 21r in each variable. Show that if the estimate 1 - ("') 2 (x) dx < M 2 (27r) n IJ I · · ·
·
.
.
.
·
/
holds for each multi-index j = 1, . . . , n ) , then
I
a = (a 1 , . . . , a n )
such that
aj
1s 0 or
m
(for every
IJ (x) - Sn (x) l :::: NCM m- 2 where N = min{N1 , . . . , Nn } and C is a constant depending on m but not on N or -1 ,
E I. j) Notice that if a sequence of continuous functions converges in mean on the interval I to a function f and simultaneously converges uniformly to '{), then f(x) = 'P(x) on I. Using this observation, prove that if a function f JR n ---+ C of period 27r in each variable belongs to c C l l (JR n , C), then the trigonometric Fourier series of f converges to f uniformly on the entire space JR. n .
X
:
:
1 3 . Fourier series of generalized functions. Every 21r-periodic function f lR ---+ C can be regarded as a function f ( s) of a point on the unit circle r (the point is fixed by the value of the arc-length parameter s, 0 :::: s :::: 21r) . Preserving the notation of Sect. 17 . 4, we consider the space V(r) on r consist ing of functions in c C = l ( r) and the space V' ( r) of generalized functions, that is, continuous linear functionals on V(r) . The value of the functional F E V' (r) on the function 'P E V( r) will be denoted F ( 'P) , so as to avoid the symbol (F, 'P) used in the present chapter to denote the Hermitian inner product (18.38). Each function f that is integrable on r can be regarded as an element of V' ( r) (a regular generalized function) acting on the function 'P E V(r) according to the formula 71"
2
f('P)
=
j f(s)'{J(s) ds . 0
Convergence of a sequence { Fn } of generalized functions in V' ( r) to a general ized function F E V' ( r) , as usual, means that for every function 'P E V( r)
18.2 Trigonometric Fourier Series
559
a) Using the fact that by Theorem 5 the relation rp( s ) = f: Ck ( rp )e i k x holds for 00
f: ck (rp) , show that in the -oo sense of convergence in the space of generalized functions V' (r) we have n 1 """" - e i k s -+ 8 as n -+ oo . L..- 27r k=n every function rp E c (r) , and, in particular, rp(O) =
-
Here 8 is the element of V' ( r) whose effect on the function rp E V( F) is defined by 8(rp) = rp(O) . b ) If f E R( F), the Fourier coefficients of the function f i n the system { e i k s } defined in the standard manner, can be written as 2 1f ck (f) = 2 f ( s )e - i k x dx = 2 f (e - i k s ) .
�
�I 0
By analogy we now define the Fourier coefficients c k (F) of any generalized function F E V' (r) by the formula Ck (F) = 2� F(e - i k • ) , which makes sense because e - i k s E V(r) . Thus to every generalized function F E V' ( r) we assign the Fourier series - oo 00 Show that 8 "' I:; 2� e i ks . - oo c) Prove the following fact, which is remarkable for its simplicity and the free dom of action that it provides: the Fourier series of every generalized function F E V' ( r ) converges to F (in the sense of convergence in the space V' ( r )) . d) Show that the Fourier series of a function F E V' ( r ) (like the function F itself, and like every convergent series of generalized functions) can be differentiated termwise any number of times. e) Starting from the equality 8 = f: 2� e i k s , find the Fourier series of 8' . - oo f) Let us now return from the circle r to the line lR and study the functions e i k s as regular generalized functions in V' (.IR) (that is, as continuous linear functionals on the space V(.IR) of functions in the class c6 oo) (.IR) of infinitely differentiable functions of compact support in .IR) . Every locally integrable function f can be regarded as an element of V' (.IR) (a regular generalized function in V' (.IR) ) whose effect on the function rp E c6 oo) (.IR, iC) is given by the rule f(rp) = J f(x)rp(x) dx. Convergence in V' (.IR) is defined in the IR standard way:
( nlim�� Fn = F) := Vrp E V(.IR) ( nlim�� Fn (rp) = F(rp) )
560
18 Fourier Series and the Fourier Transform Show that the equality
� f ei kx f 8(x - 27rk)
2
-
ex:>
=
-
oo
holds in the sense of convergence in V' (IR) . In both sides of this equality a n limiting passage is assumed as n --+ oo over symmetric partial sums 2: , and -n 8(x - xo ) , as always, denotes the 8-function of V' (R) shifted to the point xo , that is, 8(x - xo) (cp) = cp(xo) .
1 8 . 3 The Fourier Transform 18.3. 1 Representation of a Function by Means of a Fourier Integral
The Spectrum and Harmonic Analysis of a Function Let f(t) be aa.function T-periodic function, for assume examplethata periodic signalf iswithabsolutely frequencyintegrable � as a of time. We shall the function over period.theExpanding f in a Fourier series (when f is sufficiently regular, as wea know, Fourier series converges to f) and transforming that series, f(t) = ao�f) + f. a k (f) cos kwot + bk (f) sin kw t k= l = :�::>k (f)ei kwa t = Co + 2 L00 i ck i cos(kwot + arg ck ) , ( 18.78) k= l wemeanobtain a representation of f as a sum of a constant term � = co - the value of f over a period and sinusoidal components with frequencies v0and=so� on.(theInfundamental frequency), 2 vo (the second harmonic frequency), general the kth harmonic component 2jc i cos (k � t + arg c ) of the signal has frequency kvo = � ' cyclic frequency kwk 0 = 2rrkvo = �kk, amplitude 2ic k i = Ja� + b�, and phase arg c k = - arctan � The expansion of isa called periodicthefunction (signal) into a sum of simple harmonic oscillations harmonic analysis of f. The numbers {ck (f); k E Z} or {a0 (f), a k (f), bk (f); k E N} are called the spectrum of the function (signal) f. A periodic function thus has a discrete spectrum. Let us now set out (on a heuristic level) what happens to the expansion ( 18. 78) when the period T of the signal increases without bound. Simplifying the notation by writing l = f and O:k = kf, we rewrite the expansion 00 f(t) = L Ck ei k f t o
-
oo
-
-
oo
=
18.3 The Fourier Transform
as follows: f(t)
where
= f= (ck � ) e i k f t y , -oo
561
( 18.79)
l
Ck = ;l j f (t) e- i o:k t dt -l
and hence
l
Ck -l = -1 J f (t) e -IO:k t dt . 21f
1f
l
.
-l
Assuming that in the limit as --+ +oo we arrive at an arbitrary function f that is absolutely integrable over �. we introduce the auxiliary function
c(o:)
= 2�
00
j f (t) e- io:t dt ,
( 18.80)
-oo
whose values at points o: = O: k differ only slightly from the quantities formula (18.79) . In that case f (t)
�
kf
2:.:: c ( o: k ) eic� k t y ' -oo 00
f.
Ck � in ( 18.81)
where O: k = and O: k+ l - O: k = This last integral resembles a Riemann sum, and as the partition is refined, which occurs as --+ oo , we obtain
00
f(t) =
j c(o:)eio:t do: .
-
l
( 18.82)
00
Thus, following Fourier, we have arrived at the expansion of the function
f into a continuous linear combination of harmonics of variable frequency
and phase. The integral ( 18.82) will be called the Fourier integral below. It is the continuous equivalent of a Fourier series. The function c( o:) in it is the analog of the Fourier coefficient , and will be called the Fourier transform of the function f ( defined on the entire line � ) . Formula (18.80) for the Fourier transform is completely equivalent to the formula for the Fourier coefficients. It is natural to regard the function c(o:) as the spectrum of the function ( signal ) f . In contrast to the case of a periodic signal f considered above and the discrete spectrum ( of Fourier coefficients ) corresponding to it , the spectrum c(o:) of an arbitrary signal may be nonzero on whole intervals and even on the entire line ( continuous spectrum) .
562
18 Fourier Series and the Fourier Transform
{
Example 1 . Let us find the function having the following spectrum of compact support: h ' if l a l � a , c (a) = (18.83) 0 , if l a l a . Proof. By formula ( 18.82) we find, for t =f. 0
f( t ) and when t t -+ 0. D
=
a
I heiat da
-a
=
>
ei h a t � e i at t
=
2h
�
si at
'
(18.84)
= 0, we obtain f(O) = 2 ha , which equals the limit of 2 h "i� a t as
The representation of a function in the form ( 18.82) is called its Fourier integral representation. We shall discuss below the conditions under which such a representation is possible. Right now, we consider another example.
Example 2. Let P be a device having the following properties: it is a linear signal transform, that is, P 2;:: aj fJ = 2;:: aj P(fJ ) , and it preserves the
(
)
J
J
periodicity of a signal, that is, P(eiwt) = p(w) e iwt , where the coefficient p(w) depends on the frequency w of the periodic signal e iwt . We use the compact complex notation, although of course everything could be rewritten in terms of cos wt and sin wt. The function p(w) = : R(w) ei cp ( w ) is called the spectral characteristic of the device P. Its absolute value R( w ) is usually called the frequency characteristic and its argument 'P ( w ) the phase characteristic of the device P. A signal eiwt , after passing through the device, emerges transformed into the signal R (w ) ei (wt + cp (w )) , its amplitude changed as a result of the factor R (w ) and its phase shifted due to the presence of the term 'P ( w ) . Let us assume that we know the spectral characteristic p( w ) of the device P and the signal j ( t ) that enters the device; we ask how to find the signal x ( t ) = P(f) (t) that emerges from the device. Representing the signal f(t) as the Fourier integral ( 18.82) and using the linearity of the device and the integral, we find x (t)
=
P(f) (t)
In particular, if
p (w ) =
{
=
00
I c(w)p(w)eiwt dw . oo
-
� fl , l w l > fl '
1 for l w l
0 for
( 18.85)
18.3 The Fourier Transform
then
x(t)
=
n
J
-n
563
c(w)eiw t dw
and, as one can see from the spectral characteristics of the device, for lw l � n , for lw l > n
0
A device P with the spectral characteristic ( 18.85) transmits (filters) fre quencies not greater than n without distortion and truncates all of the signal involved with higher frequencies (larger than il ) . For that reason, such a de vice is called an ideal low-frequency filter (with upper frequency limit n) in radio technology. Let us now turn to the mathematical side of the matter and to a more careful study of the concepts that arise. b. Definition of the Fourier Transform and the Fourier Integral In accordance with formulas ( 18.80) and ( 18.82) we make the following defini tion. Definition 1 . The function
F[ f ] ( � )
00
2_ J f x
:=
21f
- oo
( ) e - i f; x dx
( 18.86)
is the Fourier transform of the function f : lR -+ C. The integral here is understood in the sense of the principal value
oo
Joo f(x)e-if;x x
-
d
:=
A
J f(x)e-if;x x
A -lim t +oo A -
d ,
and we assume that it exists. If f : 1R -+ C is absolutely integrable on JR, then, since l f (x)e - i x f; l = l f(x) l for x, � E JR, the Fourier transform ( 18.86) is defined, and the integral ( 1 8.86) converges absolutely and uniformly with respect to � on the entire line JR.
Definition 2. If c(�) = F[ f ] ( � ) is the Fourier transform of f : lR -+ C, then the integral assigned to f, 00
f (x)
rv
Joo
-
c(�)e i x f; d� ,
understood as a principal value, is called the Fourier integral of f.
( 1 8.87)
564
18 Fourier Series and the Fourier Transform
The Fourier coefficients and the Fourier series of a periodic function are thus the discrete analog of the Fourier transform and the Fourier integral respectively. Definition 3. The following integrals, understood at> principal values,
00 � J f(x) cos �x dx , oo 00 . 1 T. [J] (�) : = ; J f(x) sm �x dx , 00 Fe [!] (�) : =
( 18.88)
-
( 18.89)
-
are called respectively the Fourier cosine transform and the Fourier sine transform of the function f. Setting c(�) = F[/] (0 , a(�) = Fc [J] (O , and b(O = Fs [/] (0 , we obtain the relation that is already partly familiar to us from Fourier series 1 . c(O = 2 (a(�) - zb(�) ) .
As can be seen from relations ( 18.88 ) and ( 18.89 ) , a(-0 = a(�) ,
b( -0 = -b(�) .
(18.90) ( 18.91 )
Formulas ( 18.90 ) and ( 18.91 ) show that Fourier transforms are completely determined on the entire real line lR if they are known for nonnegative values of the argument. From the physical point of view this is a completely natural fact - the spectrum of a signal needs to be known for frequencies w ;:::: 0; the negative frequencies o: in ( 18.80 ) and ( 18.82 ) - result from the form in which they are written. Indeed, A
J
-A
A
A
0
c(Oei x � d� =
( J J) +
0
-A
c(�)ei x � d� =
J
( c(�)ei x � + c( -0e - i x � ) d� =
0
A
=
J
(a(�) cos x� + b(O sin x�) d� ,
0
and hence the Fourier integral ( 18.87 ) can be represented as
00 J (a(O cos x� + b(�) sin x�) d� , 0
(18.87')
18.3 The Fourier Transform
565
which is in complete agreement itwith classical form (of1 8.90) a Fourier Ifthatthe function f is real-valued, followsthe from formulas and ( series. 18.91) ( 1 8.92) sincefromin this casedefinitions a(�) and(18.88) functions on JR., as one can b(�) areandreal-valued see their ( 18.89 ) On the other hand, under the assumption( 18.86) f(x) =off(x), Eq. ( 18.92 ) can be obtained immediately from the definition the Fourier transform, if we take into account that the conjugation sign can be moved under the integral sign. This last observation allows us to conclude that ( 18.93 ) F[/]( -�) = F[f] (�) for Iteveryis alsofunction ftoJR.note -+ IC. useful that if f is a real-valued even function, that is, f(x) = f(x) = f( -x), then Fc [J] (�) = Fe[J] (� ) , Fs[J] ( �) 0 , ( 18.94 ) F[f] (�) = F[f] ( 0 = F[f] ( - � ) ; and if f is a real-valued odd function, that is, f(x) = f(x) = -f( -x), then Fe[J] (�) 0 , Fs[J] (�) = Fs[J] (� ) , ( 18.95 ) F[f] (�) = - F[f] (� ) = F[f] ( - �) ; and if f is a purely imaginary function, that is, f(x) = -f(x), then ( 18.96 ) F[/] ( - �) = - F[f] (�) . that asif f is a real-valued function, its Fourier integral ( 18.87' ) canWealsoremark be written 00 00 j y'a2 (�) + b2 (�) cos(x� + cp(�)) d� = 2 j l c(�) I cos(x� + cp(�)) d� , :
=
=
0
0
where cp(�) = - arctan !i�� = argc(�). Example Let us find the Fourier transform of f(t) f(O) = a E JR.). 3.
si� at
(assuming
18 Fourier Series and the Fourier Transform
566
A
F[f] (a)
= A-t+oo lim __!___27f J sint at e� ia �A
__!___
= A-t+oo lim 27f
d
dt =
A
+oo 2_ sin at cos at dt = J sin at cos at dt = J t
�A
27f
0
t
( sin (a + a)t + sin ( a - a)t ) dt = J 27f t t oe sin u { � sgn a ' if I a I :::; Ia I ' 1 ( =sgn (a + a) + sgn ( a - a)) J - du = 27r u 0, if lal > l a l , since we know the value of the Dirichlet integral 00 sinu d 7r ( 18 9 7 ) J -2 u Hence if we assume a ;:::: 0 and take the function f(t) = 2h s i �a t of Eq. ( 18.84) , we find, as we should have expected, that the Fourier transform is the spectrum of this function exhibited in relations (18.83). The transform function fhasin discontinuities. Example 3 is notThatabsolutely integrable on ofJR,anandabsoits Fourier the Fourier transform lutely integrable function has no discontinuities is attested by the following lemma. Lemma 1 . If the function f lR -t
+oo 0
0
U - - .
.
_
0
:
-t oo .
R
�A
�A
18.3 The Fourier Transform
establishes that the integral A
567
;7r J f(x)e -ix� dx , with respect to �; and the uniform convergence of this integral asis continuous A --+ +oo enables us to conclude that F [ / ] E C(JR., C) . 0 Example 4. Let us find the Fourier transform of the function f(t) = e- t 2 1 2 : +oo +oo 2 / t t a 2 -i e- t2 / 2 cosatdt . dt = e e J F[/](a) = J oo - oo Differentiating integrating by parts,thiswelastfindintegral that with respect to the parameter a and then d:�J (a) + aF [f] (a) = 0 , or d ln F[/](a) = -a . da - <>2 / 2 , where c is a constant which, using It follows that [ (a) = ce f ] F the Euler-Poisson integral (see Example 17 of Sect. 17.2) we find from the relation + oo c = F[/] (0) = j e - t2 1 2 dt = V2/ff . - 00 Thus we have found that F2 / 2[/] (a) = ,;2iie - <> 2 / 2 , and simultaneously shown that Fc[f] (a) = ,;2iie - <> and Fs [/] (a) 0. Normalization of the Fourier Transform We obtained the Fourier transform (18.80) and the Fourier integral (18.82) as the natural continuous analogs of the Fourier coefficients ck = 2� J11" f(x)e -i kx dx and the Fourier series I:00oo ck ei kx of a periodic function f in the trigonometric syskx ; k E Z} . This system is not orthonormal, and only the ease of tern {ei writing a trigonometric in it has caused kx ; k tradition E Z } . In ally instead of the moreFourier naturalseries orthonormal systemit00{ tokbeeiused this normalized system the Fourier series has the form -I:00 k ei kx , and the Fourier coefficients are defined by the formulas 2,., = k -J11"11" f(x)e -i kx dx. -A
-
:::=
c.
-
2,.,
568
18 Fourier Series and the Fourier Transform
The series continuous of suchtransform natural Fourier coefficients and such a Fourier would analogs be the Fourier 00 ( 18.98) f( � ) : = vk I f(x)e-i x f. dx oo and the Fourier integral 00 ( 18.99) f(x) = vk Ioo f(�)eixf. d� , whichIn the differsymmetric from thoseformulas considered(18.98) aboveandonly( 1 8.99) in thethenormalizing coefficient. Fourier "coefficient" and the Fourier "series"onlypractically coalesce,ofandthesointegral in the future we shall es sentially be interested in the properties transform ( 18.98) , calling itthetheFourier normalized Fourier transform or, where no confusion can arise, simply transform of the function f . In general the name integral operator or integral transform is customarily given to an operator A that acts on a function f according to a rule -
-
A(f) (y)
= I K(x, y)f(x) dx , X
where K(x, y) is a given function called the kernel of the integral operator, and X JR. n is the set over which the integration extends and on which the integrands are assumed to be defined. Since y is a free parameter in some set Y, it follows that A( ! ) is a function on Y. In mathematics there are many important integral transforms, and among them the Fourier transform occupies one of the most key positions. The reasons for this situation go very deep and i n volve the remarkable properties ofillustrate the transformation ( 1 8 .98) , which we shall to some extent describe and in action in the partFourier of this transform section. (18.98) . Thus, we shall study theremaining or;_ rmalized Alongthewithnotation the notation f for the normalized Fourier transform, we in troduce 00 I f(x)eif.x dx , �) = - ( 18. 100) J( : vk C
-
oo
thatFormulas is, J(�) =( 1f(8 .98)-�)and . (18.99) say that -::::.
i= f = f ,
(18.101)
18.3 The Fourier Transform
569
that other. is, the Hence integralif (18.98) transforms (18.98) and (18.99) are mutually inverse to each is the Fourier transform, then it is natural to call the Weintegral operator (18. 100) the inverse Fourier transform. discussandinjustidetail belowForcertain Fourier shall transform fy them. exampleremarkable properties of the f n l ( O = ( iO n [(o , (
r;g 11111
= y'2;f . g ' = ll f ll .
That is,of multiplication the Fourier transform maps the operator of differentiation into the operator by the independent variable; the Fourier transform ofFouritheerconvolution of functionsthe amounts to multiplying transforms; the ' s equality),the and transform preserves norm (Parseval is therefore an isometry of thebegin corresponding functionformula space. ( 1 8 . 1 01 ) . But we shall with the inversion another convenient normalization of the Fourier transform see Prob lemFor10 below. d. Sufficient Conditions for a Function to be Representable as a Fourier Integral We shall now prove a theorem that is completely analo gous inFourier both form andat content toTothepreserve theoremtheonfamiliar convergence of a trigono metric series a point. appearance of our earlier formulas and transformations to the maximum extent, we shall use the nonnormalized Fourier transform c (O in the present part of this subsec tion, together with itswhen ratherstudying cumbersome but sometimes convenient notation Afterwards, the integral Fourier transform as such, [ ] ( . f O .F wetionshalf. l as a rule work with the normalized Fourier transform 1 of the func Theorem 1 . (Convergence of the Fourier integral at a point). Let f : lR --+ C be an absolutely integrable function that is piecewise continuous on each finite
closed interval of the real axis R If the function f satisfies the Dini conditions at a point x E IR, then its Fourier integral ( ( 18.82) , (18.87) , (18.87' ) , (18.99) ) converges at that point to the value � ( f ( x _ ) + f ( x + ) ) , equal to half the sum of the left and right-hand limits of the function at that point.
By Lemma 1 the Fourier transform c(�) = .F [ f ]( � ) of the function f isweProof. continuous on lR and hence integrable on every interval [-A, A] . Just as transformed the partialofsum of the Fouri following transformations the partial Fourierer series, integral:we now carry out the
570
18 Fourier Series and the Fourier Transform
A
A x SA (x) = J c( () ei � d( = J ( 2� J f(t) e -it� dt) eix� d( = -A -A A x - t ) � d( ) dt = � J J(t) ei ( x - t )� - e -i( x - t ) A dt i( e ( = � J(t) 2n 1(x - t) 2n A 00 00 .!_ .!_ sin(x t)A = f(x + u) sinuAu du = J f(t) x - t dt oo
- oo
oo
oo
J
J
- oo
1r
- oo
=
- oo
=
J
1r
- oo
sinAu du . J ( f(x - u) + f(x + u) ) -u The ofchange in the order isoflegal. integration atintheviewsecond equality fromcontinuity the be ginning the computation In fact, of the piecewise of J , for every finite B > 0 we have the equality A A t x J ( 2� J j( t) e -i � dt) ei � d( = 2� J f(t) ( J ei(x - t � d() dt , -A -A from which as B -+ +oo, taking account of the uniform convergence of the integral J f(x) e-it� dt with respect to ( , we obtain the equality we need. We now use the value of the Dirichlet integral (18.97) and complete our transformation: 1
00
= 7r
B
B
0
B
-B
-B
)
-B
.!_
+ (f(x - u) - f(x_)) + (f(x + u) - f(x )) + sinAudu . oo
J u The resulting integral tends to zero as A -+ oo. We shall explain this and thereby finish the proof of the theorem. We represent this integral as the sum of the integrals over the interval ]0, 1] and over the interval [ 1 , +oo[. The first of these two integrals tends to zero as iAndexananameRiemann, -+ +oo in view of the BDini. Theconditions and the Riemann-Lebesgue lemma. second integral is the sum of four integralsThecorresponding to the four terms f(x - u), f(x + u), f(x_) and f(x + )· Riemann-Lebesgue lemma applies to the first two of these four integrals, and the last two can be brought into the following form, up to a constant factor: + sin Au J u du = AJ si: v dv . =
7r
0
oo
+ oo
18.3 The Fourier Transform
571
But as A +oo this last integral tends to zero, since the Dirichlet integral (18.97) converges. D Remark 1 . In the proof of Theorem 1 we have actually studied the conver gence of the integral as a principal value. But if we compare the notations (18.87) and (18.87'), it becomes obvious that it is precisely this interpretation of the integral that corresponds to convergence of the integral (18.87'). --+
From this theorem we obtain in particular
Corollary 1 . Let f : JR; --+ C be a continuous absolutely integrable function. If the function f is differentiable at each point x E JR; or has finite one sided derivatives or satisfies a Holder condition, then it is represented by its Fourier integral.
Hence for functions of these classes both equalities (18.80) and (18.82) or (18.98) and (18.99) hold, and we have thus proved the inversion formula for
the LetFouriuserconsider transformseveral for such functions. examples. Example Assume that the signal v (t) = P(f) (t) emerging from the device P considered in Example 2 is known, and we wish to find the input signal f (t) entering the device P. In Example 2 we have shown that f and v are connected by the relation 00 v( t) = J c(w)p(w) e iw t dw , -oo where c(w) F[f](w) is the spectrum of the signal F (the nonnormalized Fouri e r transform of the function f) and p is the spectral characteristic of the device P. Assuming all these functions are sufficiently regular, from the theorem just proved we conclude that then 5.
=
c(w)p(w) F[v] (w) . find c( w)(18.87). = F[f] ( w). Knowing c( w), we find usingFromthe this Fouriweer integral Example 6. Let a 0 and =
>
for x 0 , for x :::; 0 . >
Then
+oo _.!_ _1 . F[f] (�) = 21r J e - ax e il; x dx = _.!_ 21r a + �� -
0
_ _
the signal f
572
18 Fourier Series and the Fourier Transform
In adiscussing theits definition of the Fourier transform, we havesubsection. already noted number of obvious properties in Part b of the present _ (x) := f(-x), then F[f- ] (�) = F[f](-�). This is We note furtherchange that ifofJvariable an elementary - a linx l =thef(x)integral. We now take the function e + f( -x) = : 0,'!j;(x)to the= f(x) sionIfofwethenowfunction entire- f(real- x),line,which thenis an odd exten Using Theorem 1 , or more precisely the corollary to it, we find that e ' if X > 0 , 00 1 eix� d� = if X .= 0 , 27r J a + . � - oo 0 , if X < 0 ; +oo 1 ae i x � d � = e - a l x l . J ' 7r a2 + �2 e ' if X > 0 , +oo x � �ei d� = 0 ' if X = 0 , 7r J a 2 + � 2 -oo -eax if X < 0 . All thetheintegrals here arein understood inabsolute the senseconvergence, of the principal value,be although second one, view of its can also understood in thethe sense of animaginary ordinary parts improper integral. Separating real and i n these find the Laplace integrals we have encountered earlier last two integrals, we +oo J acosx� 2 + e d� = +oo � J sinx 2 2 d� = -1
()()
r= 1
2 '
r·"
0
0
a +�
18.3 The Fourier Transform
573
Example On the basis of Example 4 it is easy to find (by an elementary change of variable) that if 2 e2 1 4,;2" . e then ! (f. ) = f ( x) = e - a 2 x , v'2a Itfunctions is very instructi ve to trace the simultaneous evolution of the graphs of the f and j as the parameter a varies from 1/v'2 to 0. The more "concentrated"is one of theconnected functionswith is, thethemore "smeared" the otherprinciple is. This circumstance closely Heisenberg uncertainty in quantum mechanics. (In this connection see Problems 6 and 7.) Remark In completing the discussion of the question of the possibility of representing a function by a Fourier integral, we note that, as Examples 1 and 3 show, the conditions on f stated in Theorem 1 and its corollary are sufficient but not necessary for such a representation to be possible. 7.
�
2.
18.3.2 The Connection of the Differential and Asymptotic Properties of a Function and its Fourier Transform a. Smoothness of a Function and the Rate of Decrease of its Fourier Transform lR 1
It follows fromintegrable the Riemann-Lebesgue lemma that theat infinity. Fourier transform of any absolutely function on tends to zero ThisthehasFouri already been notedthein smoother Lemma the proved above.theWefaster now show that, like e r coefficients, function, i t s Fourier transform tends to itszero.Fourier The dual fact is that the faster a function tends to zero,Wethebegin smoother transform. with the following auxiliary proposition.
Lemma 2. Let f : lR -+ C be a continuous function having a locally piecewise continuous derivative f' on JR. Given this, a ) if the function f' is integrable on then f (x) has a limit both as x -+ -oo and as x -+ +oo; b) if the functions f and f' are integrable on then f x) -+ 0 as x -+ oo .
IR,
IR, ( Proof. Under these restrictions on the functions f and f ' the Newton-Leibniz formula holds f(x) = f (O) + J f ' (t) dt . In conditions a) the right-hand side of this equality has a limit both as x -+ +oo and as x -+ -oo. function having a belimitzero.at infinity is integrable on IR, then both of theseIf alimits mustfobviously D X
0
574
18 Fourier Series and the Fourier Transform
We now prove Proposition 1 . (Connection between the smoothness of a function and the rate of decrease of its Fourier transform. ) If f E C( k) (JR , q (k = 0, 1 , . . . ) and all the functions f, f', . . . , f( k ) are absolutely integrable on JR, then a ) for every E {0, 1 , . . . , k} n
f (n) (0 = (iO n !(�) '
(18. 102)
�) = o( �\ ) as � ---+ 0 . Proof. If k = 0, then a) holds trivially and b) follows from the Riemann Lebesgue lemma. Let k > 0. By Lemma 2 the functions f, f', . . . , f( k - l) tend to zero as x ---+ Taking this into account, we integrate by parts, b) i(
oo .
= vk (
-
oo
f ( k - l) x - � l ::'_ oo +
( )e i x
(i�) J f (k-l) ( x) e-i�x dx) = . . . 00
00
= % J f(x)e -i�x dx = (i�) k no 00 Thus return Eq. ( 1 8 . 102) is established. This is a very important relation, and we We shall to it. �e shown that i( �) = (i�) k f ( k l (O , but by the Riemann-Lebesgue lemma f ( k) (0 ---+ 0 as � ---+ 0 and hence b) is also proved. 0 b. The Rate of Decrease of a Function and the Smoothness of its Fourier Transform In view of the nearly complete identity of the direct and inverse Fourier transforms the following proposition, dual to Proposition 1 , holds. Proposition 2. (The connection between the rate of decrease of a function and the smoothness of its Fourier transform). If a locally integrable function k 0
0
0
0
-
-
f : lR ---+ C is such that the function x f(x) is absolutely integrable on JR, then a ) the Fourier transform of f belongs to C( k ) (JR, q . b ) the following equality holds: ( 18. 103 )
18.3 The Fourier Transform
575
Proof. For k = 0 relation (18. 103) holds trivially, and the continuity of f( � ) has alreadyn been proved in Lemma 1 . If k > 0, then for < k we have the estimate l x f(x) l ::::; l x k f(x) l nat infinity,x from nwhich it follows that x n f(x) is absolutely integrable. But l x ofj(x)thesee-i�integrals 1 = l x f(x) l , which enables us to in voke the uniform convergence with respect to the parameter � and successively differentiate them under the integral sign: 00 f( � ) vb J f(x) e-i� x dx ' n
- oo
00
f (O � J00 xf(x) e - i�x dx , -
- oo
By Lemma 1 this last integral is continuous on the entire real line. Hence indeed f E C k l ( JR. , C ) . (
0
c. The Space of Rapidly Decreasing Functions
We denote the set of functions f E c C 00 l (JR., C) satisfying the condition sup l x,a j C"' l (x) l < for all nonnegative and (3 by S(JR., C) or more briefly by S. Such functions are called integers rapidly decreasing functions (as x ) Thethesetstandard of rapidlyoperations decreasingof addition functionsofobviously forms a vector spaceof under functions and multiplication a function by a complex number. Example The function e - 2 or, for example, all functions of compact sup port in c600 (JR., q belong to s. Definition 4.
oo
x E IR o:
8.
)
--+ oo .
x
Lemma 3. The restriction of the Fourier transform to automorphism of S.
S is a vector-space
We first show that ( ! E S) ([ E S). ToWedothenthisremark we firstthat remark that by Proposition 2a we have f E c C oo l (JR., C) . the operation of multiplication by x"' ( 0) and the operation D of differentiation do not lead outside the class of rapidly decreasing functions. Hence, for any nonnegative integers and (3 the relation f E S implies that the function D,a(x "' f(x)) belongs to the space S. Its Proof.
=?
o:
o:
2:
576
18 Fourier Series and the Fourier Transform
Fourier transform tends to zero at infinity by the Riemann-Lebesgue lemma. But by formulas ( 18. 102) and (18. 103) Df3 ( x�)) (�) = io + f3eJCal (�) , �!3 jCa) ( �) --+ 0 � --+ oo ,
and we have shown that as that is, j E S. nowwholeshowspacethatS.S = S, that is, that the Fourier transform maps S ontoWe the Wesimple recallrelation that thef(�)direct and inverse Fourier transforms are connected by the = [( -�) . Reversing the sign of the argument of the function obviouslytransform is an operation thatS into mapsitself. the set S into itself. Hence the inverse Fourier also maps Finally, if f is an arbitrary function in S, then by what has been proved
We havefmade study above of the Fourier transform of a function : JR; aCrather defineddetailed on the real line. In particular, we have clarified the connection that exists between the regularity properties of a function and the corresponding properties of its Fourier transform. Now thattransform this question hassufficiently been theoretically answered,so weas toshallexhibit studythethefundamental Fourier only of regular functions technical properties of the Fourier transform inwe concentrated form and without technical complications. In compensation shalltransform consider and not derive only one-dimensional but practically also the multi-dimensional Fourier its basic properties independently of whatThose was discussed above. confine themselves to the one-dimensional case may assume thatwishing1 tobelow. Definition 5 . Suppose f : JR; n C is a locally integrable function on JR;n . The function f(�) := (2 7r1) n / 2 J f(x) e-i((, x ) d x ( 18. 104) is called the Fourier transform of the function f. Here we mean that x (x1 , . . . , xn ) , � = (6 , . . . , �n ), (�, x) 6 x1 + · · · + �n X n , and the integral is regarded as convergent in the following sense of principal value: a. Definitions, Notation, Examples
n
=
--+
--+
JR n
=
=
A
A
-A
-A
J rp(x1 , . . . , Xn ) dx1 · · · dxn : = A--flim+oo J · · · J rp(x1 , . . . , Xn ) dx1 · · · dxn .
JR n
18.3 The Fourier Transform
577
In thisas ncaseone-dimensional the multidimensional Fourier transform ( 18. 104) can be re garded Fourier transforms carried out with respect to eachThen, of thewhen variables x 1 , . . . , Xn . the function f is absolutely integrable, the question of the senseLetin whi= c(ha the, . . integral ( 18. 104) is to be understood does not arise at all. . , a and (3 = ((31 , . . . , f3n ) be multi-indices consisting of ) 1 n nonnegative integers (31 , j = 1, . . . , n, and suppose, as always, that va denotes the differentiation operator ax"11 8 1.��x"'nn of order I a I := a 1 + · · · + n and x�' .· = x�'1 1 . . . xnl'n . Definition 6. We denote the set of functions f E c< = ) (IR. n , C) satisfying the condition sup l x,B v a f(x) l < n EJR x for all nonnegative multi-indice8 and (3 by the f:iymbol S(IR.n , q, or by S where can arise. Such functions are said to be rapidly decreasing (aD x -+no confusion ) The setofS awithfunction the algebraic operations of functions tiplication by a complex numberof addition is obviously a vectorand space.mul Example The function e l x l 2 , where l x l 2 xi + · · · + x;;, and all the function8 in c6= ) (IR.n , IC) of compact support belong to S. Ifunif fEormly S, then integral in relation ( 18. 104) obviouslyn converges absolutely and withrulesrespect tointegral space !R. . Moreover, if f E S, � on thecanentire then by 8tandard this be di ff erentiated as desired=wil (JR.,th q.respect to any of the variables 6 , . . . , �n · Thusas many if f E times S, then f E c< Example Let us find the Fourier transform of the function exp ( - lxl 2 /2) . 's When integrating rapidly decreasing functions one can obviously use Fubini theorem and if necessary change the order of improper integrations without difficulty. In the present case, using Fubini ' s theorem and Example 4, we find 1 j e l x l 2 / 2 . e - i � , x ) dx = (27r ) n / 2 a
a1 ,
•
a
•
00
a
oo .
9.
=
-
1 0.
JRn
-
(
= n = IT _1_ J e
n d xj = IT e -�J /2 = e- 1 � 1 2 /2 . j= 1 j= 1 V2if (X) We nowassuming, state andsoprove the dbasic structural propertiesthatof thethe Fourier Fourier transform, as to avoi technical complications, transform i8 beingtoapplied to(compute) functionswith of class S. This is approximately the same as learning operate rational number8 rather than the entire space JR. all at once. The process of completion is of the same type. On this account, see Problem 5 . -
-
x] f 2 e i �j X j -
578
18 Fourier Series and the Fourier Transform
b. Linearity The linearity of the Fourier transform is obvious; it follows from the linearity of the integral. c. The Relation between Differentiation and the Fourier Transform The following formulas hold
�( 0 = (i) l<>l�<> i(�) '
(x�)) (� ) = ( i ) l<>l v<> i( � ) .
( 18. 105) ( 18. 106)
Proof. The first of these can be obtained, like formula ( 18. 102), via integra tion with a preliminary use of Fubini's theorem in the ncourse, caseFormula ofby aparts space( 18.(of!R106) of generalizes dimension > 1). relationto (the18.103) and is obtained by direct differentiation of ( 18.104) with respect parameters 6 , . . . , �n · 0 Remark In view of the obvious estimate l f (�) l :S: ( 2n1) n/2 J l f (x ) l dx < +oo , it follows from ( 18. 105) that i(O 0 as � oo for every function f E S, since D <> f E S. Next,that the simultaneous use of formulas ( 18. 105) and ( 18.106) enables us to write v !3 (x�) ) ( � ) = (i) I <> I+I/Jiev,f(o , from that if f E S, then for any nonnegative multi-indices and f3which we haveit follows �/3D<> f(� ) 0 when � oo in !Rn . Thus we have shown that n
3.
�
JRn
-+
-+
-+
(f E
d. The Inversion Formula
S)
-+
=?
(f E
a
S) .
Definition 7. The operator defined (together with its notation) by the equality ( 18. 107)
is called the inverse Fourier transform. The following Fourier inversion formula holds: �
f=f=f,
or in the form of the Fourier integral: f(x) ( 2n1) n/2 J f( � ) ei x � ) d� . =
JRn
(
,
( 18. 108) ( 18. 109)
579
18.3 The Fourier Transform
Using Fubini 's theorem one( 1 8can immediately obtain formula (18. 108) from the corresponding formula . 1 0 1 ) for the one-dimensional Fourier trans form, but, as promised, we shall give a brief independent proof of the formula. Proof. We first show that
J g(�)j(�) ei(x ,e) d� = J g(�)f(x + y) dy for any functions J, g E S(IR, C) . Both integrals are defined, since J, g E S and so by Remark we also have f, g E S. ( 1 8 . 1 10)
3 Let us transform the integral on the left-hand side of the equality to be proved:
= Jg(�) ( (27r) n/2 jJ (y)e- i (e , y) dy) ei(x ,eJ d� = = (27r) n/2 J ( Jg(�)e- i (e ,y -x) d�) f(y)dy = = J g(y - x)f(y) dy = J g(y)f(x + y) dy . 1
Rn
Rn
1
Rn
Rn
Rn
Rn
There is since no doubtandas toaretherapidly legitimacy of thefunctions. reversal Thus in the( 1 8order of integration, decreasing . 1 10) is g f nowWeverified. now remark that for every c > 0 so that, by Eq. (18. 1 10)
J g(r::�)i(�)ei (x,e) d� = J r:: -ng(yjr::) J(x + y) dy = J g(u)f(x + cu) du .
Rn
Rn
Rn
g accountintegrals of the absolute convergence withasrespect c ofTakithe nextreme in the lastandchainuniform of equalities, we find, c to g(O) J i(�)ei(x ,e) d� = f(x) J g(u) du . -+
0,
580
18 Fourier Series and the Fourier Transform
Here we set g (x ) = e - l x l 2 /2• In Example 10 we saw that g(u) = e- lul 2 /2 • Recalling the Euler-Poisson integral -Ioo e- x2 dx = V7f and u�ing Fubini ' � theorem, we conclude that I e-lul 2 12 du = (27r ) n/2 , and a� a result, we obtain Eq. ( 18. 109) . D Remark 4. In contr�t to the single equality (}8. 109) , which means that J = f, relations ( 18. 108) al�o contain the equality 1 = f . But thi� relation follows immediately from the one proved, since J(f.) = [(-f.) and � = {(;). Remark We have already seen (see Remark 3) that if f E S, then J E S, and henc� 1 �S also, that is, S S and S S. We now conclude from the relations 1 = J = f that S = S = S. e. Parseval's Equality Thi� is the name given to the relation 00
JRn
5.
C
C
( ! , g ) = ([, g) '
(18. 1 1 1 )
which in expanded form means that
I f(x)g (x) dx = I [(f. g f. df. . ) ( )
( 18. 1 1 1' )
It follows in particular from (18. 1 1 1 ) that ( 18. 1 12) 11 ! 11 2 = ( ! , f) = ([, [) = 11 111 2 . From thepreserves geometric point of view,between Eq. (18.functions 1 1 1 ) means that the Fourier transform the i n ner product (vectors of the space S), The and name hence "Parseval is an isometry of S. ' s equality" is also sometimes given to the relation I [(f,)g(f.) df. = I f(x)g(x) dx , ( 18. 1 13) which is obtained from ( 1 8 . 1 10) by setting x = 0. The main Parseval equality (18. 1 1} ) is obtained from ( 18. 1 13) by replacing g with g and using the fact that (g) = g, since � = 0 and � = g . f. The Fourier Transform and Convolution The following important relations hold ( J;9 ) = (2 7r ) n f 2 J · g ' (18.1 14) (h) = (211") - n/2 [ g (18. 1 15) *
18.3 The Fourier Transform
581
( sometimes called Borel 's formulas) , which connect the operations of convo lution multiplication of functions through the Fourier transform. Letandus prove these formulas:
Proof.
( j;g ) ( O
= ( 27r) n/2 l u g)(x)e-i(�,x) dx = = ( 27rn/2 I (I f(x - y)g(y) dy)e-i(�,x) dx = = (27r) n/2 1g(y)e-i(�,y) (lj (x - y)e- i(�,x- y) dx) dy = = ( 27r) n/2 I g(y)e-i(�,y) ( I f(u) e-i(�,u) du) dy = = I g(y)e-i(�,y) f(o dy = ( 27r) n/2 !(�)§(�) . 1
*
]Rn
1
-
-
1
1
]Rn
]Rn
]Rn
]Rn
]Rn
]Rn
]Rn
Thef,glegitimacy of the change in order of integration is not in doubt, given thatFormula E S. 1 15) can be obtained by a similar computation if we use the inversion(18.formula (18. 109) . However, Eq. ( 1 8 . 1 15) can be derived from relation (18.1 14) already proved if we recall that f = f = J , f = J, f = J, and that u · = · u = D Remark If we set J and g in place of f and g in formulas ( 1 8 . 1 14) and (18.1 15) and apply the inverse Fourier transform to both sides of the resulting equalities, we arrive at the relations J:9 = ( 2 7r ) - nf 2 (J g) , ( 1 8 . 1 14') ( 1 8 . 1 1 5') j;g = ( 2 7r ) n f 2 (J . g) . v
6.
u v,
*
v
u
*
v.
*
18.3.4 Examples of Applications
Let the Fourier transform (and some of the machinery of Fouriuser now seriesillustrate ) in action. The Wave Equation The successful use of the Fourier transform in the equations of mathematical physics is bound up ( in replaces its mathematical aspectof) primarily with the fact that the Fourier transform the operation differentiation with the algebraic operation of multiplication. a.
582
18 Fourier Series and the Fourier Transform
For example, suppose we are seeking a function u lR lR satisfying the equation where a0Fourier , . . . , antransform are constant coefficients andthisf equation is a known(assuming function.thatApply ing the to both si d es of the functions u and f are sufficiently regular), by relation (18.105) we obtain the algebraic equation :
---+
for u. After finding u(.;) = t��\ from the equation, we obtain u(x) by apply ing the inverse Fourier transform. now apply thiswave ideaequation to the search for a function u u(x, t) satisfying the Weone-dimensional (a > 0) and the initial conditions au (x, 0) = g(x) u(x, 0) = f(x) , at in lRHere x and computations in the next example weasshall notittake the time to tojustify the intermediate because, a rule, is easier simply find theto required function and verify directly that it solves the problem posed than justify and overcome all functions, the technicalwhichdifficulties that arise along the way.playAs itanhappens, generalized have already been mentioned, essential role in thet astheoretical struggle withoutthesea Fouri difficulties. Thus, regarding a parameter, we carry erhand transform on x onentiation both sides of the equation. Then, assuming on the one that differ respect to theonparameter integral sign is permitted and usingwith formula (18.105) the other under hand, the we obtain =
JR.
from which we find
t) = A (,;) cosa.;t + B(.;) sina.;t . By the initial conditions, we have u (,;,
u� ( .; , O)
u(.; , o ) = =
!(.;) = A ( .; ) ,
(u D ( .; , O)
=
a B(.;) .
g(.;) = ,;
18.3 The Fourier Transform
Thus,
583
Multiplying this equality by v�27r eix� and integrating with respect to � the inverse Fourier transform - and using formula ( 18. 105) wein short, obtaintaking immediately t 1 1 u(x, t ) = 2 (f(x - at ) + f( x + at)) + 2 I (g(x - aT ) + g(x + a r ) ) dr . b. The Heat Equation Another element of the machinery of Fourier trans forms (specifically, formulas (example, 1 8 . 1 14') and (18. 1 15')) which remained in the background in the preceding itself quite clearly when we seek a function u = u(x, t), x E �R_n , t manifests 0, that satisfies the heat equation 0
2::
au
2 at = a .1u (a > 0) !R.n . u(x, 0) = f (x) vx1 + . . . + uxn ·
and the initial condition 82 82 on all of Here, as always .1 Carrying out a Fourier transform with respect to the variable x E �R_n , we find by (18.105) the ordinary equation = =
from which it follows that
�
u(� , t) c(�)e- a21 � 1 2 t ' 2 wewherefind1 � 1 �� + · · · + �;. Taking into account the relation U(� , 0) = f(�), Now applying the inverse Fourier transform, taking account of ( 1 8. 1 14') , we obtain =
=
I
u(x, t ) = (2rr) - n / 2 f(y)Eo (Y - x, t) dy ,
where Eo (x, t) is the function whose Fourier transform with respect to x is inverse Fourier respect to � of10.theMaking functionan ee--a21a21 �� 11 22 tt . isTheessentially already transform known to with us from Example
584
18 Fourier Series and the Fourier Thansform
obvious change of variable, we find
( )e Setting we find the fundamental solution E(x, t) (2a v'1rt) - n e - � (t > 0) , ofSect.the17.4) heat, and equation, which was already familiar to us (see Example 15 of the formula -..fo n - � n/ 2 (211") av'i E(x, t) = (211") - n f 2 E0 (x, t), Eo (x, t)
1
=
4a t
•
lx12
=
u(x, t) =
(f E)(x, t) *
for the solution satisfying the initial condition u(x, 0) = f (x) . c . The Poisson Summation Formula This is the name given to the following relation � L
:
00
=
-t
00
00
rp(n)einx ' ( 18 . 1 17)
L (x + 211" ) = L n = - oo m = - oo
R
=
ck (f)
:=
00
2�
d
�J
=
00
+
00
f(x )e - i k x x = L n = - oo 0 � n+l) 2 ( 1
2�
� J
+
oo
d
211"n)e - i kx x =
J
n =2:: - oo � 2 � n
- oo
18.3 The Fourier Transform
585
f is a smooth 2n-periodic function and so its Fourier series converges to itButat every point E R Hence, at every point E JR. we have the relation = = = i nx = M:: 1 cp(n) einx . D 2:: ( + 2nn) = f(x) = 2:: en ( ! ) eM:: 2:: 2 2 1f 1f = = = n -= n -= n -= As can beforseenfunctions from theof proof, relations ( 1 8 . 1 16) and ( 1 8 . 1 1 7) by notoRemark means hold only class S. But if does happen to belong S, then Eq. ( 18. 1 1 7) can be differentiated termwise with respect to any number of times, yielding as a corollary new relations between , and . x
x
x
V
7.
V
1
cp
•
•
x
•
d. Kotel'nikov's Theorem (Whittaker-Shannon Sampling Theo rem)25
This example,series basedandlikethetheFourier preceding one hason aa direct beautifulrelation combito nation of the Fourier integral, the theoryappearing of information transmission in abecause communication channel.capabilities To keep itof from artificial, we recall that of the limited our sense For organs, we aretheableearto"hears" perceivein signals onlyfrom in a20certain rangekHz.of frequencies. example, the range Hz to 20 Thus, no matter what the signals are, we, like a filter (see Subsect. 18.3. 1 ) cut out only(having a bounded part ofspectrum). their spectra and perceive them as band-limited signals a bounded For that signal reason,f (t)we (where shall assume from the outset that the transmitted orthereceived t is time, < t < ) is band-limited, spectrum being nonzero only for frequencies whose magnitudes do not exceed a certainfunction criticalthevalue a > 0. Thus J(w) 0 for l w I > a, and so for a band-limited representation = f (t) vk J J(w)eiwt dw -= reduces to the integral over just the interval [ - a, a]: ( 1 8 . 1 18) f (t) vk J J(w)eiwt dw . - oo
=
oo
=
=
a
-a
2 5 V.A. Kotel'nikov ( b. 1908) - Soviet scholar, a well-known specialist in the theory of radio communication. J.M. Whittaker ( 1905-1984) - British mathematician who worked mainly in complex analysis. C.E. Shannon ( 1916-2001) - American mathematician and engineer, one of the founders of information theory and inventor of the term "bit" as an abbreviation of "binary digit."
18 Fourier Series and the Fourier Transform
586
On the closed interval [ - a , a] we expand the function J(w) in a Fourier senes J(w) L ck (j) ei rraw k (18.1 19) - oo { ei rraw k ; k E Z} which is orthogonal and complete in that iinterval. n the system Taking account of formula ( 1 8 . 1 18), we find the following simple expression for the coefficients a ck (f) of this series: J2;2a f ( - -ka ) (18. 120) ck ( f) 2a1 f f (w ) e rrw k dw Substituting the series ( 1 8. 1 19) into the integral (18. 1 18) , taking account of relations ( 18. 120) , we find a 1 f J2; j (t) _ ( 2a f J ( �a k ) ei w t - i rrak w ) dw = J2; k= -oo 1 oo = L f ( � k ) f eiw ( t-%k ) dw . 2a = k -oo Calculating these elementary integrals, we arrive at Kotel 'nikov 's formula oo -a �k ) . (18.121) f ( t ) =Loo f ( a k) sina(t a ( t - -7r k ) k (18.121) shows that, in order to reconstruct a message described byquency aFormula band-limited function f ( t ) whose spectrum is concentrated in the fre range l w l a, it suffices to transmit over the channel only the values marker values) of the function at equal time intervals /a. f ( kThis) (called proposition, together with formula (18.121) is due to V.A. Ko tel ' nikov and is called Kotel 'nikov 's theorem or the sampling theorem. Remark The interpolation formula (18.121) itself was known in mathe ' nikov ' s 1933 paper, but this paper was the first to point matics before Kotel out the fundamental significance of theoverexpansion (18.121) for the theory of transmission of continuous messages a communication of the derivation of formula ( 1 8 . 121) given above is also duechannel. to KotelThe' nikidea ov. In reality transmission and receiving time ofwea communication isRemark also limited, so thattheinstead of the entire series (18.121) take one of its partial sums I: . Special research has been devoted to estimating the errors that thereby arise. =
�
=
:= -
�
-a
00
=
-· 'a
-a
7r
a
-a
=
:::;
iJ.
8.
9.
N
-N
�
iJ.
=
1r
18.3 The Fourier Transform
587
Remark Ifcommunication it is known howf(t)muchovertimea given is required to transmitchannel, one sample value of the communication it is easy to estimate the number of such communications that can be transmitted in parallel over that channel. In other words, the( and possibility arisesas ofa function estimat ing the transmitting capability of the channel moreover, ofspectrum the informational of the communications that will affect the of the signalsaturation f(t ) ) . 1 0.
18.3.5
Problems and Exercises
a) Write out the proof of relations ( 18.93)-( 18.96) in detail. b) Regarding the Fourier transform as a mapping f >-+ j, show that it has the following frequently used properties: 1.
f(at) >-+ �1 f-( -;;w )
( the change of scale rule ) ;
f(t - to) >-+ f(w) e - i w to
{
( time shift of the input signal - the Fourier preimage - or the translation theorem)
f(w)2 cos wto , [f(t + to) ± f(t - to)] >-+ f(w )2 sin wto ; ±i 0t f(t) e w >-+ f(w ± wo) ( frequency shift of the Fourier transform ) ; 1 -
- + wo )] , f(t) cos wot >-+ 2 [f(w - wo) + f(w - + wo)] f( t) sin wot >-+ 21 [f(w - wo) - f(w
( amplitude modulation of a harmonic signal ) ; wot >-+ 1 - f(w - - wo) sin 2 2
f(t)
4 [2f(w) -
{
- + wo)] . - f(w
c ) Find the Fourier transforms ( or, as we say, the Fourier images) of the fol lowing functions: 1 for l t l :S A , 2A
JIA (t) =
( the
rectangular pulse) ;
IlA
Q
for l t l > A
(t) cos wot
18
588
Fourier Series and the Fourier Transform
(a harmonic signal modulated by a rectangular pulse) ; Ih (t + 2A) + Ih (t - 2A) (two rectangular pulses of the same polarity) ; lh (t - A) - liA (t + A) (two rectangular pulses of opposite polarity) ; � (1 - � ) for l t l � A , AA (t) = for l t l > A 0 (a triangular pulse) ; cos ae and sin ae ( a > 0) ; IW ! and IW ! e - a i t i ( a > 0) . d) Find the Fourier preimages of the following functions: wA , 2 I. -- sm 2 w A 2 . 2 wA . smc smc - ,
{
•
7r wA where sine ;- := � is the sample function ( cardinal sine) . e ) Using the preceding results, find the values o f the following integrals, which we have already encountered: 00 oo 00 I I sin 2 x 2 x , d cos x dx , sin x 2 d x . X2
si x
7r
l
- oo
- oo
- oo
f) Verify that the Fourier integral of a function f(t) can be written in any of the following forms: 00
f(t) "'
I - oo
f(w) e i t w dw = 2�
00
00
I - oo
dw
I - oo
i x t dx =
f(x) e - w (
- )
00
=
� I dw 0
00
I
f(x) cos 2w(x - t) dx .
- oo
Let f = f(x, y) be a solution of the two-dimensional Laplace equation � + = 0 in the half-plane y 2: 0 satisfying the conditions f(x, 0) = g(x) and f(x, y) -+ 0 as y -+ +oo for every x E JR. a) Verify that the Fourier transform [(t;., y) of f on the variable x has the form 2.
�
g( t;,) e- yi{ i .
b) Find the Fourier preimage of the function e- Y i el on the variable t;.. c) Now obtain the representation of the function f as a Poisson integral
1
f(x, y) = ;
00
I - oo
y ) dt;. , (x - t;.) 2 + y 2 g( t;,
which we have met in Example 5 of Sect.
17.4.
18.3 The Fourier Transform
589
We recall that the nth moment of the function f : lR -+
3.
-
oo
bution, that is, f(x) 2 0 and J f(x:) dx = 1 , then xo = M1 (f) is the matheoo matical expectation of a random variable x with the distribution f and the varioo ance CJ 2 : = J ( x - x0) 2 f ( x) dx of this random variable can be represented as oo CJ 2 = M2 (f) - M[ (f) . Consider the Fourier transform -
-
j f(x)e- i�x dx 00
f(EJ =
of the function f. By expanding e - i � x in a series, show that n a) f(f.) = f; ( - i) n"f. n U ) C if, for example, f E S. =O n b) Mn (f) = (i) n j\ n l (0) , n = 0, 1, . . . . c) Now let f be real-valued, and let f(f.) = A(E,)ei'P ( O , where A( f.) is the absolute value of f(EJ and cp(EJ is its argument; then A( f.) = A( -EJ and cp( - f.) = -cp(E,) . To 00 normalize the problem, assume that J f(x) dx = 1 . Verify that in that case oo -
and 4. a) Verify that the function e- a lxl ( a > 0) , like all its derivatives, which are defined for x =/= 0 , decreases at infinity faster than any negative power of lx l and yet this function does not belong to the class S. b) Verify that the Fourier transform of this function is infinitely differentiable on JR, but does not belong to S (and all because e- a lx l is not differentiable at x = 0) . 5. a) Show that the functions of class S are dense in the space R 2 (1R n ,
590
18 Fourier Series and the Fourier Transform
c) Show that a vector-space structure and an inner product can be introduced in a natural way on L 2 , and in these structures the Fourier transform L 2 �L 2 turns out to be a linear isometry of L 2 onto itself. d) Using the example of the function f(x) = Vb l+ x 2 , one can see that if f E R 2 (JR., q we do not necessarily have f E R(JR., C). Nevertheless, if f E R2 (JR., q , then, since f i s locally integrable, one can consider the function
A
fA (�) =
vk I f(x) e - i�x dx . -A
Verify that fA E C(JR., C) and fA E R 2 (lR., C). e) Prove that fA converges in L 2 to some element f E L 2 and I fA ll as A --7 +oo (this is Plancherel's theorem 2 6 )
--7
ll fll = ll f ll
6. The uncertainty principle. Let cp(x) and '1/J(p) be functions of class 8 (or elements DO DO of the space L 2 of Problem 5), with 'ljJ = ip and J l cp l 2 ( x ) dx = J l 'l/JI 2 (p) dp = 1. I n this case the functions I'PI 2 and l 'l/JI 2 can be regarded as probability densitie:s for random variables x and p respectively. a) Show that by a shift in the argument of
00
I l axcp(x) + cp' (x) l 2 dx � 0
-
DO
and, using Parseval's equality and the formula ip' (p) = ipip(p) , show that a 2 M2 (
M.
Plancherel ( 1885-1967) - Swiss mathematician.
18.3 The Fourier Transform
591
(called Heisenberg 's2 7 uncertainty principle) , is mathematically the same as the relation between M2 ( cp ) and M2 ('l/J) found above. The next three problems give an elementary picture of the Fourier transform of generalized functions. 7.
{
a) Using Example 1 , find the spectrum of the signal expressed by the functions
L1a (t)
=
1 2a for l t l S a ,
O
for lt l
>a.
b) Examine the variation of the function .d, ( t) and its spectrum as a -+ +0 and tell what, in your opinion, should be regarded as the spectrum of a unit pulse, expressed by the 8-function. c) Using Example 2, now find the signal cp ( t ) emerging from an ideal low frequency filter (with upper frequency limit a ) in response to a unit pulse o(t). d) Using the result just obtained, now explain the physical meaning of the terms in the Kotel'nikov series ( 18. 121 ) and propose a theoretical scheme for transmitting a band-limited signal f(t) , based on Kotel'nikov's formula ( 18. 121 ) .
The space of L. Schwartz. Verify that a) If cp E S and P is a polynomial, then (P cp) E S; b) if cp E S, then D"'cp E S and D f3 (PD"cp) E S, where a and (3 are nonnegative multi-indices and P is a polynomial. c) We introduce the following notion of convergence in S. A sequence { cp k } of functions 'P k E S converges to zero if for all nonnegative multi-indices a and (3 the sequence of functions { x 13 D"cp(x) } converges uniformly to zero on lRn . The relation 'P k -+ cp E S will mean that ( cp - 'P k ) -+ 0 in S. The vector space S of rapidly decreasing functions with this convergence is called the Schwartz space. Show that if 'P k -+ cp in S, then (j)k -+ (j5 in S as k -+ oo. Thus the Fourier transform is a continuous linear operator on the Schwartz space. 8.
·
9. The space S' of tempered distributions. The continuous linear functionals defined on the space S of rapidly decreasing functions are called tempered distributions. The vector space of such functionals (the conjugate of S) is denoted S' . The value of the functional F E S' on a function cp E S will be denoted F( cp). a) Let P : lRn -+ C be a polynomial in n variables and f : lRn -+ C a locally integrable function admitting the estimate l f(x) l S I P(x) l at infinity (that is, it may increase as x -+ oo, but only moderately: not faster than power growth) . Show that f can then be regarded as a (regular) element of S' if we set
f(cp)
=
J f(x)cp(x) dx
lll. n
(cp E S) .
2 7 W. Heisenberg ( 1901-1976 ) - German physicist , one of the founders of quantum mechanics.
592
18 Fourier Series and the Fourier Transform
b) Multiplication of a tempered distribution F E S' by an ordinary function f : JR n -t C is defined, as always, by the relation (JF) (�) := F(f�) . Verify that for tempered distributions multiplication is well defined, not only by functions f E S, but also by polynomials P : JR n -t C. c) Differentiation of tempered distributions F E S' is defined in the traditional way : (D a F) (�) : = ( - 1 ) 1 a i F(D a �). Show that this is correctly defined, that is, if F E S', then D a F E S' for every nonnegative integer multi-index a = ( a 1 , . . . , a n ) d) If f and � are sufficiently regular functions (for example, functions in S) , then, as relation (18.1 13) shows, the following equality holds:
f( �) =
IIRn f(x)�(x) dx JRnI f(x) ij5(x) dx =
=
J ( i/5) .
This equality (Parseval's equality) is made the basis of the definition of the Fourier transform F of a tempered distribution F E S'. By definition we set F( �) : = F( i/5) . Due to the invariance of S under the Fourier transform, this definition is correct for every element F E S' . Show that i t i s not correct for generalized functions i n V' (JRn ) mapping the space V(JR n ) of smooth functions of compact support. This fact explains the role of the Schwartz space S in the theory of the Fourier transform and its application to generalized functions. e) In Problem 7 we acquired a preliminary idea of the Fourier transform of the 8-function. The Fourier transform of the 8-function could have been sought directly from the definition of the Fourier transform of a regular function. In that case we would have found that ?( � ) -
u
_
( 27r ) n / 2 lll.In 1
'(X) e -i ( E. x ) dx - ( 27r1) n 2 . _
u
/
Now show that when we seek the Fourier transform of the tempered distribution 8 E S' (JR n ) correctly, that is, starting from the equality 8( � ) = 8( ij5) , the result (still the same) is that 8(ij5) = i/5 ( 0 ) = 2 rr �n / 2 • (One can renormalize the Fourier ( transform so that this constant equals 1; see Problem 10) . f) Convergence in S' , as always in generalized functions, is understood in the following sense: (Fn -t F) in S' as n -t oo : = (V� E S (Fn (�) -t F(�) as n -t oo )) . Verify the Fourier inversion formula (the Fourier integral formula) for the 8function: A A 8(x)
=
lim
A -+ + oo
(
1
(
21r ) n 12 1 . . . 1 8(� ) ei x .o d� .
_
-A
-A
g) Let 8(x - x0) , as usual, denote the shift of the 8-function to the point x0 , that is, 8(x - Xo ) (�) = �(xo ) . Verify that the series
"L 8(x - n ) 00
n = - cx:> n converges in S' (JR ). (Here 8 E S' (JR n ) and
n
E Z.)
18.3 The Fourier Transform
593
h) Using the possibility of differentiating a convergent series of generalized func tions termwise and taking account of the equality from Problem 13f of Sect. 18.2, DO show that if F = 2:.::: 8(x - n) , then n = - oo
F = y'2;
DO
L
)
8(x - 21r n .
n = - oo
i) Using the relation F( VJ ) = F(ij5) , obtain the Poisson summation formula from the preceding result. j) Prove the following relation (the 0-formula) DO
L
2
e-tn =
n= - oo
�1r L e DO
-
t
,.-
2
- -n
'
2
(t
>
0) '
n = - oo
which plays an important role in the theory of elliptic functions and the theory of heat conduction. 10.
If the Fourier transform F[f] of a function f : JR. --+ C is defined by the formulas DO
] ( v) := F[f] (v) := J f ( t) e- 2rrivt dt , - DO
many of the formulas relating to the Fourier transform become particularly simple and elegant. a) Verify that i(u) =
vh:-1 ( 2: ) .
b) Show that .f[F[f]] (t) = f( -t), that is, DO
f(t) =
J ](v) e2rrivt d v .
- DO
This is the most natural form of the expansion of f in harmonics of different frequencies and in this expansion is the frequency spectrum of f. c) Verify that J = 1 and i = 8. d) Verify that the Poisson summation formula (18.1 16) now assumes the par ticularly elegant form
v,
] ( v)
L VJ ( n ) = L
n = - oo
n = - oo
(j> (n) .
1 9 Asymptotic Expansions
The majority ofbyphenomena thatofwenumerical have to parameters deal with canhaving be characterized mathematically a certain set rather com plicated interrelations. However the description of a phenomenon as a ruleor becomes significantly simpler if it is known that some of these parameters some combination of them is very large or, contrariwise, very small. Example In describing relative motions occurring with speeds v that are much than the speed of lig3htof( lSect. v l c) we may use, instead of the Lorentzsmaller transformations (Example 1 .3) 1.
�
t - (�)x ' t' = Jl - (�) 2
x' = -vr:xl=--==;:;vt( �;:;:):;;<::< 2 '
the Galilean transformation since vjc Example
x' = x - vt ,
�
2.
t' = t '
0.
The period
of oscillations of a pendulum is connected with the maximal deviation 2 = sinangle2 !t'fof (see from its equilibrium position via the parameter k Sect. 6.4) . If the oscillations are smalll, that is, 0, we obtain the simple formula cp 0
cp 0
T�
�
2rr{f
for the period of such oscillations. Supposeposition a restoring a particle toisthereturning toExample its equilibrium and thatforcetheacting force isonproportional displaceit ment (a spring with spring constant k, for example). Suppose also that the 3.
m
596
19 Asymptotic Expansions
resisting force ofoftheproportionality medium is proportional to the square of the velocity (with coefficient a). The equation of motion in that case has the following form (see Sect. 5.6): m!i + ax 2 + kx = 0 . If the mediumby"rarities", a 0 and one may assume that the motion is approximated the motionthendescribed by the equation m!i + kx = 0 (harmonic oscillations with frequencey {f), and if the medium "condenses", then a and, dividing by a, we find in the limit the equation :i;2 = 0, that is, x(t) const. Example 4 . If 1r(x) is the number of primes not larger than x E JR., then, as is known for large x the quantity 1r(x) can be found with small relative(see errorSect.by the3.2),formula 7r(x) � -1nx- . Example It would be difficult to find more trivial, yet nevertheless impor tant relations than sinx � x or ln(l + x) � x , inThese whichrelations the relative smallerifasdesired, x approaches can beerrormadebecomes more precise namely0 (see Sect. 5.3) . -+
-+ oo ,
=
X
5.
1 . � x - -x smx 3! by adjoining one or more of the following terms obtained from the Taylor series. Thus theofproblem is to findbeinga clear, convenient, andspecifics essentially correct description a phenomenon studied using the of the situa tion that arises when some parameter (or combination of parameters) that characterizes the phenomenon is small (tends to zero) or, contrariwise, large (tendsHence, to infinity). weofarethisoncetypeagaiaren essentially discussing the theory of limits. Problems called asymptotic problems. They arise, as one can The see, solution in practically allasymptotic areas of mathematics and consists natural sciofethence.following of an problem usually stages: passing to thesimplified limit anddescription finding theof(main term of the)estimating asymptotics, that is, a convenient the phenomenon; the error that arises i n using the asymptotic formula so found, and determining its range of applicability; then sharpening the main term of the asymptotics, 3
19.1 Asymptotic Formulas and Asymptotic Series
597
to theequally processalgorithmic of adjoiningin the nextcase).term in Taylor 's formula (but faranalogous from being every methods solving asymptotic problemsof(called asymptotic methods) are The usually closelyofconnected with the specfics a problem. Among the few rather general and at the same time elementary asymptotic methods finds 's formula, one of the most important relations in differential one Taylor calculus. The present chaptermethods should ofgivanalysis. e the reader a beginning picture of the elementary asymptotic the firsttosection we shall introducemethods; the general concepts andshalldefini tithem onsInrelating elementary asymptotic in the second we use 's method of constructing the asymptotic expan in discussing Laplace sion of Laplace transforms. This method, which was discovered by Laplace incomponent his research on saddle-point the limit theorems oflater probability theory, is an important of the method developed by Riemann, usually discussed in a course of complex analysis. Further i n formation on various asymptotic methodsTheseof anlysis can becontain foundaninextensive the specialized books cited in the bibli o graphy. books also bibliography on this circle of questions. 19. 1 Asymptotic Formulas and Asymptotic Series 19. 1 . 1 Basic Definitions
a. Asymptotic Estimates and Asymptotic Equalities For the sake of completeness we begin with some recollections and clarifications. Definition 1. Let f X -+ Y and g X -+ Y be real- or complex- or in general Then thevector-valued relations functions defined on a set X and let be a base in X. xEX f = O(g) or f(x) = O (g(x) ) f = O(g) or f(x) = O (g(x) ) over the base f = o(g) or f(x) = o (g(x) ) over the base mean by a:(x) definition that in thebounded equalityonl f(x)X,l ultimately a:(x) l g(x) l , the real-valued function is respectively bounded over the base and infinitesimal over the base Theserelation relations are usually called asymptotic estimates (of f). The f "' g or f(x) "' g(x) over the base :
:
B
B
B,
B.
B
=
B ,
598
19 Asymptotic Expansions
which bycalled definition means that f(x) = g(x) + o (g(x)) over the base B, is usually asymptotic equivalence or asymptotic equality 1 of the functions over the base B. Asymptotic estimates and asymptotic equalities unite in the term asymp totic formulas. Wherevernotations it is not important to indicate the argument of a function the abbreviated f = o(g), f = O(g), or f g are used, and we shall makeIf fsystematic use of this abbreviation. = O(g) and simultaneously g = 0( ! ), we write f ::::: g and say that f andIng are quantities of the same order over the given base. what we are going to be doing below, Y =
C
3
COS X z
z
z
3
X
z
z
C
z
1.
o C:x )
1
x
---t 0 as x ---t +oo . Thiscomputational circumstance,importance as we shallwhensee one below,considers leads totheasymptotic seriesbutthat have relative error not the absolute error; for that reason these series are often divergent, in contrast to classicalbeingseries,approximated for which theandabsolute value of the difference between theto function the nth partial sum of the series tends zero as n ---t +oo. ( l:x )
1 It is also useful to keep in mind the symbol equivalence.
�
often used to denote asymptotic
19. 1 Asymptotic Formulas and Asymptotic Series
599
Let us consider some examples of ways of obtaining asymptotic formulas. Example The labor involved in computing the values of n! or ln n! increase asunder n E that N increases. We shall use the fact that n is large, however, and obtain assumption a convenient asymptotic formula for computing ln n! approximately. It follows from the obvious relations n n+1 n k n n k+1 J ln x dx = L J ln x dx < L lnk < L J ln x dx = J ln x dx k = 2k - 1 k =1 k=2 k 1 2 that n n+1 2 0 < ln n! - J ln x dx < J ln x dx + J ln x dx < ln2 ( n + 1 ) . n 1 1 But n J ln x dx = n ( ln n - 1 ) + 1 = n ln n - (n - 1 ) , 1 and therefore as n --+ oo n Inn! = J ln x dx + O ( ln2 ( n + 1 ) ) = 1 = nlnn - ( n - 1 ) + O ( lnn ) = nlnn + O ( n) . = (ton lnzeron) when --+ +oo, the relative error of the formula ln n!Sincen ln0 n(n)tends as n --+n +oo. Example We shall show that as x --+ +oo the function x t en dt ( n E JR.) fn (x) = J t 1 is+ooasymptotically equivalent to the function 9n (x) x - n ex . Since 9n (x) --+ as x --+ +oo, applying L ' Hopital ' s rule we find - n ex- n 1 = 1 . x fn (x) = lim f� (x) = lim lim x --++ = 9n (x) x--+ + = g� (x) x --+ cxo x - ne x - nx - e x Example Let us find the asymptotic behavior of the function x t e dt f(x) = J t 6.
o
�
7.
=
8.
1
600
19 Asymptotic Expansions
more precisely. It differs from the exponential integral x Ei (x) = J tet dt oo onlyIntegrating by a constant term. we obtain by parts, x et x 2et t t t x x e ( e ) 1 e j j dt = + - + - dt = f(x) = l + -
t t2 1 1 t3 X t t t x ( e e 3! et dt = 1! e ) 2! j = -t + 2t + 3t l 1 + 4 1 t X ( n - 1 ) ! ) x j n! et ( 0! 1! 2! t . . = e t + -t 2 + -t 3 + + t n l 1 + -n + 1 dt . t 1 This last integral, as shown in Example 7, is O(x -( nn+ 1 l ex ) as x ---+ +oo. Including in the term O(x -(n+ 1 l ex ) the constant -e k2::= 1 (k - 1 ) ! obtained when t = 1 is substituted, we find that t 1 1 t2
-
-
-
·
-
f(x) = ex t (k �k 1 ) ! + a c:: 1 ) as X ---+ +oo . k= 1 The error 0 ( x��r ) in the approximate equality f(x) t (k X-k 1 ) ! ex k= 1 �
is asymptotically infinitesimal compared with each term of the sum, including the last. As the same time, as x ---+ + oo each successive term of the sum is infinitesimal its predecessor; therefore it is natural the continuallycompared sharper with sequence of such formulas as a series generatedto bywritef: !( x ) _ eX � (k X-k 1 ) ! . k= 1 this series obviously diverges for every value of x E JR, so thatWewenote cannotthatwrite f ( X ) = ex � ( k -k 1 ) ! roo.J
�
� k= 1 x
19. 1 Asymptotic Formulas and Asymptotic Series
601
Thus we ofareadealing here with ina new and clearly useful asymptotic in terpretation series connected, contrast with the classical case, with the relative rather than the absolute error of approximation of the function. The sopartial sums of such a series, in contrast tofunction the classical case, points are usedas not much to approximate the values of the at specific to(which describe collective passage in question in thetheirpresent examplebehavior occursunder as x the+oo)limiting . -+
b. Asymptotic Sequences and Asymptotic Series Definition 2.
A sequence of asymptotic formulas
f(x) f(x)
7/Jo(x) + o (7/Jo(x)) , 7/Jo(x) + ¢1 (x) + o (¢1 (x)) ,
f(x)
7/lo(x) + 7/J 1 (x) + · · · + 7/Jn (x) + o ( 7/Jn (x) )
that areasvalithed over a base in the set X where the functions are defined, is written relation B
j ( X ) � 7/Jo ( X ) + 7j; 1 ( X ) + · · · + 7/Jn ( X ) + . . .
or, more briefly, as f (x) f in the given base B.
�
CXJ
7/J (x). It is called an asymptotic expansion of k =O k 2:.::
expansions we always haveIt is clearo 7/Jfrom this= 7/Jdefinition that in asymptotic ( n (x)) n + l (x) + o (¢n + l (x)) over the base and hence for any n = 0, 1 , 2, . . . we have 7/Jn + l ( x) = o (7/Jn ( x)) over the base is, each successive term of thein comparison expansion contributes its correction, which isthatasymptotically more precise with its predecessor. Asymptotic expansions usually arise in the form of a linear combination B,
B,
c0tp0(x) + c 1 tp 1 (x) + . . . + Cn iPn (x) + . . . of functions of some sequence { IPn ( x)} that is convenient for the specific
problem.
19 Asymptotic Expansions
602
Let X be a set with a base B defined on it. The sequence {
Definition 3.
identically zero on any element of Remark The condition that (pp2 2<> . . .<>PPnn<>. . . , . ; d) themultiplication sequence {g(x)
8
2.
B
9.
C
•
•
X
•
oo
00 ;
· · ·
B,
f(x) � Co<po(x) + CI
isrespect calledtoantheasymptotic expansion or asymptotic series of the function f with asymptotic sequence {
3.
19. 1 . 2 General Facts about Asymptotic Series
When we speakonly of thein asymptotic behavior of a function over a base we are interested the naturefunctions of the limiting behavior of the function, so that if two generally different f and g are equal on some element of the base they have the same asymptotic behavior over and should be considered equal in the Moreover, asymptotic sense. if we fix toin carry advanceout some asymptoticexpansion, sequencewe{
B
B,
19. 1 Asymptotic Formulas and Asymptotic Series
603
with the there limitedwillpossibilities of that any such system of functions { 'Pn }. To be specific, be functions are infinitesimal with respect to every term 'Pn of the given asymptotic system. Example Let 'Pn (x) = x1n , n = 0, 1, . . . ; then e - x = o ( rpn (x)) as x +oo. For that reason it is natural to adopt the following definitions. Definition 5. If { 'Pn ( x)} is an asymptotic sequence over the base B, a func tion f such that f(x) = o ( rpn (x) ) over B for each n = 0, 1 , . . . is called an asymptotic zero with respect to { 'Pn ( x)}. Definition 6 . Functions f and are asymptotically equal over the base B respect to a sequence of functions { 'Pn } that is asymptotic over B if the diwithfference f - is an asymptotic zero with respect to { 'Pn }. Proposition 1 . (Uniqueness of an asymptotic expansion). Let { 'Pn } be an asymptotic sequence of functions over a base B . ) If a function f admits an asymptotic expansion with respect to the sequence { 'Pn } over B, then that expansion is unique. b) If the functions f and admit an asymptotic expansion in the system { 'Pn }, then these expansions are the same if and only if the functions f and are asymptotically equal over B with respect to { 'Pn } Proof. a) Suppose the function rp is not identically zero on any element of B. We shall show that if f(x) = o ( rp(x) ) over B, and at the same time f(x) = crp(x) + o ( rp(x) ) over B, then c 0. Indeed, l f(x) l l crp(x) l - i o (rp(x)) I = l c l l rp(x) l - o (l rp(x) l) over B, and soButif ifl eif(x) > 0, there exists B1 E B at each point of which l f(x) l � l rp(x) l . o ( rp(x) ) over B, then there exists B2 E B at each point of which1 l1f(x) l :::; l�l l rp(x) l . Hence at each point x E B1 n B2 we would have to � l rp(x) l :::; ¥ 1 rp(x) l or, assuming l ei =/:- 0, 3 l rp(x) l :::; 2 l rp(x) l . But this is have =1- 0 attheeven one pointexpansion of B1 n Bof2a. function f with respect impossible ifusrp(x) Now let consider asymptotic to theLetsequence { 'Pn }. rpo(x) and corp f(x) = corpo(x) +o ( ) f(x) = (x) +o ( rpo(x) ) over B. Sub tracting the second equality from the first, we find that 0 = (co - co)rpo(x) + o ( rp0(x) ) over B. But 0 = o ( rp0(x) ) over B and so, by what has been proved, co - co = 0. we havef proved that c0{ = c},0 , then . . . , Cn - 1 = Cn - 1 for two expansions of the Iffunction in the system by the equalities 'Pn f(x) = Corpo(x) + · · · + Cn - 1 'Pn - 1 (x) + Cn 'Pn (x) + o ( rpn (x)) , --t
1 0.
g
g
a
g
g
�
=
=
�
f(x) = corpo(x) + · · · + Cn - 1 'Pn - 1 (x) + Cn 'Pn (x) + o ( rpn (x)) we find in the same way that Cn = Cn·
604
19 Asymptotic Expansions
By induction we now conclude that a) is true. b) If f(x) Co<po(x) + · · · + Cn
B
B,
=
B.
=
f(x) = f(O) + 2.1! f'(O)x + · · · + ..!:_n! f ( n ) (O)xn + o(xn ) asthexsystem --t 0. But even the function x 1 1 2 cannot be expanded asymptotically in 1, x, x2 , Thus one must not identify an asymptotic sequence
and asymptotic an asymptoticin it.expansion withmanyanymore canonical basetypesandoftheasymptotic expansionbeof any There are possible havior thanofcanthebeasymptotic described behavior by any fixed asymptoticis notsequence, soanthatexpan the description of a function so much n in termsOneofcannot, a preassigned asymptotic system as it theis theindefinite search forintegral such aofsiosystem. for example, when computing anofelementary function, functions, require in advance that thenotresultbe anbe elementary a composi tion certain elementary because it may function at all. The search for asymptotic formulas, like the computation of indefinite interest onlythan to thetheextent the result is simpler and more integrals, accessible istoofinvestigation originalthatexpression. b. Admissible Operations with Asymptotic Formulas The elementary arithmetic properties of the symbols o and 0 ( such properties as o(g) + o(g) = o(g), o(g) +O(g) = O(g) +O(g) = O(g), and the like ) have been studied along with the theory of limits ( Proposition 4 of Sect. 3.2). The following obvious proposition follows from these properties and the definition of an asymptotic expansion. Proposition 2 . ( Linearity of asymptotic expansions ) . If the functions f and .
•
• •
00
00
g admit asymptotic expansions f L=O an iPn and g L=O bn
c::'.
c::'.
00
19.1 Asymptotic Formulas and Asymptotic Series
605
Further expansionscases. and asymptotic formulas in general will properties involve moreof asymptotic and more specialized Proposition 3 . ( Integration of asymptotic equalities ) . Let f be a continuous
function on the interval I = [a, w[ (or I =]w, a]). a) If the function g( x) is continuous and nonnegative on I and the integral w I g( x) dx diverges, then the relations a
f(x) = O(g (x) ) , f(x) = o(g (x) ) , f(x) '"'" g(x)
as I
imply respectively that
F(x) = O(G(x) ) , F(x) = o(G(x)) , where
X
F(x) = J f(t) d t a
and
and
3
x -+ w
F(x) '"'" G(x) ,
X
G(x) = J g(t) d t . a
b) If the functions 'Pn ( x), n = 0, 1 , . . . , which are continuous and positive on I = [a, w[ form an asymptotic sequence as I 3 x -+ w and the integrals w
X
expansion f ( x) c:::o 2::: Cn 'Pn ( x) as I 3 x -+ w with respect to the asymptotic n =O sequence { 'Pn ( x)} of b) , then :F( x) has the asymptotic expansion :F( x) c:::o 00
"E
P . n=O Cn n (x) Proof. a) If f(x) = O ( g (x) ) as I x -+ w, there exists x0 E I and a constant M such that l f (x) l .S Mg (x) for x E [xo , w[. It follows that I f f(t) dt l .S I :? f(t) dt l + M J g(t) dt = o( J g(t) dt) . To prove the other two relations one can usex L 'Hopital 's rule (as in Example 7) , taking account of the relation G(x) = I g (t) dt -+ as I x -+ w. 3
a
a
�
a
oo
a
As a result, we find lim F(x) lim F'(x) = lim f(x) . I3 x -+w G (x) I3 x -+w G' ( ) I3 x -+w g( ) X
X
3
606
19 Asymptotic Expansions
1 , . . . ) applying L ' Hopital ' s rule (x) as I 3 x ----t w ( again,b) Since we findtPnthat tP (x) 'P (x) tf>� (x) I3limx -+w lf>nn+(IX ) I3limx-+w tf>�+(lX ) I3limx -+w 'Pnn+(lX ) c) The function rn (x) in the relation f(x) cotpo(x) + CI 'P I (x) + · · · + Cn 'Pn (x) + rn (x) , being the difference of continuousw functions on I, is itself continuous on I, and we obviously have Rn (x) J rn (t) dt ----t as I 3 x ----t w. But rn (x) o ( cpn (x) ) as I 3 x w and tPn (x) ----t as I 3 x ----t w. Therefore, again by L'Hopital ' s rule, it follows that in the equality :F(x) co lf>o(x) + c 1 tf> 1 (x) + · · · + Cn lf>n (x) + Rn (x) the quantity Rn (x) is o (tPn (x)) as I 3 x ----t w . D Remark Differentiation of asymptotic equalities and asymptotic series is generally not legitimate. The function f(x) = e- x sin(ex ) is continuously differentiable onasExample lR and is an asymptotic zero with respect to the asymptotic sequence { xln } x ----t +oo. The derivatives of the functions }n , up to a constant factor, agai nonlyhavefaithels toform . However thezero;function f'(x) = -ex sin( ex ) + cos( ex ) x\ asymptotic ' not be an it doesn expansion with respect to the sequence } } as xt ----teven+oo.have an asymptotic ----t 0
n =
0,
=
= 0 .
=
=
=
----t
X
0
0
=
=
5.
11.
{ n
19.1.3 Asymptotic Power Series
Intheyconclusion, let us examine asymptotic powerinseries in some detail, form, since are encountered relatively often, although a rather generalized as was the case in expansions Example 8.with respect to the sequence { xn ; 1 , . . . } , We shall study which is asymptotic as x ----t and with respect to { ln ; = 1 , . . . } , which ischange asymptotic as x ----t oo. Since these are both thex same object up to the of variable x � , we state the next proposition only for expansions with respect to the firstin sequence andexpansions then notewith the specifics ofthecertain of the formulations given the case of respect to second sequence. =
Proposition 4.
0
n
0,
n =
Let 0 be a limit point of E and let
f(x) '::::' ao + a 1 x + a2 x2 + · · ·
,
as E 3 x ----t O .
0,
19.1 Asymptotic Formulas and Asymptotic Series
607
x ---+ 0, a) (a f + f3g) '::::' L=O (aan + f3bn )xn n b) (f · g)(x) '::::' L=O Cn Xn , where Cn = aobn + a 1 bn - 1 + · · · + an bo , = n 0, 1, c) if bo -=/:- 0, then ( i ) (x) L; dn x n , where the coefficients d n can be =O n found from the recurrence relations n an = 2..: b k dn - k , . . . k =O d) if E is a deleted neighborhood or one-sided neighborhood of 0 and f is Then as E
3
00
;
00
0
0
0
;
c::::
n
00
continuous on E, then X
J0 f( t ) dt
c::::
a 1 x2 + . . . + --x an - 1 n + . . . a0x + 2 n
e) if in addition to the assumptions of d) we also have f E then
a� = (
n
+ 1)an + I ,
n
f ' ( x)
c::::
a� + a� x + · · ·
= 0, 1, . . . .
,
C ( l ) (E)
and
a) Thistheis properties a special case of Proposition 2. b) Using of o( ) (see Proposition 4 of Sect. 3. 2 ), we find that Proof.
(f . g)(x) = f(x) . g(x) = = (ao + a 1 x + · · · + an x n + o(xn ))(bo + b 1 x + · · · + bn x n + o(xn )) = = (aobo) + (aob l + a 1 bo)x + · · · + (aobn + a 1 bn - l + · · · + an bo)xn + o(xn ) as Ec) Ifx b---+ -=1-0. 0, then g(x) -=1- 0 for close to zero, and therefore we can x 0 consider the ratio ;(�j = h(x). Let us verify that nif the coefficients d0, . . . , dn inin the representation h( x) = do + d 1 x + ·n· · + dn x + rn ( x) have been chosen accordance with c), then rn (x) = o(x ) as E x ---+ 0. From the identity f (x) = g(x)h(x), we find that ao + a 1 x + · · · + an x n + o(xn ) = = (bo + b 1 x + · · · + bn xn + o(xn))(do + d 1 x + · · · + dn x n + rn (x)) = = (bodo) + (bod l + b 1 do )x + · · · + (bodn + b 1 dn - l + · · · + bn do)xn + + born (x) + o ( rn (x) ) + o(x n ) , 3
3
608
19 Asymptotic Expansions
from which it follows that o(xn ) born (x)+o(rn (x))+o(xn ), or rn (x) o(xn ) as Ed) This x -+ 0, since bo -1=- 0. follows 0 from part c ) of Proposition 3 if we set = 0 there and recall that I f(t) dt I0 f(t) dt . e) Since the function f'(x) is continuous on ]0, x] (or [x, 0[) and bounded ( it tends to a� as x -+ 0 ) , the integal I f'(t) dt exists. Obviously f (x) = a0 + 0 I0 f' (t) dt , since f(x) -+ a0 as x -+ 0. Substituting the asymptotic expansion of f'(x) into this equality and using what was proved in d) , we find that a� - 1 xn + . . . a� x2 + . . . + -f(x) -::::: a0 + a0' x + 2 It now follows from the uniqueness of asymptotic expansions ( Proposition 1) that a� ( + 1)an , 0, 1 , . . . . D Corollary 1 . If U is a neighborhood (or one-sided neighborhood) of infinity in � and the function f is continuous in U and has the asymptotic expansion =
3
-
w
X
=
X
=
X
X
=
n
n
n =
a1 + a2 + · · · + an + f(x) -::::: a0 + 2 xn x x
then the integral F
00
(x) =
over an interval contained in expansion:
j ( (t) X
!
-
ao
·
-
·
as
·
3
U x -+
oo ,
�1 ) dt
U converges and has the following asymptotic
The convergence of the integral is obvious, since a a f ( t) - ao -1 "' -22 as U t -+ t t It remains only to integrate the asymptotic expansion a a a a f ( t) - a0 - -1 -::::: -22 + 33 + + nn . as U t -+ t t t t citing, for example, Proposition 3d ) . D Proof.
oo .
3
-
�
·
·
·
-
·
3
·
oo ,
If in addition to the hypotheses of Corollary 1 it is known that and f' admits the asymptotic expansion
Corollary 2 .
f E cC l l (U) f
,
a� + a� + . . . + -a' + . . . (x) -::::: a0 + x x2 1
n
as
3
U x -+
00 ,
19.1 Asymptotic Formulas and Asymptotic Series
609
then this expansion can be obtained by formally differentiating the expansion of the function f, and a� = - ( n - 1)an- 1 ,
n=
2 , 3, . . .
Proof. Since J' (x) = a� + � + 0( 1/x 2 ) as f(x) = f(xo) +
X
U
and a� = a� = 0 . ::7
x -+ oo , we have
J f' (t) dt = a�x + a� ln x + 0 ( 1 ) xo
U
as ::7 x -+ oo; and since f(x) � a 0 + 7" + � + · · · and the sequence x, ln x, 1 , � ' � ' . . . is an asymptotic sequence as ::7 x -+ oo , Proposition 1 enables us to conclude that a� = a� = 0. Now, integrating the expansion J ' (x) � � + � + · · · , by Corollary 1 we obtain the expansion of f(x) , and by the uniqueness of the expansion we arrive at the relations a� = - ( n - 1)an _ 1 for n = 2 , 3, . . . . D
U
19. 1 .4 Problems and Exercises
a) Let h(z) = I: a n Z- n for lz l > R, z E C. Show that then h(z) � I: a n Z- n n=O n=O as C 3 z -+ oo. b) Assuming that the required solution y(x) of the equation y' (x) + y 2 (x) = sin -fx has an asymptotic expansion y(x) � f C n X - n as x -+ oo, find the first three n=O terms of this expansion. c) Prove that if f(z) = f a n z n for lz l < r, z E C, and g(z) � b 1 z + b2 z 2 + · · as n=O C 3 z -+ 0, then the function f o g is defined in some deleted neighborhood of 0 E C and (f o g) (z) � c0 + c1z + c 2 z 2 + as C 3 z -+ 0, where the coefficients c0 , c 1 , . . . can be obtained by substituting the series in the series, just as for convergent power series. =
=
1.
·
·
2.
·
·
Show the following. a) If f is a continuous, positive, monotonic function for x
t f( k ) k =O b) c)
k
�
0, then
= ]O f(x) dx + o (f(n) ) + 0(1) as n -+ oo ;
£: f: = ln n + c + o(1) as n -+ oo. £:= l k" (ln k) /3 n "+��� n ) �' as n -+ oo for a > - 1 .
k =l
2:
19 Asymptotic Expansions
610
3. Through integration by parts find the asymptotic expansions of the following functions as x -+ +oo: + oo a) r. (x) = I t • - l e - t dt - the incomplete gamma function;
b) erf (x)
00 I e-x 00
-
2
dx =
c) F(x)
=
X
=
Jrr
2 I e - t dt - the probability error function (we recall that X
-x V7f is the Euler-Poisson integral) . +oo " t
I �� dt if a > 0.
X
4. Using the result of the preceding problem, find the asymptotic expansions of the following functions as x -+ +oo: 00 a) Si (x) = f si t dt - the sine integral (we recall that I s i � x dx � is the
0
Dirichlet integral) . b) C(x)
=
�
J0 cos � e dt , S(x) J0 sin � e dt - the Fresnel integrals (we recall 00 2 =
that I cos x 2 dx = I sin x dx + oo
0
=
0
0
=
� yif) .
The following generalization of the concept of an expansion in an asymptotic sequence { 'P n (x) } introduced by Poincare and studied above is due to Erdelyi? Let X be a set, B a base in X, { 'P n (x) } an asymptotic sequence of functions on X. If the functions f(x) , '1/Jo (x) , 'I/J 1 (x) , 'I/J2 (x) , . . . are such that the equality 5.
f(x) = t 'I/Jk (x) + o ( cp n (x) ) over the base k =O holds for every n = 0, 1 , . . . , we write
B
00
{ 'P n (x) } over the base B , n =O and we say that we have the asymptotic expansion of the function f over the base B in the sense of Erdelyi. a) Please note that in Problem 4 you obtained the asymptotic expansion in the sense of Erdelyi if you assume 'P n (x) = x - n , n = 0, 1 , . . . . b) Show that asymptotic expansions in the sense of Erdelyi do not have the property of uniqueness (the functions '1/J n can be changed) . c ) Show that i f a set X, a base B i n X, a function f o n X, and sequences {J.t n (x) } and { 'P n (x) } , the second of which is asymptotic over the base B, are given, then the expansion f(x)
f(x)
�
�
L '1/Jn (x) ,
00
,L: a n J.t n (x) , {cpn (x) } over the base B ,
n =O
where a n are numerical coefficients, is either impossible or unique. 2 A. Erdelyi (1908- 1977 ) - Hungarian/British mathematician.
19.1 Asymptotic Formulas and Asymptotic Series
61 1
6. Uniform asymptotic estimates. Let X be a set and l3x a base in X, and let and g(x, y) be ( vector-valued ) functions defined on X and depending on the parameter y E Y. Set 1/(x, y) l = a(x, y) i g (x, y)l. We say that the asymptotic relations
f(x, y)
f(x, y) = o(g (x, y) ) , f(x, y) = o (g(x, y) ) , f(x, y) "' g(x, y) are uniform with respect to the parameter y on the set Y if ( respectively ) a( x, y) � 0 on Y over the base Bx ; a(x, y) is ultimately bounded over the base !3, uniformly with respect to y E Y; and finally f = a · g + o ( g ), where a(x, y) � 1 on Y over the base l3x . Show that if we introduce the base l3 = {!3, x Y} in X x Y whose elements are the direct products of the elements B, of the base Bx and the set Y, then these definitions are equivalent respectively to the following:
f(x, y) o(g (x, y) ) , J(x, y) o (g (x, y) ) , J(x, y) "' g(x, y) =
=
over the base !3.
7. Uniform asymptotic expansions. The asymptotic expansion 00
f(x, y) l: an (y)
over the base Bx in X holds uniformly on Y in the equalities
n f(x, y) 2: a k (y)
n =
�
00
[ J (x, y) dy � ([ an (Y) )
dy
over the base Bx .
b ) Let Y = [c, dJ C JR. Assume that the function f(x, y) is continuously differ entiable with respect to y on the closed interval Y for each fixed x E X and for some Yo E Y admits the asymptotic expansion
f(x, yo) l: a,. (yo)
over the base Bx .
Prove that if the asymptotic expansion
8f a n (Y)
00
over the base Bx
612
19 Asymptotic Expansions
holds uniformly with respect to y E Y with coefficients a n (Y) that are continuous in y, n = 0, 1 , . . . , then the original function f(x, y ) has an asymptotic expansion j(x, y) � L a n (y)cp n (x) over the base Bx that is uniform with respect to y E Y, n =O its coefficients a n (y) , n = 0, 1, . . . are smooth functions of y on the interval Y and � ( y ) = On (y) . oc
Let p(x) be a smooth function that is positive on the closed interval c :::; x :::; d. a ) Solve the equation � (x, .X) = .\ 2 p(x)u(x, .X) in the case when p(x) = 1 on [c, d] . b ) Let 0 < m :::; p(x) :::; M < +oo on [c, d] and let u(c, .X) = 1 , �� (c, .X) = 0. Estimate the quantity u(x, .X) from above and below for x E [c, d] . c ) Assuming that ln u(x, .X) � f; cn (x) .\ 1 -n as .X --+ + oo , where eo (x) , c1 (x), . . . n =O are smooth functions and, using the fact that ( �) 1 = :!!__ ( .!!.... ) 2 show that � 2 (x) = p(x) and ( c� - 1 + 'f c� · c� - k ) (x) = 0. k =O 8.
II
u
-
1
u
'
1 9 . 2 The Asymptotics of Integrals (Laplace's Method) 1 9 . 2 . 1 The Idea of Laplace's Method
In this subsection we shall discuss Laplace ' s method - one of the few reason ably general methods of constructing the asymptotics of an integral depend ing on a parameter. We confine our attention to integrals of the form b
F(>-.) J f(x)e>. S (x) dx , =
a
(19.1)
where S(x) is a real-valued function and ).. is a parameter. Such integrals are usually called Laplace integrals.
Example
1.
The Laplace transform +co
L(f) (�)
=
J f(x)e -t;x dx 0
is a special case of a Laplace integral.
Example b
2.
Laplace himself applied his method to integrals of the form
J f(x)cpn (x) dx, where
a
n
E N and cp(x) > 0 on ]a, b[. Such an integral is also a
special case of a general Laplace integral (19. 1 ) , since cpn (x) = exp ( n ln cp(x)).
19.2 The Asymptotics of Integrals (Laplace's Method}
613
We shall be interested in the asymptotics of the integral ( 19. 1 ) for large values of the parameter A, more precisely as A -+ +oo, A E So as not to become distracted with secondary issues when describing the basic idea of Laplace ' s method, we shall assume that [a, b] = I is a finite closed interval in the integral ( 19. 1 ) , that the functions f(x) and S(x) are smooth on I, and that S(x) has a unique, strict maximum S(x0) at the point xo E I. Then the function exp (AS(x)) also has a strict maximum at xo, which rises higher above the other values of this function on the interval I as the value of the parameter A increases. As a result , if f(x) ¢. 0 in a neighborhood of xo, the entire integral ( 19. 1) can be replaced by the integral over an arbitrarily small neighborhood of x0, thereby admitting a relative error that tends to zero as A -+ +oo. This observation is called the localization principle. Reversing the historical sequence of events, one might say that this localization principle for Laplace integrals resembles the principal of local action of approximate identities and the 8-function. Now that the integral is being taken over only a small neighborhood of x0, the functions f(x) and S(x) can be replaced by the main terms of their Taylor expansions as I 3 x -+ Xo. It remains to find the asymptotics of the resulting canonical integral, which can be done without any particular difficulty. It is in the sequential execution of these steps that the essence of Laplace ' s method of finding the asymptotics of an integral is to be found.
�-
Example 3. Let x0 = a, S'(a) =/=- 0, and f(a) =/=- 0, which happens, for example, when the function S(x) is monotonically decreasing on [a, b]. Under these conditions f(x) = f(a) + o(1) and S(x) = S(a) + (x - a)S'(a) + o(1), as I 3 x -+ a. Carrying out the idea of Laplace ' s method, for a small c > 0 and A -+ +oo, we find that
F(A)
a+c
rv
J f(x) e>.S(x ) dx a
g
rv
f(a) e >.S( a ) J e >. t S ' ( a ) dt = 0 !( a ) e >.S( a ) >.S ' ( a ) c ) . = - AB ' (a) ( 1 - e
S'(a) < 0, it follows that in the case in question e >.S( a ) as A -+ +oo . ( 19.2) F(A) "' - f(a) AB' (a) Example 4 - Let a < xo < b. Then S'(xo) = 0, and we assume that S"(xo) =f. 0, that is, S"(x0) < 0, since xo is a maximum. Using the expansions f(x) = f(xo) + o(x - xo) and S(x) = S(xo) + �S"(x0)(x - x0) 2 + o((x - x0) 2 ), which hold as x -+ x0, we find that for small Since
19 Asymptotic Expansions
614
e > 0 and
A
---+
+oo
F(A) "'
��
g
J f(x) e>.S(x ) dx "' f(xo) e>.S(xo ) J e � >.S" (xo ) t2 dt .
xo - c
-g Making the change of variable �AS"(x0) t 2 = -u2 ( since S"(x0) < 0) , we obtain cp (>. , g ) g t2 xo Jg e � >.S" ( ) dt = AS'�xo) - cp (>. , g ) >.S"(-2-x 0-e) +oo as A +oo. where cp( A, e ) = V - ---+
f
---+
Taking account of the equality
+oo
J e - u2 du
- oo
=
.Jii ,
we now find the principal term of the asymptotics of the Laplace integral in this case: 27r (19.3) F(A) ,...., xo ) e >.S( xo ) as A\ ---+ +oo .
AB"(xo) J( If x0 = a , but S'(x0) = 0 and S"(x0) < 0, then, reasoning as in
Example 5. Example 4, we find this time that
a+g
g xo x ) ) >.S( >.S( J e �>.S" (xo) t2 dt ' F(A) ,...., J f(x) e dx ,...., f(xo) e 0 a and so
AS" (xo) J(xo ) 2 7f
F(A) ,...., -21
e
>.S( xo )
as
A ---+ \
+oo .
(19.4)
We have now obtained on a heuristic level the three very useful formulas ( 19.2) - ( 1 9.4) involving the asymptotics of the Laplace integral (19. 1 ) . It i s clear from these considerations that Laplace ' s method can b e used successfully in the study of the asymptotics of any integral
j f(x, A) x
d as
X
A
---+
+oo
(19.5)
provided ( a ) the localization principle holds for the integral, that is, the in tegral can be replaced by one equivalent to it as A ---+ +oo extending over
19.2 The Asymptotics of Integrals (Laplace's Method)
615
arbitrarily small neighborhoods of the distinguished points, and ( b ) the in tegrand in the localized integral can be replaced by a simpler one for which the asymptotics is on the one hand the same as that of the integral being investigated and on the other hand easy to find. If, for example, the function S(x) in the integral ( 1 9 . 1 ) has several local maxima x0, X I , . . . , Xn on the closed interval [a, b], then, using the additivity of the integral, we replace it with small relative error by the sum of similar integrals taken over neighborhoods U(x1 ) of the maxima x0, x i , . . . , X n so small that each contains only one such point . The asymptotic behavior of the integral f ( x )e >- S ( x ) dx as A --+ + oo ,
J
U ( xj )
as already mentioned, is independent of the size of the neighborhood U(x1 ) itself, and hence the asymptotic expansion of this integral as A --+ + oo is denoted F(A, x1 ) and called the contribution of the point Xj to the asymptotics of the integral ( 19. 1 ) . In its general formulation the localization principle thus means that the asymptotic behavior of the integral ( 19.5) is obtained as the sum 2::: F(A, Xj )
j
of the contributions of all the points of the integrand that are critical in some respect . For the integral ( 19. 1) these points are the maxima of the function S(x), and, as one can see from formulas ( 19.2)-( 19.4) , the main contribution comes entirely from the local maximum points at which the absolute maximum of S(x) on [a, b] is attained. In the following subsections of this section we shall develop the gen eral considerations stated here and then consider some useful applications of Laplace ' s method. For many applications what we have already discussed is sufficient. It will also be shown below how to obtain not only the main term of the asymptotics, but also the entire asymptotic series. 19.2.2 The Localization Principle for a Laplace Integral Lemma 1. ( Exponential estimate ) .
Let M = sup
a<x
S(x ) < oo,
and suppose
that for some value Ao > 0 the integral ( 19. 1) converges absolutely. Then it converges absolutely for every A ;:::: Ao and the following estimate holds for such values of A : b
I F(A) I ::; J i f ( x )e >- S where A E JR.
a
(x)
l dx ::; Ae >-M ,
( 19.6)
616
19 Asymptotic Expansions
Proof. Indeed, for A 2: Ao ,
II
l II
b
I F(A) I =
l
b
f(x) e>.S (x) dx =
a
a
b
:::; e ( >. - >. o) M
f(x) e>. o S (x) e ( >. - >. o) S (x) dx :::;
I
b
(
lf(x) e>. o S (x) I dx = e->. o M
a
I lf(x)e>.o S (x) I dx) e>.M . D a
Suppose the integral ( 19. 1 ) converges absolutely for some value A = Ao , and suppose that in the interior or on the boundary of the interval I there is a point x0 at which S(xo) = sup S(x) = M . If f(x) and S(x) are continuous at xo and a< x 0 and every sufficiently small neighborhood UI (xo) of xo in I we have the estimate
Lemma 2 . (Estimate of the contribution of a maximum point).
II
l
f( x) e>.S (x) dx :=:: Be>. ( S (xo) - e: )
U1 (xo)
(19.7)
with a constant B > 0, valid for A 2: max{..\0 , 0} . Proof. For a fixed c > 0 let us take any neighborhood U1 (x0) inside which l f (x) l 2: � l f (xo) l and S(xo ) - c :::; S(x) :::; S(xo ) . Assuming that f is real valued, we can now conclude that f is of constant sign inside U1 (x) . This enables us to write for A 2: max{ Ao , 0}
II
l I
f( x) e>.S (x) dx =
lf( x) l e>.S (x) dx
UI (xo)
U1 (xo)
2:
I
UI (xo)
�
2:
l f(xo ) l e>. ( S (xo) - e: ) dx
=
Be>. ( S (x o ) - e: ) . D
Suppose the integral ( 19. 1) con verges absolutely for a value A = Ao , and suppose that inside or on the bound ary of the interval I of integration the function S(x) has a unique point x0 of absolute maximum, that is, outside every neighborhood U(x0) of the point xo we have sup S(x) < S(xo) . I\ U (xo) If the functions f (x) and S(x) are continuous at xo and f(x0) :/=: 0, then Proposition 1 . (Localization principle) .
F(A)
=
Fu1 (xo) (A) ( 1 + O ( r = ) ) as A ---7 + oo ,
(19.8)
19.2 The Asymptotics of Integrals (Laplace's Method) where
617
Ur(x0) is an arbitrary neighborhood of x0 in I, Fui (x 0 ) (>.)
:=
J
UI (xo)
f (x) e >. S (x) dx ,
and 0 ().. - oo ) denotes a function that is o ( ).. - n ) as )..
� +oo for every E N . Proof. It follows from Lemma 2 that if the neighborhood U1(x0) is sufficiently small, then the following inequality holds ultimately as ).. � +oo for every E>O x0
n
I FU1 (xo ) (>.) [ > e>. ( S (xo) - c: ) · (19.9) At the same time, by Lemma 1 for every neighborhood U(x0) of the point we have the estimate
J
I\ U (xo)
l f (x) l e >. S (x) dx � Ae >.Jl
as )..
� +oo ,
(19. 10)
11 =
sup S(x) < S(x0). x EI\U (xo) Comparing this estimate with inequality ( 19.9) , it is easy to conclude that inequality ( 19.9) holds ultimately as ).. +oo for every neighborhood Ur(x0) of xa. It now remains only t o write
where A > 0 and
�
and, citing estimates (19.9) and (19. 10) , conclude that ( 19.8) holds. D Thus it is now established that with a relative error of the order 0 (>. - oo ) as +oo when estimating the asymptotic behavior of the Laplace integral, one can replace it by the integral over an arbitrarily small neighborhood Ur(x0) of the point x0 where the absolute maximum of S(x) occurs on the interval I of integration.
).. �
19.2.3 Canonical Integrals and their Asymptotics Lemma 3. ( Canonical form of the function in the neighborhood of a critical point ) . If the real-valued function S(x) has smoothness cC n+ k ) in a neighbor hood (or one-sided neighborhood) of a point x 0 E � ' and
S'(xo) = · · · = s C n - l ) (xo)
=
0,
s C n l (xo) =/=- 0 ,
and k E N or k = oo, then there exist neighborhoods (or one-sided neighbor hoods) Ix of x o and Iy of O in � and a diffeomorphism r.p E C k ) (Iy , Ix ) such that
618
19 Asymptotic Expansions S (
= S(x0) +
Here
s
y n , when y E Iy and
= Xo
and
1
s
= sgn S ( n l (x0) .
(0) = ( I S (nn!l (xo ) )
1/ n
l Proof. Using Taylor ' s formula with the integral form of the remainder,
S(x)
=
1
� j J s(n) (xo + t(x - Xo ) ) ( 1 - t) n 1 dt '
( -x n S(xo ) + �, n-
-
0
we represent the difference S(x) - S(x0 ) in the form
S(x) - S(xo )
= (x - xo )nr(x) ,
where the function 1
r(x)
= ( n � 1) ! J S(n) (xo + t(x - Xo ) ) ( 1 - t) n- 1 dt , 0
by virtue of the theorem on differentiation of an integral with respect to n the parameter x, belongs to class C (k) , and r(x0) = � n . S( l (x0 ) -1- 0. Hence the function y = 'l/;(x) = (x - x0 ) v' l r(x) l also belongs to C( k) in some neighborhood (or one-sided neighborhood) Ix of x0 and is even monotonic, since s ( xo ) n '1/J' (x o ) = v' lr( xo ) l = c n�; l f / -1- 0 0
In this case the function '1/J on Ix has an inverse '1/J - 1 =
Remark 1 . The cases n ones of most interest.
= 1 or n = 2 and k = 1 or k = oo are usually the
Suppose the interval of integration I = [a, b] in the integral (19. 1 ) is finite and the following conditions hold: a ) f, S E C(I, lR) ; b) max S ( x) is attained only at the one point x0 E I; xE I n c ) S E C( ) (UI (x0 ) , JR) in some neighborhood U1 (x0 ) of x0 {inside the interval I); d) sC n l (x0) -1- 0 and if 1 < n, then S(ll (x0 ) = . . . = sC n - 1 l (x0 ) = 0.
Proposition 2 . (Reduction) .
19.2 The Asymptotics of Integrals (Laplace's Method)
619
Then as A --+ +oo the integral (19. 1 ) can be replaced by an integral of the form n R(A) = r ( y ) e - y dy
e>.S(xo) I
>.
Iy
with a relative error defined by the localization principle ( 19.8) , where Iy = [ - c-, c-] or Iy = [O , c-], c is an arbitrarily small positive number, and the func tion r has the same degree of smoothness Iy that f has in a neighborhood of
xo.
Proof Using the localization principle, we replace the integral ( 1 9 . 1 ) with the integral over a neighborhood I = UI (xo) of xo in which the hypotheses of Lemma 3 hold. Making the change of variable x = cp( y ), we obtain
x
I f(x) e>.S(x) dx = ( I f ( cp( y ) ) cp (y ) e- >.yn dy)e>.S(xo) . '
Iy
Ix
(19. 1 1 )
The negative sign i n the exponent ( - A Y n ) comes from the fact that by hypothesis x0 = cp( O ) is a maximum. D
The asymptotic behavior of the canonical integrals to which the Laplace integral ( 19 . 1 ) reduces in the main cases is given by the following lemma.
Let a > 0, (3 > 0, 0 < a � oo, and f E C ( [O, a] , IR ) . Then with respect to the asymptotics of the integral
Lemma 4. (Watson 3 ) .
a
W(A) = I xf3 - l f(x) e ->. x "' dx 0 as A --+ +oo, the following assertions hold: a) The main term of the asymptotics of ( 19.12 ) has the form
1 .x- a + O ( A - --u# +-I ) , W(A) = -f(O)F((3/a) if it is known that f(x) = f(O) + O(x) as x --+ 0 . b) If f(x) = ao + a 1 x + · · · + an x n + O(x n + l ) as x --+ 0, then W(A) = � t a k r ( k : f3 ) .x - � + O (.x- n +g+ 1 ) . k =O Q
3
{3
G.H. Watson (1886- 1965) - British mathematician.
( 19. 12 )
( 19. 13 )
( 19.14 )
620
19 Asymptotic Expansions
c) If f is infinitely differentiable at x = 0, then the following asymptotic expansion holds: + f3) � f C kl ( ) r , (19. 15) W(A) ':::' �a � a L.., k =O which can be differentiated any number of times with respect to A. Proof. We represent the integral ( 19. 12) as a sum of integrals over the interval ]0, c-] and [c-, a[, where c is an arbitrarily small positive number. By Lemma 1
k! o r (
I J xf3-l f (x )e- >-.xa dx l a
E
and therefore
:<::::
k
Ae - >-. .:a =
O ( r00 )
E
W(A) = J xf3- l f ( x )e - >-.xa dx + O ( r = )
as
as
A ---+ +oo ,
A ---+ +oo .
0
n In case b ) we have f ( x ) I: a k x k + rn ( x ) , where rn E C[O, c] and k =O i rn ( x ) i :<:::: Cxn + l on the interval [O, c-]. Hence E E n W(A) = L a k x k +f3- l e ->-.xa dx + c(A) xn+ f3 e - >-.xa dx + o(A - ) , k =O 0 0
J
where c(A) is bounded as A ---+ +oo. By Lemma 1 , as A ---+ +oo, E + k + >-.xa l f3 x k + f3 - l e ->-.xa dx + O(A - 00 ) dx = x - e
oo J
J
.
0
0
But
oo
J
oo k J x +f3- l e - >-.xa dx = � r ( k : {3 ) A - � , + 0
from which formula ( 19. 14) and the special case of it, formula ( 19. 13) , now follow. The expansion ( 19. 15) now follows from ( 19. 14) and Taylor ' s formula. The possibility of differentiating ( 19. 15) with respect to A follows from the fact that the derivative of the integral ( 19. 12) with respect to the parameter A is an integral of the same type as ( 19. 12) and for W' ( A ) one can use formula ( 19. 15) to present explicitly an asymptotic expansion as A ---+ +oo that is the same as the one obtained by formal differentiation of the original expansion (19.15). D
621
19.2 The Asymptotics of Integrals (Laplace's Method) Example
6.
Consider the Laplace transform
F(>.) =
+oo
j0 f(x) e- >-x dx ,
which we have already encountered in Example 1. If this integral converges absolutely for some value ,\ = ).. 0 and the function f is infinitely differentiable at x = 0, then by formula (19.15) we find that
00 f ( k l r < k + I ) F(>.) L (o) =O k c::::
as
>. � +oo .
19.2.4 The Principal Term of the Asymptotics of a Laplace Integral Theorem 1 . (A typical principal term of the asymptotics ) .
Suppose the in terval of integration I = [a, b] in the integral (19.1) is finite, f, S E C(I, JR) , and max S ( x) is attained only at one point x0 E I . x EI Suppose it is also known that f(x0) -=/:- 0, f(x) = f(xo) + O(x - xo) for I 3 x � x0, and the function S belongs to C ( k ) in a neighborhood of x0 . The following statements hold. a) If xo = a, k = 2, and S'(x0) -=f. 0 (that is, S'(x0 ) < 0}, then
S( xo r 1 0 r 1 F(>.) = -Sf(xo) ' (xo) e>- l [1 + ( )]
b) if a < xo
as
,\ � + oo ;
(19.2')
< b, k = 3, and S"(xo) -=f. 0 (that is, S"(xo) < 0}, then
) if
x0 = a, k = 3, S'(a) = 0, and S"(a) -=f. 0 (that is, S"(a) < 0}, then F(,\) = - 2S"(xo ) f(x0 ) e>-S ( xo l r 1 1 2 [1 + 0 (>.- 1 1 2 )] as ,\ � +oo (19 4') c
7r
·
·
Proof Using the localization principle and making the change of variable
x = cp(y) shown in Lemma 3, according to the reduction in Proposition 2, we
arrive at the following relations: c
a) F(,\) = e>- S( xo ) ( (f o cp)(y)cp'(y) e->- Y dy + O( r 00 ) ) ; 0
j
19 Asymptotic Expansions
622
E:
b) F(>.. ) = e>.s(xo) ( l(f o .y2 dy + O(>.. - =) ) = = e>.S(xo) ( I ( (f o .y2 dy + 0 (>.. - oo ) ) ; E:
- E:
0
E:
F(>..) = e>.s(xo) ( l (f o .y2 dy + O(r=) ) . Under the requirements stated above, the function (f o
)
0
D
Now let us consider some examples o f the application o f this theorem. Example 7. The asymptotics of the gamma function. The function +oo dt > =
r(>.. + 1) 1 t )..e - t
(>.. -1)
0
can be represented as a Laplace integral +oo e e
r(>.. + 1) = I -t >. n t dt , 0
and if for integral
I
>.. > 0 we make the change of variable t = >..x , we arrive at the +oo + l >.,A + r(>.. 1) = I e - >.(x -ln x) d ' 0
x
which can be studied using the methods of the theorem. The function S x ) ln x - x has a unique maximum x on the interval 0, oo , and S" ( 1 ) == By the localization principle (Proposition and assertion b) of Theorem we conclude that
( = =1 1) -1.1, r(>.. + 1) = � (�) ).. [ 1 + 0 (>.. - 112)] as >.. ---+ + In particular, recalling that F(n + 1) = n! for n E N, we obtain the classical Stirling 's formula4 n! = �(n/e)n[1 + O(n-112)] as n ---+ nEN. ---] + [
oo
oo ,
4 See also Problem 10 of Sect. 7.3.
0
19.2 The Asymptotics of Integrals (Laplace's Method) Example
8.
623
The asymptotics of the Bessel function
11" In (x) = � J ex cos e cos nO dO , 0 where n E N. Here f ( O ) = cos nO, S(O) = cos O, omax :<;; x :<;; 7r S(O) = S(O) = 1, S'(O) = 0, and S"(O) = - 1, so that by assertion c ) of Theorem 1 ex [1 + O(x - 1 1 2 )] as x -t +oo . In (x) = y� 21TX Example 9. Let f E C{ l l ( [a, b], IR ) , S E c< 2 l ( [a, b], IR ) , with S(x) > 0 on [a, b], and a:<;;max S(x) is attained only at the one point x0 E [a, b]. If f (xo) of- 0, x :<;; b S'(x0) = 0, and S"(x0) of- 0, then, rewriting the integral .
b
.F (A) = j f (x) [S(x) f dx a
in the form of a Laplace integral b
n
( )
.F (A) = J f (x) e>. i S x dx ,
a on the basis of assertions b ) and c ) of Theorem
1, we find that as A -t +oo [S(xo)]>.+1 / 2 A - 1 / 2 [1 + O(A - 1 / 2 )] '
.F (A) = f (xo) -S�'�xo) = 1 if a < x0 < b and = 1/2 if x0 = a or x0 = b. c
where
c
Example
1 0.
c
The asymptotics of the Legendre polynomials
11"
j
Pn (x) = � (x + � cos O ) n dO 0 in the domain x > 1 as n -t oo, n E N , can be obtained as a special case of the preceding example when f 1, S(O) = x + � cos O , 0max :<;; 6 :<;; 11" S(O) = S(O) = x + � , S'(O) = 0 , 8"(0) = - � . =
Thus,
624
19 Asymptotic Expansions
1 9 . 2 . 5 *Asymptotic Expansions of Laplace Integrals
Theorem 1 gives only the principal terms of the characteristic asymptotics of a Laplace integral (19.1) and even that under the condition that f(x0) -:/- 0. On the whole this is of course a typical situation, and for that reason Theorem 1 is undoubtedly a valuable result . However, Watson ' s lemma shows that the asymptotics of a Laplace integral can sometimes be brought to an asymptotic expansion. Such a possibility is especially important when f(xo) = 0 and Theorem 1 gives no result . It is naturally impossible to get rid of the hypothesis f(x0) -:/- 0 com pletely, without replacing it with anything, while remaining within the limits of Laplace ' s method: after all, if f(x) = 0 in a neighborhood of a maximum x0 of the function S(x) or if f (x) tends to zero very rapidly as x -+ xo , then x0 may not be responsible for the asymptotics of the integral. Now that we have arrived at a certain type of asymptotic sequence { e>. c A -pk } , (Po < p 1 < · · · ) as A -+ + oo as a result of the considerations we have studied, we can speak of an asymptotic zero in relation to such a sequence and, without assum ing that f(x0) -:/- 0, we can state the localization principle as follows: Up to an asymptotic zero with respect to the asymptotic sequence { e>.S (xo) A -pk } (Po < p 1 < · · · ) the asymptotic behavior of the Laplace integral (19.1) as A -+ +oo equals the asymptotics of the portion of this integral taken over an arbitrarily small neighborhood of the point x0 , provided that this point is the unique maximum of the function S(x) on the interval of integration. However, we shall not go back and re-examine these questions in order to sharpen them. Rather, assuming f and S belong to cC = l , we shall give a derivation of the corresponding asymptotic expansions using Lemma 1 on the exponential estimate, Lemma 3 on the change of variable, and Watson ' s lemma (Lemma 4) .
Let I = [a, b] be a finite interval, f, S E C(J, �), and assume max S(x) is attained only at the point x0 E I xE I and f , S E C ( oo ) ( UI (x0 ) , �) in some neighborhood U1 (x0) of x0 . Then in re lation to the asymptotics of the integral (19.1) the following assertions hold. a) If x0 = a, sC m l (a) -:/- 0, sUl (a) = 0 for 1 ::; j < m, then
Theorem 2 . (Asymptotic expansion) .
00
F(A) ::::::' A - l fm e>.S ( a ) L a k A - k / m as A -+
k =O
where
+ oo ,
(- 1) k+ l mk r k + 1 h (x , a ) d k ( f(x) h (x , a )) l ( � ) ( ldxl ) x=a ' k! h (x, a) = ( S( a ) S(x) ) - / m j S' (x) . -
(19.16)
19.2 The Asymptotics of Integrals (Laplace's Method) b)
625
If a < Xo < b, S ( 2m l (xo) =1- 0, and sUl (xo) = 0 for 1 � j < 2m, then (19.17) F(.>.. ) ::::: ).. - l / 2m e.X S( xo ) L Ck A - k / m as ).. ---+ +oo ' k =O CXJ
where
2 k + l 2 k 2k + 1 d 2k Ck = 2 (-1) (2k)( !2m ) r ( � ) ( h(x, xo) dx ) ( f(x)h(x, xo)) i x = xo ' h (x, xo) = ( S(xo) - S(x)) 1 - 2!, /S'(x) . ( n l (xo)(x - xo) n as x ---+ x0, then the c) If f ( n l (xo) =f. 0 and f(x) �f . n main term of the asymptotics in the cases a) and b) respectively has the form rv
F(.>.. ) = ]:_m .>.. - � e.X S( a ) r ( n m+ 1 ) ( I Smm!(a) l ) � x X [ � f ( n l (a) + 0 (.>.. - � )] , (19.18) ! + 1 ) ( (2m) ! ) w X F(.>.. ) = ]:__m ).. - W e .X S( xo ) r ( n2m I S2m (xo) l X [ � f ( n ) (xo) + O (.>.. - W )] . (19.19) ! The expansions (19.16) and (19.17) can be differentiated with respect
d) to ).. any number of times.
Proof. It follows from Lemma 1 that under these hypotheses the integral (19.1) can be replaced by an integral over an arbitrarily small neighborhood of xo up to a quantity of the form e .X S( xo ) Q(.>.. - =) as ).. ---+ oo. Making the change o f variable x =
e -.X S ( xo )
ju
o
(19.20)
Iy
where Iy = [O , E], a = m if x0 = a, and Iy = [ - E, E] , a = 2m if a < x0 < b. The neighborhood in which the change of variable x =
626
19 Asymptotic Expansions If Iy =
[
]
- c- , c- ,
that is, in the case
( 19.20 ) into the form
c
a < xo <
b, we bring the integral
e>.S( xo ) ![(! .y 2 m dy ' 0
(19.21)
o
o
and, once again applying Watson ' s lemma, we obtain the expansion (19.17). The possibility of differentiating the expansions (19.16) and (19.17) fol lows from the fact that under our assumptions the integral (19.1) can be differentiated with respect to >. to yield a new integral satisfying the hy potheses of the theorem. We write out the expansions (19.16) and (19.17) for it , and we can verify immediately that these expansions really are the same as those obtained by formal differentiation of the expansions (19.16) and ( 19.17) for the original integrals. We now take up the formulas for the coefficients a k and c k . By Watson ' s lemma a k = k !� �:f (O) r � , where if> (y) = (f o
( )
S(
d k iJ>
d k k+ 1 dy k (0) = ( -m) ( h (x, a) dx ) ( f(x) h(x, a) ) l x =a , 1 where h (x, a) = ( S(a) - S(x)) - � /S'(x). Formulas for the coefficients C k can be obtained similarly by applying Watson ' s lemma to the integral ( 19.21). Setting '1/J(y) = f (. -+ +oo, c 1 '1/J ( n\( o ) r ( n + 1 ) .x - W . / 'I/J (y)e - >.y2"' dy _ 2m n=O n. 2m O But, 'I/J ( 2k+ 1 ) (0) 0 since '1/J(y) is an even function; therefore this last �
f
=
asymptotic expansion can be rewritten as
c / '1/J( y) e - >.y 2"' dy 0
�
1 � 'I/J ( 2k) (O) r ( 2k + 1 ) >. _ 2�;:; 1 _ 2m 2m � k =O (2k) !
·
19.2 The Asymptotics of Integrals (Laplace's Method)
627
It remains only to note that 1/1 ( 2 k ) (O) = 2tJ> ( 2 k ) (O), where tl>( y ) = f (
( ! o
n:�
C xo) (
and
(n) ( ! o
n!
..I..
a
•
It now remains only to substitute these expressions into the integrals
( 19.20 ) and 19.21 ) respectively and use formula (19.13 ) from Watson ' s
lemma. D
(
n
Remark 2. We again get formula ( 19. 2') from formula (19. 18) when = and m = 1 . Similarly, when = 0 and m = 1 formula (19. 19) yields ( 19.3' ) again. Finally, Eq. ( 19.4' ) comes from Eq. (19. 18) with = 0 and m = 2 . All of this, of course, assumes the hypotheses of Theorem 2.
n
0
n
Remark 3. Theorem 2 applies to the case where the function S(x) has a unique maximum on the interval I = [a, b] . If there are several such points x� , . . . , Xn , the integral (19. 1 ) is partitioned into a sum of such integrals, each of whose asymptotics is described by Theorem 2. That is, in this case the n asymptotic behavior is obtained as the sum L: F(>., Xj ) of the contributions
j= l
of these maximum points. It is easy to see that when this happens, some or even all of the terms may cancel one another.
Example
11.
If s E
c C = l (IR, IR) and S(x) --+ - oo as X --+ oo, then
F(>.)
=
00
J S'(x) e>-.S (x) dx
-
oo
=
0
for
)..
>0.
Hence, in this case such an interference of the contributions must necessarily occur. From the formal point of view this example may seem unconvincing,
628
19 Asymptotic Expansions
since previously we had been considering the case of a finite interval of in tegration. However, those doubts are removed by the following important remark.
Remark 4. To simplify what were already very cumbersome statements in Theorems 1 and 2 we assumed that the interval of integration I was finite and that the integral (19. 1 ) was a proper integral. In fact , however, if the inequality sup S(x) < S(x 0 ) holds outside an interval U(x0 ) of the maxiI\ U (xo ) mum point x 0 E I, then Lemma 1 enables us to conclude that the integrals over intervals strictly outside of U ( x 0 ) are exponentially small in comparison with e >- S (xo ) as A -+ + oo (naturally, under the assumption that the integral ( 19. 1 ) converges absolutely for at least one value A = A 0 ) . Thus both Theorem 1 and Theorem 2 are also applicable to improper integrals if the conditions just mentioned are met. Remark 5. Due to their cumbersome nature, the formulas for the coefficients obtained in Theorem 2 can normally be used only for obtaining the first few terms of the asymptotics needed in specific computations. It is extremely rare that one can obtain the general form of the asymptotic expansion of even a simpler function than appears in Theorem 2 from these formulas for the coefficients a k and C k . Nevertheless, such situations do arise. To clarify the formulas themselves, let us consider the following examples. Example
12.
The asymptotic behavior of the function Erf ( x )
as
+oo
=
I X
- 2 e u
du
x -+ + oo is easy to obtain through integration by parts: X
X
from which, after obvious estimates, it follows that Erf ( x )
c::::
e- x 2 00 ( - 1 ) k (2k - 1 ) ! !
-- �
2x k=O L...J
2k
x -2 k as x -+ + oo .
Let us now obtain this expansion from Theorem 2. By the change of variable u = xt we arrive at the representation +oo
Erf (x) = x
I 1
2 2 e- x t dt .
( 19.22)
19.2 The Asymptotics of Integrals (Laplace's Method)
629
Setting >. = x 2 here and denoting the variable of integration, as in Theorem 2, by x, we reduce the problem to finding the asymptotic behavior of the integral
F(>.)
=
j e - Ax2 dx , 00
(19.23)
1
since Erf (x) = xF(x 2 ). When Remark 4 is taken into account, the integral (19.23) satisfies the hypotheses of Theorem 2: S(x) = -x 2 , S'(x) = - x < 0 for 1 � x < + oo ,
-2, 8(1) Thus, xo
8'(1)
=
=
2
- 1. 1, m 1, f(x) 1, h(x, ) -�x , h(x, ) d� -�x d� . Hence, ( _�x d� ) 0 ( - 2� ) - 2� = ( - ! ) x - 1 , .! 2 1 _£_ 1 ...!.. _ ...!.. _.!!_ ( - ...!.. (_ - 2x d x ) ( - 2x ) - 2x dx 2x ) - ( - 2 ) (-1)x - 3 ' ( _�x d� ) 2 ( - 2� ) ( - 2� d� ) 1 ( (- ! ) 2 (-1)x - 3 ) ( - ! ) 3 ( - 1)( - 3)x - 5 , = a =
=
a
=
=
=
a
=
-
=
=
=
_ ( 2k - 1 ) !! x -( 2k + 1 ) · ( _-12x_ A_ dx ) k ( ...!.2x . ) - � Setting x 1, we find that k 1 ak ( - 1)k ! + r ( k + 1 ) ( ( 2k2 k-+ 11) !! ) (- 1 ) k (2k2 k-+ 11) !! . Now writing out the asymptotic expansion (19.16) for the integral (19.23) taking account of the relations Erf (x) = xF(x 2 ), we obtain the expansion (19.22) for the function Erf (x) as x -+ + oo . _
=
Example
_
=
1 3.
_
=
In Example 7, starting from the representation
+ oo 1 A+ F(>. + 1) .). J e - A ( x - Jn x ) dx ' 0 =
(19. 24 )
we obtained the principal term of the asymptotics of the function r ( >. + 1) as >. -+ +oo . Let us now sharpen the formula obtained earlier, using Theorem 2b) . To simplify the notation a bit, let us replace x by x + 1 in the integral (19. 24 ). We then find that
+ oo 1 A A+ .). e J e A ( Jn ( l + x )- x ) d x F(>. + 1) -1 =
19 Asymptotic Expansions
630
and the question reduces to studying the asymptotics of the integral +oo
F(>.) = J e >. (in(l + x) -x) dx
( 19.25)
-1 as ).. +oo. Here S(x) = ln ( 1 + x) - x, S'(x) = 1 �x - 1 , S'(O) = 0, that is, X o = 0, S"(x) = - 1 ;x )2 , S"(O) = - 1 -/=- 0. That is, taking account of ( Remark 4, we see that the hypotheses b ) of Theorem 2 are satisfied, where we must also set f(x) 1 and m = 1 , since S"(O) -/=- 0. In this case the function h(x, x0) = h (x) has the following form: 1+X - ln ( 1 + x)) 1 2 . h(x) = - --(x X ---*
=
I
If we wish to find the first two terms of the asymptotics, we need to compute the following at x = 0:
(h (x) dd) 0 (h (x) ) = h(x) , (h(x) dxd ) 1 (h(x)) = h(x) ddxh (x) , (h(x) dxd ) 2 (h(x)) = (h (x) dxd ) (h (x) ddxh (x) ) = dh d2 h = h (x) [ ( ) 2 (x) + h(x) 2 (x)] . dx dx
This computation, as one can see, is easily done if we find the values h(O), h'(O), h"(O), which in turn can be obtained from the Taylor expansion of h (x), x ;:::: 0 in a neighborhood of 0: h (x) = - 1 +X x [x - (x - �2 x2 + �3 x3 - �4 x2 + O(x 5 ))] 1 1 2 = 1+x � 2 � 3 � 4 = _ [ x - 3 x + 4 x + O (x5 )] 1 / 2 = X 2 1+x 1 = - J2 [1 - � x + � x 2 O(x 3 )] / 2 = 3 4 1+x = - J2 (1 - � x + 2._ x 2 + O (x 3 ) ) = 3 36 J2 5 1 = - J2 - 3 x + J2 x 2 + 0 ( x 3 ) . 36 Thus, h(O) = - �, h' (O) = - :{f , h "(O) = 1 � , 8 +
(h (x) dxd ) o (h (x) ) I x =O = - J21 ,
631
19.2 The Asymptotics of Integrals (Laplace's Method) 1 ( h (x) dd 1 ( h (x)) l x =O = - 3 , ( h (x) d ) 2 ( h (x)) l x =O = ' 12
�
)
�(
co = - 2r( ) - �)
(
=
�
v'2ii ,
22 3 1 1 1 1 ..tii r . ) r = = 4 ( ) c1 = -2 ( ) · 2! 2 2 2 12 J2 12 1 2 J2 Hence, as A -+ oo,
that is, as
A -+ +oo, (19.26)
It is useful to keep in mind that the asymptotic expansions (19. 16) and ( 19. 17) can also be found by following the proof of Theorem 2 without invok ing the expressions for the coefficients shown in the statement of Theorem 2. As an example, we once again obtain the asymptotics of the integral (19.25) , but in a slightly different way. Using the localization principle and making a change of variable x = r.p(y) in a neighborhood of zero such that 0 = r.p(O), S ( r.p( y)) = ln( 1 + r.p(y)) - r.p(y) = - y 2 , we reduce the problem to studying the asymptotics of the integral E:
j
- E:
r.p'( y) e - >. y 2 dy =
E:
j '¢(y) e - >.y2 dy , 0
where '1/J(y) = r.p' (y) + r.p' ( -y) . The asymptotic expansion of this last integral can be obtained from Watson ' s lemma
( )
E:
oo 'ljJ( k l (O) k + 1 - k+l / A ( ) 2 as A -+ + oo ' r -'1/J(y) e - >. y 2 dy � -21 '"' k! 2 � k =O 0 which by the relations '¢ ( 2 k+l ) (0) = 0, 'ljJ( 2 k ) (0) = 2 r.p ( 2 k+l ) (0) yields the
J
asymptotic series
( )
00 k+l '"' r.p ( 2 ) (0) r k + -1 A - ( k+l/ 2) 2 =O (2k! ) k�
()
00 r.p ( 2 k+l ) (0) A- k = A - 1 1 2 r -21 '"' . � k!22 k k=O
632
19 Asymptotic Expansions
Thus for the integral (19.25) we obtain the following asymptotic expansion
00 'P ( 2k + l ) (0) .x - k (19.27) F(.A) �- .x - 1 / 2 Y11f" "' ' 6 k =O k!2 2k where x = cp(y) is a smooth function such that x - ln ( 1 + x) = y 2 in a neighborhood of zero ( for both x and y). If we wish to know the first two terms of the asymptotics, we must put the specific values cp ' (0) and cp ( 3 ) (0) into formula (19.27).
It may be of some use to illustrate the following device for computing these values, which can be used generally to obtain the Taylor expansion of an inverse function from the expansion of the direct function. Assuming x > 0 and y > 0, from the relation
x - ln(1 + x) = y2
we obtain successively
But x "' V'i,y as y ---+ 0 (x ---+ 0) , and therefore, using the representation of x already found, one can continue this computation and find that as y ---+ 0 V2 y ( ..;2y + 3 V2 yx + O(y3 ) ) - V2 y(..;2y) 2 + O(y4 ) = x = V'i,y + 3 12 V2 Y3 + O(y4 ) = = .t.y + 32 y 2 + g2 Y2 X - (j 3 = .,f2y + �3 y 2 + �9 y 2 ( .,f2y) V2 6 y + O(y4 ) = V2 y3 + O(y4 ) . = V'i,y + 32 y 2 + 18 Thus for the quantities cp'(O) and cpC 3 l (O) of interest to us we find the following values: cp' (0) = V'i, , 'P ( 3 ) (0) = '{; . Substituting them into formula ( 19.27), we find that F(.A) = .x - I / 2 v'27f(1 + 112 ..\ - l + O(.A - 2 ) ) as .A ---t + oo , from which we again obtain formula (19.26). r.; y
_
19.2 The Asymptotics of Integrals (Laplace's Method)
633
In conclusion we shall make two more remarks on the problems discussed in this section.
Remark 6. (Laplace ' s method in the multidimensional case) . We note that Laplace ' s method can also be successfully applied in studying the asymptotics of multiple Laplace integrals F( >.. ) =
J f(x)e>-.S (x) dx ,
X
in which x E lR n , X is a domain in JR n , and f and S are real-valued functions in X. Lemma 1 on the exponential estimate holds for such integrals, and by this lemma the study of the asymptotics of such an integral reduces to studying the asymptotics of a part of it
J
f(x)e>-. S (x) dx ,
U (x o ) taken over a neighborhood of a maximum point x 0 of the function S(x) . If this is a nondegenerate maximum, that is, S"(x 0 ) 0, then by Morse ' s lemma (see Sect. 8.6 of Part 1) there exists a change of variable x =
=f.
j (f o -. i yi 2 dy , I
which in the case of smooth functions f and S can be studied by applying Fubini ' s theorem and using Watson ' s lemma proved above (see Problems 8-1 1 in this connection) .
Remark 7. (The stationary phase method) . In a wider interpretation, Laplace ' s method, as we have already noted, consists of the following: 1° a certain localization principle (Lemma 1 on the exponential estimate) , 2 ° a method of locally reducing an integral to canonical form (Morse ' s lemma) , and 3 ° a description of the asymptotics of canonical integrals (Watson ' s lemma) . We have met the idea of localization previously in our study of approxi mate identities, and also in studying Fourier series and the Fourier transform (the Riemann-Lebesgue lemma, smoothness of a function and the rate at which its Fourier transform decreases, convergence of Fourier series and in tegrals) .
634
19 Asymptotic Expansions Integrals of the form F(>.)
=
J f(x) ei>.S (x) dx ,
X
where x E JR. n , called Fourier integrals, occupy an important place in mathe matics and its applications. A Fourier integral differs from a Laplace integral only in the modest factor i in the exponent. This leads, however, to the relation l ei>.S ( x ) I = 1 when >. and S(x) are real, and hence the idea of a domi nant maximum is not applicable to the study of the asymptotics of a Fourier integral. Let X = [a, b] C JR.\ f E Cb oo ) ( [a, b] , JR.) , ( that is, f is of compact support on [a, b] ) , S E C ( oo l ( [a, b] , JR) and S'(x) =/=- 0 on [a, b] . Integrating by parts and using the Riemann-Lebesgue lemma ( see Prob lem 12) , we find that b
J a
f(x) ei>.S ( x ) dx = _.!._ i>.
b
J Sf'(x)(x) dei>.S (x) = a
= _ ;_ J � (L ) (x)e i>.S ( x ) dx = b
IA
= �A
b
J a
a
dx
ft (x)e i >.S ( x ) dx
S'
=
.
..
=� >, n
b
J fn (x) ei>.S(x ) dx = a
= o(>. - n ) as >. -t oo .
Thus if S'(O) =/=- 0 on the closed interval [a, b] , then because of the con stantly increasing frequencies of oscillation of the function ei>.S ( x ) as >. --+ oo, the Fourier integral over the closed interval [a, b] turns out to be a quantity of type 0 ( >. - oo ) . The function S(x) in the Fourier integral is called the phase function. Thus the Fourier integral has its own localization principle called the stationary phase principle. According to this principle, the asymptotic behavior of the Fourier integral as >. --+ oo ( when f E Cb 00 ) ) is the same as the asymptotics of the Fourier integral taken over a neighborhood U(x 0 ) of a stationary point x0 of the phase function ( that is, a point x 0 at which S'(x0 ) 0) up to a quantity 0 ( >. - oo ) . After this, by a change of variable the question reduces to the canonical integral =
E ( >. )
=
E
J J(x) ei>.xa dx 0
19.2 The Asymptotics of Integrals ( Laplace's Method )
635
whose asymptotic behavior is described by a special lemma of Erdelyi, which plays the same role for the Fourier integral that Watson ' s lemma plays for the Laplace integral. This scheme for investigating the asymptotics of a Fourier integral is called the stationary phase method. The nature of the localization principle in the stationary phase method is completely different from its nature in the case of the Laplace integral, but the general scheme of Laplace ' s method, as one can see, remains applicable even here. Certain details relating to the stationary phase method will be found in Problems 12-17. 19.2.6 Problems and Exercises
Laplace 's method in the one-dimensional case. 1 . a) For a > 0 the function h ( x ) = e - >. x" attains its maximum when x = 0. Here h ( x ) is a quantity of order 1 in a 8-neighborhood of x = 0 of size 8 = 0(>. - l / " ) .
Using Lemma 1 , show that i f 0 < 8 < 1 , then the integral
W (>.)
=
a
J
x(3 - I f ( x ) e- >. x" dx ,
c( >. ,o)
<> 6-1
where c(.>., 8 ) = >. has order O ( e - A >. 6 ) as >. -+ + oo , A being a positive constant. b ) Prove that if the function f is continuous at x = 0, then
W (>.) = a - 1 F(/3 / a) [f (O) + o(1)] >. - (3 /a as >. -+ + oo .
c ) In Theorem 1 a ) , the hypothesis f ( x ) = f ( xo ) + O ( x - xo ) can be weakened and replaced by the condition that f be continuous at xo . Show that when this is done the same principal term of the asymptotics is obtained, but in general not Eq. ( 1 9 . 2 ) itself, in which O ( x - xo ) is now replaced by o( 1). 2.
'
a) The Bernoulli numbers B2 k are defined by the relations
lt l < 2 7r . It is known that
Show that
( r'r )
( r') r
(x)
=
ln x +
1-) t !"" ( .!.t - 1 - e- t e d t . - x
0
1
- ( x ) � ln x - - 2x
oo B2 k x -2 k L k= 2k O
as
x
-+ + oo .
19 Asymptotic Expansions
636
b) Prove that as x --+ +oo
This asymptotic expansion is called Stirling 's series. c) Using Stirling's series, obtain the first two terms of the asymptotics of r(x+1) as x --+ +oo and compare your result with what was obtained in Example 13. d) Following the method of Example 13 and independently of it using Stirling's series, show that
r(x + 1 )
=
�(; r ( 1 + 1 �x + 28�x 2 + o(:3))
(
)
(
as x --+ +oo .
)
a) Let f E c [o, aJ, � , S E C ( ll [o, aJ, � , S(x) > 0 on [O , a] , and suppose S(x) attains its maximum at x = 0, with S' (O) =I 0. Show that if f(O) =I 0, then
3.
a
I (A) : = I f(x)S >.. (x) dx rv 0
)..�\��) s>-- +1 (0) as )..
--+
+oo .
b) Obtain the asymptotic expansion
00 2:::a k ).. - ( " +1 ) as ).. k =O if it is known in addition that J, S c< 00 ) ( [0, a], �) . I (A) s>-- + 1 (0) �
--+
+oo ,
E
4.
a) Show that
7r / 2
I sinn 0
t
dt =
�( 1 + ( - )) as O
n
1
n --+
+oo .
b) Express this integral in terms of Eulerian integrals and show that for n E N "t equa ls ( 2(n2 n-)1! )! ! ! · 27r .
1
.
-
)
2 !! 1 ( (2(2n) c) Obtam Walhs. , formula 1r - nhm . _ n . . n 1 ) " --+ oo d) Find the second term in the asymptotic expansion of the original integral as
n --+
_
·
+oo.
a) Show that J ( 1 - x 2 ) n dx rv � as n --+ +oo. 1
5.
-1
b) Find the next term in the asymptotics of this integral. 6. Show that if a
>
0, then as x --+ +oo
l
+ oo
0
t
( )
a ..!. -at t dt rv vfhx _l_ 2 ;,:;- a exp -; x a x
19.2 The Asymptotics of Integrals ( Laplace's Method )
637
a) Find the principal term of the asymptotics of the integral + oo (1 + t f e - n t dt as n -+ + oo .
7.
I 0
b ) Using this result and the identity k!n - k
=
+ oo n t k J e - t dt, show that 0
Laplace 's method in the multidimensional case. The exponential estimate lemma. Let M = sup S(x) , and suppose that for some x ED .X = .Xo the integral 8.
F(.X)
=
I f(x)e>.S( x ) dx
D c JR n
converges absolutely. Show that it then conveges absolutely for .X ?: .Xo and
lf(.X) I
� I l f (x) e>. s( x ) l dx � Ae>.M
(.X ?: .Xo) ,
D
where A is a positive constant.
9. Morse 's lemma. Let xo be a nondegenerate critical point of the function X E IR. n , defined and belonging to class in a neighborhood of xo . Then
S(x) , there exist neighborhoods U and V of x = xo and y = 0 and a diffeomorphism
c
s (
=
:
S(xo) + � t vi ( Yi ) 2
,
j=l
where det
•
.
Asymptotics of a canonical integral. a) Let t = (h , . . . , t n ) , V = { t E IR. n l l t i l � o, j = 1 , 2, . . . , n } , and a E c
where t'
=
(t 2 , . . . , t n ) and
!11
>
0. Show that
.X -+ + oo . This expansion is uniform in t' E V' = c
and a k E
I; a k (t')_x- ( k + 2 ) as k =O {t' E IR. n- l lt i l � o, j = 2, . . . , n } Ft (.X, t ' )
�
l
00
1
638
19 Asymptotic Expansions
-
b) Multiplying F1 (>.. , t') by e 9 t � and justifying the termwise integration of the corresponding asymptotic expansion, obtain the asymptotic expansion of the function
F2 ( >.. , t " )
=
8 J- 8 F1 (>.. , t')e -
� t2 2 2
d t 2 as >.. --+ +oo ,
where t " = (h , . . . , t n ) , v2 > 0. c) Prove that for the function
A ( >.. ) where Vj
>
=
8 8 !- 8 -8! a(t 1 , . . . , tn ) e ·
=
·
-�
f: Vj t�
J='
dt1 . . . t n ,
0, j = 1 , . . . , n , the following asymptotic expansion holds:
A ( >.. ) where a o
·
�
>.. - n /2 L a k >.. - k as >.. --+ +oo , k=O
V v;2.� ):n a(O) .
1 1 . The asymptotics of the Laplace integral in the multidimensional case. a) Let D be a closed bounded domain in ne , J, S E C(D, JR.), and suppose max S(x) xED i s attained only at one interior point x 0 of D. Let f and S b e c C oo ) i n some neighborhood of xo and det S " (xo) =P 0. Prove that if the integral ( * ) converges absolutely for some value >.. = >.. o , then
F ( >.. )
�
e.\ S(xo ) >.. - n /2 L a k >.. - k as >.. --+ +oo ' k=O
and this expansion can be differentiated with respect to >.. any number of times, and its principal term has the form
F ( >.. ) = e .\ S(xo ) >.. -n /2
(27r) n ( (xo) + O ( >.. - 1 )) I det S"(xo) l f
·
b) Verify that if instead of the relation J, S E c C oo ) all we know is that f E C and s E c C 3 ) in a neighborhood of Xo, then the principal term of the asymptotics as >.. --+ +oo remains the same with 0 ( >.. - 1 ) replaced by o(1) as >.. --+ +oo.
The stationary phase method in the one-dimensional case. 1 2 . Generalization of the Riemann-Lebesgue lemma. a) Prove the following gener alization of the Riemann-Lebesgue lemma. Let S E C (l ) ( la, bJ , lR ) and S'(x) # 0 on [a, b] = : I . Then for every function f that is absolutely integrable on the interval I the following relation holds:
F ( >.. )
=
b
J f(x ei.\ S (x) dx --+ 0 as >.. --+ oo , )
>.. E lR .
19.2 The Asymptotics of Integrals (Laplace's Method) b) Verify that if it is known in addition that f c. -+ oo F(>.)
=
)
(
k � ( i >.) - ( k + 1 ) -1_ j_ f(x) L....S ' (x) dx S ' (x) k=O
lb
c< n +1 l (J, JR) and S
E
+
639 E
o(>. - ( n + 1 ) ) .
a
c) Write out the principal term of the asymptotics of the function F(>.) as >. -+ oo, >. E JR. E c< 2 l [c, b] , but d) Show that if S E c<=l (I, JR) and t E c< 2 l [a, c] , t [a , � �.� f (j_ c< 2 l [a, b] , then the function F(>.) is not necessarily o (>. - 1 ) as >. -+ oo . e) Prove that when j, s E c<=J (I, JR) , the function F(>.) admits an asymptotic series expansion as >. -+ oo . f) Find asymptotic expansions as >. -+ oo , >. E JR, for the following integrals: e J(1 + x) - " '1/!i (x, >.) dx, j = 1 , 2, 3, if a > 0 and '1/! 1 = e L\ x , '1/!2 = cos >.x, and 0 '1/!3 = sin >.x.
l
l
13. The localization principle. a) Let I = [a, b] c JR, f c<=l (I, JR) , and S' (x) i- 0 on I. Prove that in this case
F(>.)
:=
b
I f(x)ei>. S ( x) dx a
=
o l >- 1 - oo
(
)
E
ca =l (I, JR) , S
as >. -+ 00
E
.
b) Suppose f E ca =l (J, JR) , s E c<=l (J, JR), and X 1 , , X m is a finite set of stationary points of S(x) outside which S' (x) i- 0 on I. We denote by F(>., xi ) the integral of the function f(x)e i >. S ( x ) over a neighborhood U(xi ) of the point Xj , j = 1 , . . . , m, not containing any other critical points in its closure. Prove that . • •
F(>.)
=
f F(>., Xj ) o ( l >- 1 -oo ) as >. -+ +
j =1
00 .
14. Asymptotics of the Fourier integml in the one-dimensional case. a) In a reason ably general situation finding the asymptotics of a one-dimensional Fourier integral can be reduced through the localization principle to describing the asymptotics of the canonical integral a E(>.) = x{J - 1 f(x)e i >. x"' dx ,
I 0
for which the following lemma holds. E r d e l y i's l e m m a. Let a :::0: 1 , (J k = 0, 1, 2, . . . . Then E(>.)
c:::
00
>
0, f
E
(
c<=l [o , aJ , lR and j< k l (a)
L a k >. - .!!:.±.1! "' as >. -+ + oo .
k=O
)
=
0,
19 Asymptotic Expansions
640
)
(
where
k + f] i � -"¥- t < k l ( o ) ak = ! r e , a a k! and this expansion can be differentiated any number of times with respect to A. Using Erdelyi's lemma, prove the following assertion. Let I = [ x o - t5, xo + .5] be a finite closed interval, let f, S E c<=l (J, ffi.) with f E Co (J, ffi.) , and let S have a unique stationary point xo on I, where S' (xo) = 0 but S" (xo) /=- 0. Then as A --+ + oo x o+J , -k F- ( "'' x 0 ) .J( x ) ei >. S(x) dx - ei �sg n S 11 (x o ) ei >. S ( x oh" - � � L a k" k=O xo - 8 and the principal term of the asymptotics has the form
I
·-
F(A, xo)
�
2 71'
=
A I S " (xo ) l
e i( � sg n S " (x o ) +>- S (x o )) ( ! ( xo ) + O(A - 1 ) )
·
b) Consider the Bessel function of integer order n � 0: 7r
Jn (x) Show that
=
� I cos(x sin ip -
n
ip) dip .
0
f2[ (
ln (x) = y � cos x -
n1r
1r
2 - 4
)
+
]
O(x - 1 ) as x --+ +oo
.
The stationary phase method in the multidimensional case 1 5 . The localization principle. a) Prove the following assertion. Let D be a domain in ffi.n , f E c6 = l (D, ffi.) , S E c<=l (D, ffi.) , grad S(x) /=- 0 for x E supp f, and
F(A) :=
I f ( x )ei>. S( x) dx .
D
Then for every k E N there exists a positive constant A(k) such that the estimate IF(A) I ::; A(k)A - k holds for A � 1 , and hence F(A) = O(A - = ) as A --+ + oo . b) Suppose as before that f E c6 =l (D, ffi.) , S E c<=l (D, ffi.) , but S has a finite number of critical points x 1 , . . . , Xm outside which grad S(x) /=- 0. We denote by F(A, XJ ) the integral of the function f(x) e i >. S (x) over a neighborhood U( x J ) of Xj whose closure contains no critical points except Xj . Prove that F(A)
=
L F(A, XJ ) j=l
+
O(A -= )
as
A --+ + oo
.
19.2 The Asymptotics of Integrals (Laplace's Method)
641
16. Reduction to a canonical integral. If xo is a nondegenerate cricial point of the function S E cC=) (D, JR) defined in a domain D c JRn , then by Morse's lemma (see Problem 9) there exists a local change of variable x = tp(y) such that Xo = tp(O) , s ( tp(y) ) = S(xo) + ! Ej (yl ) 2 ' where Ej = ± 1 , y = (y 1 ' ' y n ) , and det tp' (y) > j l 0. Using the localization principle (Problem 15) , now show that if f E c6 =) ( D, JR) , S E cC=) (D, lR) , and S has at most a finite number of critical points in D, all nondegenerate, then the study of the asymptotics of the integral ( * * ) reduces to studying the asymptotics of the special integral
�
,
8
• . .
8
I I
-8
. , .
-8
'1/J(y 1 , . . . , y n )e
!f .
z: "J (y1 ) 2 n
j=I
.
dy 1 . . . dy n .
17. Asymptotics of a Fourier integral in the multidimensional case. a) Using Erdelyi's lemma (Problem 14a) ) and the scenario described in Problem 10, prove that if D is a domain in lR n , f, S E cC=) (D, JR) , supp f is a compact subset of D, and Xo is the only critical point of S in D and is nondegenerate, then for the integral ( * * ) the following asymptotic expansion holds as A ---t +oo:
= i \ A) :::' A - n / 2 e i >. S ( x o ) L a k A - k ' k=O which can be differentiated any number of times with respect to A. The main term of the asymptotics has the form F(A)
=
c;)
n/2
[
exp iAS(xo) + i: sgn s " (xo) x
]
X
l det S" (xo) l - 1 1 2 [/(xo) + O(A - 1 )] as A ---t +oo .
Here S" (x) is the symmetric and by hypothesis nonsingular matrix of second derivatives of the function S at xo (the Hessian) , and sgn S" (xo ) is the signature of this matrix (or the quadratic form corresponding to it) , that is, the difference v+ - v_ between the number of positive and negative eigenvalues of the matrix S" (xo ) .
Topics and Q uestions for Midterm Examinat ions
1 . Series and Integrals Depending on a Parameter 1. The Cauchy criterion for convergence of a series. The comparison theorem and the basic sufficient conditions for convergence (majorant , integral, Abelco Dirichlet) . The series ((s) = L n-8 • n= l 2. Uniform convergence of families and series of functions. The Cauchy criterion and the basic sufficient conditions for uniform convergence of a series of functions (M-test , Abel-Dirichlet) . 3 . Sufficient conditions for two limiting passages to commute. Continuity, integration, and differentiation and passage to the limit . 4. The region of convergence and the nature of convergence of a power series. The Cauchy-Hadamard formula. Abel ' s (second) theorem. Taylor ex pansions of the basic elementary functions. Euler ' s formula. Differentiation and integration of a power series.
5. Improper integrals. The Cauchy criterion and the basic sufficient con ditions for convergence (M-test , Abel-Dirichlet) . 6 . Uniform convergence of an improper integral depending on a param eter. The Cauchy criterion and the basic sufficient conditions for uniform convergence (majorant , Abel-Dirichlet) . 7 . Continuity, differentiation, and integration of a proper integral depend ing on a parameter.
8. Continuity, differentiation, and integration of an improper integral de pending on a parameter. The Dirichlet integral. 9. The Eulerian integrals. Domains of definition, differential properties, reduction formulas, various representations, interconnections. The Poisson integral.
10. Approximate identities. The theorem on convergence of the convolu tion. The classical Weierstrass theorem on uniform approximation of a con tinuous function by an algebraic polynomial.
644
Topics and Questions for Midterm Examinations
2 . Problems Recommended as Midterm Questions Problem
1. P is a polynomial. Compute (et 1.: ) P( x).
tAxo is a solution of the Veri f y that the vector-valued function e Cauchy x = Ax, x(O) = xo. (Here x = Ax is a system of equations defined byproblem the matrix A.) Problem 3 . Find up to order o(1/n3) the asymptotics of the positive roots A 1 < A2 < · · · < An < · · · of the equation sinx + 1/x = 0 as n -+ Problem 4. a) Show that ln 2 = 1 - 1/2 + 1/3 - · · ·. How many terms of this series must be taken to determine ln 2 within w - 3? b) Verify that ! ln ��� = t + �t 3 + � t 5 + · · ·. Using this expansion it becomes convenient to compute ln x by setting x = ���. c) Setting t = 1/3 in b), obtain the equality 1 1 1 (1) 1 (1)5 2 ln 2 = 3 + 3 3 3 + 5 3 + . . . How this manywithtermstheofresult this series must one take to find ln 2 within w-3? Com pare of a). This is one of the methods of improving convergence. Problem 5 . Verify that in the sense of Abel summation a) 1 - 1 + 1 · · · = ! · b) 2::: sinktp = ! · ! 'P, 'P � 21rn, n E Z. c) ! + 2::: cosktp = 0, 'P � 27rn, n E Z. Problem 6 . Prove Hadamard ' s lemma: a) If f E C( 1 l (U(xo)), then f(x) f(xo) + cp(x)(x - x0), where 'P E C(U(xo)) and cp(xo) = f'(xa ). b) If f E C(n ) (U(xo)), then f(x) = f(xo) + �1. f'(xo)(x - xo) + · · · + + (n 1 1)! f( n - I )(xo)(x - xa ) n - I + cp(x)(x - x0) n , where 'P E C(U(xo)) and cp(xo) = ,h J Cn l(xo). Problem 2 .
oo .
00
k= l
00
k= l
_
2. Problems Recommended as Midterm Questions
645
c) What do these relations look like in coordinate form, when x = (x 1 , x n ) , that is, when f is a function of n variables? .
.
.
•
Problem 7. a) Verify that the function
Jo (x) =
.!_ 7r
1
J
cos xt
0 v'f=t2
dt
y" �y' y
satisfies Bessel ' s equation + + = 0. b) Tty to solve this equation using power series. c) Find the power-series expansion of the function J0 (x) .
+Joo +Joo
Problem 8. Verify that the following asymptotic expansions hold a)
r (a ' x) : =
b)
Erf (x) : =
as x -+ +oo.
X
X
oo r (a) � r (a oo
t O!. - l e - t dt � e -01. """"'
k= l
k + 1)
1 '- x 2 t2 e - dt � 2 v ne - """"'
x a -k '
1
� r (3/2 k)x 2 k- l
k= l
_
Problem 9. a) Following Euler, obtain the result that the series 1 - 1 !x +
2!x 2 - 3!x 3 + · · · is connected with the function S(x) : =
+Joo 0
e- t dt . 1 + xt
--
b ) Does this series converge? c) Does it give the asymptotic expansion of S(x) as x -+ 0? Problem 10. a) A linear device A whose characteristics are constant over time responds to a signal J(t) in the form of a <>-function by giving out the signal (function) E(t) . What will the response of this device be to an input signal f(t) , -oo < t < +oo? b) Can the input signal f always be recovered uniquely from the trans formed signal f := Af?
646
Topics and Questions for Midterm Examinations
3 . Integral Calculus ( Several Variables) 1. Riemann integral on an n-dimensional interval. Lebesgue criterion for ex istence of the integral. 2 . Darboux criterion for existence of the integral of a real-valued function on an n-dimensional interval. 3. Integral over a set . Jordan measure (content) of a set and its geomet ric meaning. Lebesgue criterion for existence of the integral over a Jordan measurable set . Linearity and additivity of the integral. 4. Estimates of the integral. 5 . Reduction of a multiple integral to an iterated integral. Fubini ' s theo rem and its most important corollaries. 6. Formula for change of variables in a multiple integral. Invariance of measure and the integral. 7. Improper multiple integrals: basic definitions, majorant criterion for convergence, canonical integrals. Computation of the Euler�Poisson integral. 8 . Surfaces of dimension k in JR n and basic methods of defining them. Abstract k-dimen�ional manifolds. Boundary of a k-dimensional manifold as a (k - I )-dimensional manifold without boundary. 9. Orientable and nonorientable manifolds. Methods of defining the ori entation of an abstract manifold and a (hyper )surface in JRn . Orientability of the boundary of an orientable manifold. Orientation in duced on the boundary from the manifold. 10. Tangent vectors and the tangent space to a manifold at a point. Interpretation of a tangent vector as a differential operator. 1 1 . Differential forms in a region D C JR n . Examples: differential of a function, work form, flux form. Coordinate expression of a differential form. Exterior derivative operator. 1 2 . Mapping of objects and the adjoint mapping of function� on the�e ob jects. Transformation of points and vectors of tangent spaces at these points under a smooth mapping. Transfer of functions and differential forms under a smooth mapping. A recipe for carrying out the transfer of forms in coordinate form. 13. Commutation of transfer of differential form� with exterior multipli cation and differentiation. Differential forms on a manifold. Invariance ( un ambiguou� nature) of operations on differential forms. 14. A scheme for computing work and flux. Integral of a k-form over a k-dimensional smooth oriented surface, taking account of orientation. Inde pendence of the integral of the choice of parametrization. General definition
4. Problems Recommended for Studying the Midterm Topics
647
of the integral of a differential k-form over a k-dimensional compact oriented manifold. 15. Green ' s formula on a square, its derivation, interpretation, and ex pression in the language of integrals of the corresponding differential forms. The general Stokes formula. Reduction to a k-dimensional interval and proof for a k-dimensional interval. The classical integral formulas of analysis as particular versions of the general Stokes formula. 16. The volume element on JR. n and on a surface. Dependence of the volume element on orientation. The integral of first kind and its independence of orientation. Area and mass of a material surface as an integral of first kind. Expression of the volume element of a k-dimensional surface S k C JR. n in local parameters and the expression of the volume element of a hypersurface s n - l c JR. n in Cartesian coordinates of the ambient space. 1 7. Basic differential operators of field theory ( grad, curl, div ) and their connection with the exterior derivative operator d in oriented Euclidean space
JR_3 .
18. Expression of work and flux of a field as integrals of first kind. The basic integral formulas of field theory in JR. 3 as the vector expression of the classical integral formulas of analysis. 19. A potential field and its potential. Exact and closed forms. A necessary differential condition for a form to be exact and for a vector field to be a potential field. Its sufficiency in a simply connected domain. Integral criterion for exactness of 1-forms and vector fields.
20. Local exactness of a closed form ( the Poincare lemma ) . Global anal ysis. Homology and cohomology. De Rham ' s theorem ( statement ) . 2 1 . Examples of the application of the Stokes ( Gauss-Ostrogradskii ) for mula: derivation of the basic equations of the mechanics of continuous media. Physical meaning of the gradient, curl, and divergence. 22. Hamilton ' s nabla operator and work with it . The gradient , curl, and divergence in triorthogonal curvilinear coordinates.
4. Problems Recommended for Studying the Midterm Topics The numbers followed by closing parentheses below refer to the topics 1�-22 just listed. The closing parentheses dashes are followed by section numbers ( for example 13.4 means Sect . 4 of Chapt . 13 ) , which in turn are separated by a dash from the numbers of the problems from the section related to the topic from the list above.
648
Topics and Questions for Midterm Examinations
1) 11.1-2,3; 2) 11.1-4; 3) 11.2-1,3,4; 4) 11.3-1,2,3,4; 5) 11.4-6,7 and 13.2-6; 6) 11.5-9 and 12.5-5,6; 7) 11.6-1,5,7; 8) 12.1-2,3 and 12.4-1,4; 9) 12.2-1,2,3,4 and 12.5-11; 10) 15.3-1,2; 11) 12.5-9 and 15.3-3; 12) 15.3-4; 13) 12.5-8,10; 14) 13.1-3,4,5,9; 15) 13.1-1,10,13,14; 16) 12.4-10 and 13.2-5; 17) 14.1-1,2; 18) 14.2-1,2,3,4,8; 19) 14.3-7,13,14; 20 ) 14.311,12; 21) 13.3-1 and 14.1-8; 22) 14.1-4,5,6.
Examinat ion Topics
1. Series and Integrals Depending on a Parameter 1. Cauchy criterion for convergence of a series. Comparison theorem and the basic sufficient conditions for convergence (majorant , integral, Abeloo Dirichlet) . The series ((s) = I: n-8 •
n=l
2 . Uniform convergence of families and series of functions. Cauchy crite rion and the basic sufficient conditions for uniform convergence of a series of functions (M-test , Abel-Dirichlet) . 3 . Sufficient conditions for commutativity of two limiting passages. Con tinuity, integration, and differentiation and passage to the limit . 4. Region of convergence and the nature of convergence of a power series. Cauchy-Hadamard formula. Abel ' s (second) theorem. Taylor expansions of the basic elementary functions. Euler ' s formula. Differentiation and integra tion of a power series. 5. Improper integrals. Cauchy criterion and the basic sufficient conditions for convergence (majorant , Abel-Dirichlet) . 6 . Uniform convergence of an improper integral depending on a parame ter. Cauchy criterion and the basic sufficient conditions for uniform conver gence ( M-test , A bel-Dirichlet) . 7. Continuity, differentiation, and integration of a proper integral depend ing on a parameter. 8. Continuity, differentiation, and integration of an improper integral de pending on a parameter. Dirichlet integral. 9. Eulerian integrals. Domains of definition, differential properties, reduc tion formulas, various representations, interconnections. Poisson integral. 10. Approximate identities. Theorem on convergence of the convolution. Classical Weierstrass theorem on uniform approximation of a continuous function by an algebraic polynomial.
650
Examination Topics
1 1 . Vector spaces with an inner product . Continuity of the inner product and algebraic properties connected with it . Orthogonal and orthonormal sys tems of vectors. Pythagorean theorem. Fourier coefficients and Fourier series. Examples of inner products and orthogonal systems in spaces of functions. 12. Orthogonal complement. Extremal property of Fourier coefficients. Bessel ' s inequality and convergence of the Fourier series. Conditions for com pleteness of an orthonormal system. Method of least squares. 13. Classical (trigonometric) Fourier series in real and complex form. Riemann-Lebesgue lemma. Localization principle and convergence of a Fourier series at a point. Example: expansion of cos( a x) in a Fourier series and the expansion of sin(7rx) /7TX in an infinite product . 14. Smoothness of a function, rate of decrease of its Fourier coefficients, and rate of convergence of its Fourier series. 1 5 . Completeness of the trigonometric system and mean convergence of a trigonometric Fourier series. 16. Fourier transform and the Fourier integral (the inversion formula) . Example: computation of f for f(x) : = exp( -a 2 x 2 ) . 17. Fourier transform and the derivative operator. Smoothness of a func tion and the rate of decrease of its Fourier transform. Parseval ' s equality. The Fourier transform as an isometry of the space of rapidly decreasing functions. 18. Fourier transform and convolution. Solution of the one-dimensional heat equation. 19. Recovery of a transmitted signal from the spectral function of a device and the signal received. Kotel ' nikov ' s formula. 20. Asymptotic sequences and asymptotic series. Example: asymptotic expansion of Ei ( x). Difference between convergent and asymptotic series. Asymptotic Laplace integral (principal term) . Stirling ' s formula.
2. Integral Calculus (Several Variables)
651
2. Integral Calculus ( Several Variables) 1 . Riemann integral on an n-dimensional interval. Lebesgue criterion for ex istence of the integral. 2 . Darboux criterion for the existence of the integral of a real-valued function on an n-dimensional interval. 3. Integral over a set. Jordan measure (content) of a set and its geomet ric meaning. Lebesgue criterion for existence of the integral over a Jordan measurable set . Linearity and additivity of the integral. 4. Estimates of the integral. 5. Reduction of a multiple integral to an iterated integral. Fubini ' s theo rem and its most important corollaries. 6. Formula for change of variables in a multiple integral. Invariance of measure and the integral. 7. Improper multiple integrals: basic definitions, the majorant criterion for convergence, canonical integrals. Computation of the Euler-Poisson integral. 8. Surfaces of dimension k in !R n and the basic methods of defining them. Abstract k-dimensional manifolds. Boundary of a k-dimensional manifold as a (k - I)-dimensional manifold without boundary. 9. Orientable and nonorientable manifolds. Methods of defining the ori entation of an abstract manifold and a (hyper )surface in !Rn . Orientability of the boundary of an orientable manifold. Orientation on the boundary induced from the manifold. 10. Tangent vectors and the tangent space to a manifold at a point . Interpretation of a tangent vector as a differential operator. 1 1 . Differential forms in a region D C !R n . Examples: differential of a function, work form, flux form. Coordinate expression of a differential form. Exterior derivative operator. 12. Mapping of objects and the adjoint mapping of functions on these ob jects. Transformation of points and vectors of tangent spaces at these points under a smooth mapping. Transfer of functions and differential forms under a smooth mapping. A recipe for carrying out the transfer of forms in coordinate form. 13. Commutation of the transfer of differential forms with exterior mul tiplication and differentiation. Differential forms on a manifold. lnvariance (unambiguous nature) of operations on differential forms. 14. A scheme for computing work and flux. Integral of a k-form over a k-dimensional smooth oriented surface. Taking account of orientation. Inde pendence of the integral of the choice of parametrization. General definition
652
Examination Topics
of the integral of a differential k-form over a k-dimensional compact oriented manifold. 1 5 . Green's formula on a square, its derivation, interpretation, and ex pression in the language of integrals of the corresponding differential forms. General Stokes formula. Reduction to a k-dimensional interval and proof for a k-dimensional interval. Classical integral formulas of analysis as particular versions of the general Stokes formula. 1 6 . Volume element on JR n and on a surface. Dependence of volume ele ment on orientation. The integral of first kind and its independence of ori entation. Area and mass of a material surface as an integral of first kind. Expression of volume element of a k-dimensional surface S k c JRn in local parameters and expression of volume element of a hypersurface s n - l c ]Rn in Cartesian coordinates of the ambient space.
17. Basic differential operators of field theory ( grad, curl, div ) and their connection with the exterior derivative operator d in oriented Euclidean space JR3 0
18. Expression of work and flux of a field as integrals of first kind. Basic integral formulas of field theory in JR3 as the vector expression of the classical integral formulas of analysis. 19. A potential field and its potential. Exact and closed forms. A necessary differential condition for a form to be exact and for a vector field to be a potential field. Its sufficiency in a simply connected domain. Integral criterion for exactness of 1-forms and vector fields. 20. Examples of the application of the Stokes ( Gauss-Ostrogradskii ) for mula: derivation of the basic equations of the mechanics of continuous media. Physical meaning of the gradient, curl, and divergence.
References
1 . Classic Works 1 . 1 . Primary Sources
Newton, 1 . : - a. ( 1687): Philosophire Naturalis Principia Mathematica. Jussu Societatis Regire ac typis Josephi Streati, London. English translation from the 3rd edition (1726): University of California Press, Berkeley, CA (1999) . - b. (1967-1981): The Mathematical Papers of Isaac Newton, D. T. Whiteside, ed. , Cambridge University Press. Leibniz, G. W. (1971): Mathematische Schriften. C. I. Gerhardt , ed. , G. Olms, Hildesheim. 1 . 2 . Major Comprehensive Expository Works
Euler, L. - a. (1748) : Introductio in Analysin Infinitorum. M. M. Bousquet, Lausanne. English translation: Springer-Verlag, Berlin - Heidelberg - New York (1988-
1990) .
- b. (1755) : Institutiones Calculi Differentialis. Impensis Academire lmperialis Scientiarum, Petropoli. English translation: Springer, Berlin - Heidelberg New York (2000) . - c. (1768-1770): Institutionum Calculi Integralis. Impensis Academire Impe rialis Scientiarum, Petropoli. Cauchy, A.-L. - a. (1989) : Analyse Algebrique. Jacques Gabay, Sceaux. - b. (1840- 1844) : Lec;ons de Calcul Differential et de Calcul Integral. Bachelier, Paris. 1 . 3 . Classical courses of analysis from the first half of the twentieth century
Courant, R. (1988) : Differential and Integral Calculus. Translated from the Ger man. Vol. 1 reprint of the second edition (1937) . Vol. 2 reprint of the 1936 original. Wiley Classics Library. A Wiley-Intercience Publication. John Wiley & Sons, Inc., New York.
654
References
de la Vallee Poussin, Ch.-J. ( 1954, 1957 ) : Cours d'Analyse Infinitesimale. ( Tome 1 1 1 ed. , Tome 2 9 ed. , revue et augmentee avec la collaboration de Fernand Simonart. ) Librairie universitaire, Lou vain. English translation of an earlier edition, Dover Publications, New York ( 1946 ) . Goursat, E. ( 1992 ) : Cours d'Analyse Mathematiques. ( Vol. 1 reprint of the 4th ed 1924, Vol. 2 reprint of the 4th ed 1925 ) Les Grands Classiques Gauthier-Villars. Jacques Gabay, Sceaux. English translation, Dover Publ. Inc., New York ( 1959 ) .
2 . Textbooks
5
Apostol, T. M. ( 1974 ) : Mathematical Analysis. 2nd ed. World Student Series Edi tion. Addison-Wesley Publishing Co. , Reading, Mass.-London-Don Mills, Ont. Courant, R. , John F. ( 1999 ) : Introduction to Calculus and Analysis. Vol. I. Reprint of the 1989 edition. Classics in Mathematics Springer-Verlag, Berlin. Courant, R., John F. ( 1989 ) : Introduction to Calculus and Analysis. Vol. II. With the assistance of Albert A. Blank and Alan Solomon. Reprint of the 1974 edi tion. Springer-Verlag, New York. Nikolskii, S. M. ( 1990 ) : A Course of Mathematical Analysis. Vols. 1, 2. Nauka, Moscow. English translation of an earlier adition: Mir, Moscow ( 1985 ) . Rudin, W. ( 1976 ) : Principals of Mathematical Analysis. McGraw-Hill, New York. Rudin, W. ( 1987 ) : Real and Complex Analysis. 3rd ed. , McGraw-Hill, New York. Spivak, M. ( 1965 ) : Calculus on Manifolds: a Modern Approach to the Classical Theorems of Advanced Calculus. W. A. Benjamin, New York. Whittaker, E. T., Watson, J. N. ( 1979 ) : A Course of Modern Analysis. AMS Press, New York.
3. C lassroom Materials Biler, P. , Witkowski, A. ( 1990 ) : Problems in Mathematical Analysis. Monographs and Textbooks in Pure and Applied Mathematics, 132. Marcel Dekker, New York. Demidovich, B. P. ( 1990 ) : A Collection of Problems and Exercises in Mathematical Analysis. Nauka, Moscow. English translation of an earlier edition, Gordon and Breach, New York - London - Paris ( 1969 ) . Gelbaum, B. ( 1982 ) : Problems in Analysis. Problem Books in Mathemetics. Springer-Verlag, New York-Berlin, 1982. Gelbaum, B., Olmsted, J. ( 1964 ) : Counterexamples in Analysis. Holden-Day, San Francisco. Makarov, B. M., Goluzina, M. G., Lodkin, A. A., Podkorytov, A. N. ( 1992 ) : Selected Problems in Real Analysis. Nauka, Moscow. English translation: Translations of Mathematical Monographs 107, American Mathematical Society, Providence ( 1992 ) . 5
For the convenience of the Western reader the bibliography of the English edition has substantially been revised.
4. Further Reading
655
P6lya, G., Szego, G. ( 1970/71): Aufgaben und Lehrsatze aus der Analysis. Springer Verlag, Berlin - Heidelberg - New York. English translation: Springer-Verlag, Berlin - Heidelberg - New York ( 1972-76).
4. Further Reading Arnol'd, V. I. - a. (1989a) : Huygens and Barrow, Newton and Hooke: Pioneers in Mathemat ical Analysis and Catastrophe Theory, from Evolvents to Quasicrystals. Nauka, Moscow. English translation: Birkhauser, Boston (1990) . - b. (1989b) : Mathematical Methods of Classical Mechanics. Nauka, Moscow. English translation: Springer-Verlag, Berlin - Heidelberg - New York (1997) . Avez, A. ( 1986) : Differential Calculus. Translated from the French. A Wiley Intercience Publication. John Wiley & Sons, Ltd., Chichister. Bourbaki, N. ( 1969): Elements d'Histoire des Mathematiques. 2e edition revue, cor rigee, augmentee. Hermann, Paris. English translation: Springer-Verlag, Berlin - Heidelberg - New York (1994) . Cartan, H. ( 1977) : Cours de Calcul Differentiel. Hermann, Paris. English translation of an earlier edition: Differential Calculus. Exercised by C.Buttin, F.Riedeau and J.L.Verley. Houghton Miffiin Co. , Boston, Mass. (1971 ) . de Bruijn, N. G. (1958) : Asymptotic methods i n analysis. North-Holland, Amster dam. Dieudonne, J. (1969) : Foundations of Modern Analysis. Enlarged and corrected printing. Academic Press, New York. Dubrovin, B. A., Novikov, S. P. , Fomenko, A. T. (1986) : Modern Geometry - Meth ods and Applications. Nauka, Moscow. English translation: Springer-Verlag, Berlin - Heidelberg - New York (1992) . Einstein, A. (1982) : Ideas and Opinions. Three Rivers Press, New York. Contains translations of the papers "Principles of Research" (original German title: "Mo tive des Forschens" ) , pp. 224-227, and "Physics and Reality," pp. 290-323. Evgrafov, M. A. (1979) : Asymptotic Estimates and Entire Functions. 3rd ed. Nauka, Moscow. English transalation from the first Russian edition: Gordon & Breach, New York ( 1961) . Fedoryuk, M . V . (1977) The Saddle-Point Method. Nauka, Moscow (Russian) . Feynman, R. , Leighton, R. , Sands, M. (1963-1965): The Feynman Lectures on Physics, Vol. 1: Modern Natural Science. The Laws of Mechanics. Addison Wesley, Reading, Massachusetts. Gel'fand, I. M. ( 1998) : Lectures on Linear Algebra. Dobrosvet, Moscow. English translation of an earlier edition: Dover, New York (1989) . Halmos, P. ( 1974) : Finite-dimensional Vector Spaces. Springer-Verlag, Berlin - Hei delberg -- New York. Jost, J. (2003) : Postmodern Analysis. 2nd ed. Universitext. Berlin: Springer. Klein, F. ( 1926) : Vorlesungen iiber die Entwicklung der Mathematik im 19 Jahrhun dert. Springer-Verlag, Berlin Kolmogorov, A. N., Fomin, S. V. (1989) : Elements of the Theory of Functions and Functional Analysis. 6th ed. , revised, Nauka, Moscow. English translation of an earlier edition: Graylock Press, Rochester, New York (1957) .
656
References
Kostrikin, A. l., Manin, Yu. l. ( 1986 ) : Linear Algebra and Geometry. Nauka, Moscow. English translation: Gordon and Breach, New York ( 1989 ) . Landau, L. D., Lifshits, E. M. ( 1988 ) : Field Theory. 7th ed. , revised, Nauka, Moscow. English translation of an earlier edition: Pergamon Press, Oxford-New York ( 1975 ) . Lax, P. D., Burstein S. Z., Lax A. ( 1972 ) : Calculus with Applications and Comput ing. Vol. I. Notes based on a course given at New York University. Courant Institute of Mathematical Sciences, New York University, New York. Manin, Yu. l. ( 1979 ) : Mathematics and Physics. Znanie, Moscow. English translation: Birkhauser, Boston ( 1979 ) . Milnor, J. ( 1963 ) : Morse Theory. Princeton University Press. Narasimhan, R. ( 1968 ) : Analysis on Real and Complex Manifolds. Masson, Paris. Olver, F. W. J. ( 1997 ) : Asymptotics and Special Functions. Reprint. AKP Classics. Wellesley, MA: A K Peters. Pham, F. ( 1992 ) : Geometrie et calcul differentiel sur les varietes. [Course, studies and exercises for Masters in mathematics] Inter Editions, Paris. Poincare, H. ( 1982 ) : The Foundations of Science. Authorized translation by George Bruce Halstead, and an introduction by Josiah Royce, preface to the UPA edition by L. Pearce Williams. University Press of America, Washington, DC. Pontryagin, L. S. ( 1974 ) : Ordinary Differential Equations. Nauka, Moscow. En glish translation of an earlier edition: Addison-Wesley, Reading, Massachusetts ( 1962 ) . Shilov, G. E. - a. ( 1996 ) : Elementary Real and Complex Analysis. Revised English edition translated from Russian and edited by Richard A. Silverman. Corrected reprint of the 1973 English edition. Dover Publications Inc., Mineola, NY. - b. ( 1969 ) : Mathematical Analysis. Functions of one variable. Pts. 1 and 2. Moscow: Nauka, ( Russian ) . - c. ( 1972 ) : Mathematical Analysis. Functions of several variables ( 3 pts. ) . Moscow: N auka, ( Russian ) . Schwartz, L. ( 1998 ) : Analyse. Hermann, Paris. Weyl, H. ( 1926 ) : Die heutige Erkenntnislage in der Mathematik. Weltkreis-Verlag, Erlangen. Russian translations of eighteen of Weyl's essays, with an essay on Weyl: Mathematical Thought. Nauka, Moscow ( 1989 ) . Zel'dovich, Ya. B., Myshkis, A. D. ( 1967 ) : Elements of Applied Mathematics. Nauka, Moscow. English translation: Mir, Moscow ( 1976 ) .
Index of B asic Not at ion
LOGICAL SYMBOLS ==} - logical consequence (implication) � - Logical equivalence
·-
.-
}
equality by definition; colon on the side of the object defined
SETS
E - closure of the set E 8E - boundary of the set E E E \ 8E - interior of the set E B(x, r) - ball of radius r with center at x S(x, r) - sphere of radius r with center at x 0
:=
SPACES (X, d) - metric space with metric d (X, T ) - topological space with system T of open sets ]Rn (C n ) n-dimensional real (complex) space JR 1 = lR (C 1 = q - set of real (complex) numbers x = ( x 1 , . . . , xn ) - coordinate expression of a point of n-dimensional space C ( X , Y) - set (space) of continuous functions on X with values in Y C[a, b] - abbreviation for C([a, b] , JR) or C([a, b] , C ) C ( k ) (X, Y) - set of mappings from X into Y that are k times continu ously differentiable C ( k ) [a, b] - abbreviation for C ( k ) ([a, b] , JR ) or C ( k ) ([a, b] , C) . Cp[a, b] - space C[a, b] endowed with norm II/Il P C2 [a, b] - space C[a, b] with Hermitian inner product (!, g) of functions or mean-square deviation norm
658
Index of Basic Notation
R ( E ) - set (space) of functions that are Riemann integrable over the set E R[a, b] - space R ( E ) when E = [a, b] R(E) - space of classes of Riemann integrable functions on E that are equal almost everywhere on E Rp(E) (Rp(E)) - space R(E) endowed with norm R2 (E) (R2 (E)) - space R(E) endowed with Hermitian inner product ( ! , g ) or mean-square deviation norm Rp [a, b], R2 [a, b] - spaces Rp ( E ) and R2 ( E ) when E = [a, b] .C(X; Y ) , (.C(X1 , . . . , Xn ; Y)) - space of linear (n-linear) mappings from X (from (X1 X · · · X Xn )) into Y TMP or TM(p) , TpM, Tp(M) - tangent space to the surface (manifold) M at the point p E M S - Schwartz space of rapidly decreasing functions V( G) - space of fundamental functions of compact support in the do main ·G V' (G) - space of generalized functions on the domain G V - an abbreviation for V( G) when G = JRn V' - an abbreviation for V' (G) when G = JRn
I JI P
M ETRICS , NORM S , I N N E R PRO D U CTS
1
d (x , x 2 ) - distance between points x 1 and x 2 in the metric space (X, d) lx l , l l x l l - absolute value (norm) of a vector x E X in a normed vector space - norm of the linear (multilinear) operator , p :2:: 1 - integral norm of the function dx := ( J
A I AI 1 P I J I P I J I P (x) ) 1 l f l 2 -- mean-square deviation norm ( I / I l P when = 2) E
p
(a, b) - Hermitian inner product of the vectors a and b
f
(!, g) := EJ(! · g) (x) dx - Hermitian inner product of the functions f and g
a · b - inner product of a and b in JR 3 a x b -- vector (cross) product of vectors a and b in JR3 (a, b , c ) - scalar triple product of vectors a, b, c in JR 3
F U NCTIONS
g o f - composition of functions f and g
f - 1 - inverse of the function f f (x) - value of the function f at the points x; a function of x
Index of Basic Notation
659
f(x l , . . . , xn ) � value of the function f at the point x = (x 1 , . . . , xn ) E X in the n-dimensional space X ; a function depending on n variables x l , . . . , xn . supp f � support of the function f J f (x) -- jump of the function f at the point x {ft : t E T} � a family of functions depending on the parameter t E T {!n ; n E N} or {!n } � a sequence of functions ft --+ f on E convergence of the family of functions Ut ; t E T} to the B function f on the set E over the base B in T ft =4 f on E uniform convergence of the family of functions Ut ; t E T} B to the function f on the set E over the base B in T �
�
f = o(g) over B f = O(g) over B j g or j � g over "'
}
B
asymptotic formulas (the symbols of comparative asymptotic behavior of the functions f and g over the base
B
00
f(x) 2::: 'Pn (x) over B � expansion in an asymptotic series n= l V(x) � Dirichlet function �
exp(A) � exponential of a linear operator A B(o:, /3) � Euler beta function T(o:) � Euler gamma function XE characteristic function of the set E �
DIFFERENTIAL CALCULUS
f'(x) , fx (x) , df(x) , D f(x) � tangent mapping to f (differential of f) at the point x . *fr , a;j(x) , D;j(x) � partial derivative (partial differential) of a function f depending on variables x 1 , . . . , x n at the point x = (x 1 , . . . , x n ) with respect to the variable x i Dv f (x) � derivative of the function f with respect to the vector v at the point x
'V
� Hamilton ' s nabla operator grad f � gradient of the function f div A � divergence of the vector field A curl B � curl of the vector field B INTEGRAL CALCULUS
f.L(E) � measure of the set E
Index of Basic Notation
660
I J(x) dx E
I f(x 1 , . . . , x n ) dx 1 . . . dx n E
I . . . I f(x l ' . . . ' x n ) dx l . . . dx n E
,}
integral of the function f over the set E C JRn
I dy I f(x, y) dx - iterated integral
y
X
I P dx + Q dy + R dz
1
I F . ds , I (F , ds) 1
1
curvilinear integral (of second kind) or the work of the field F = (P, Q, R) along the path '"'(
,}
I f ds - curvilinear integral (of first kind) of the function f along the curve '"Y
1
II P dy A dz + Q dz A dx + R dx A dy s
II F · du , s
II (F, du) , x
integral (of second kind) over the surface S in JR3 ; flux of the field F = (P, Q, R) across the surface S
III f da - surface integral (of first kind) of f over the surface s
D I F F ERENTIAL FORMS
w (wP) - a differential form (of degree p) wP A wq - exterior product of forms wP and wq dw - (exterior) derivative of the form w I w - integral of the form w over the surface (manifold) M
M
w} : = (F, · ) - work form w{r : = (V, · ) - flux form
S
Subject Index
8-function, 281 8-neighborhood, 5, 1 1 c-grid, 16 k-dimensional volume, 185 k-path, 254 n-dimensional disk, 1 79, 322 nth moment, 404 p-- cycle, 358 p-- form, 197 TJ space, 14 T2 space, 14 0-formula, 593
A Abel summation, 394 Abel's transformation, 376 Abel-Dirichlet test, 376, 377, 379 Absolute convergence - of a series, 374-376 - of an improper integral, 420 - of functions, 375 Adiabatic, 223 Adiabatic constant, 224 Adjoint mapping, 317 Admissible set, 117 Alexander horned sphere, 162 Algebra - exterior, 319 - graded, 314 - Grassmann, 319 - Lie, 72 - of forms, 313 -- skew-symmetric, 314 - of functions, 401 - - complex, 401 real, 401 self-adjoint, 405 separating points, 401
Almost everywhere, 1 1 1 Alternation, 314 Amplitude, 560 Amplitude modulation, 587 Analysis, harmonic, 560 Angular velocity, 69-70 Approximate identity, 455 Area - as the integral of a form, 227-230 - Minkowski outer, 194 - of a k-dimensional surface, 187, 229 - of a piecewise-smooth surface, 229 - of a sphere in lie, 192, 444 Asymptotic equality, 598 Asymptotic equivalence, 598 Asymptotic estimate, 597 - uniform, 6 1 1 Asymptotic expansion - in the sense of Erdelyi, 610 - uniform, 6 1 1 Asymptotic formula, 598 Asymptotic methods, 597 Asymptotic problem, 596 Asymptotic sequence, 601 Asymptotic series, 601 , 602 - general, 601 - in the sense of Erdelyi, 610 - in the sense of Poincare, 602 - power, 606-609 Asymptotic zero, 603 Asymptotics, 596, 597 - of a Bessel function, 640 - of a Laplace integral, 638 - of canonical integral, 633 - of Legendre polynomials, 623 - of the Fourier integral, 635, 639 - of the gamma function, 622, 629
662
Subject Index
- of the Laplace integral, 635 - of the probability error function, 628 Atlas analytic, 325 of a surface (manifold) , 162, 320 - of class c ( k ) ' 325 - orienting, 174, 328 - smooth, 325 Averaging of a function using a kernel, 494 B
Ball - closed, 5 - in JR n , volume, 444 - in a metric space, 5 Band-limited signal, 585 Base - in the set of partitions, 108 - of a topology, 10 Basis of a vector space, 500 Bernoulli integral, 309 Bernoulli numbers, 635 - generating function, 635 Bernoulli polynomials, 555 Bessel function, 392, 410 - asymptotics, 640 Bessel's equation, 392, 410, 414, 434 Bessel's inequality, 508-509 - for trigonometric system, 534 Beta function, 437-439 Bicompact set, 15 Borel's formulas, 581 Boundary - of a p-dimensional cube, 357 of a half-space, 1 78 - of a manifold, 321 - of a surface, 178 Boundary cycle, 358 Boundary point, 6, 13, 321 Boundedness - total, 397 - uniform, 397 - uniform, of a family of functions, 371 Brachistochrone, 92-94 Bracket, Poisson, 348 Bundle of tangent paths, 347
c
Canonical embedding, 349 Canonical integral - asymptotics, 633 Cardinal sine, 588 Carnot cycle, 226 Category of a set, 28 Cauchy integral, 309 Cauchy-Bunyakovskii inequality, 452, 505 Cauchy-Riemann equations, 310 Chain - of charts, 329 - - contradictory, 329 - - disorienting, 329 - of singular cubes, 357 Change of variable in an integral, 135-143, 156-158 Characteristic - frequency, 562 - phase, 562 - spectral, 562 Chart - local, 161, 320 - parameter domain, 161 - range, 161 - range of, 320 Charts, consistent, 17 4, 328 Chebyshev metric, 3 Chebyshev polynomials, 525 Chebyshev-Laguerre polynomials, 525 Circulation of a field along a curve, 233, 277 Class - orientation, 328 Closed ball, 5 Closed form, 352 Closed set, 13 - in a metric space, 5, 6 - in a topological space, 13 Closure, 6, 13 Coefficient - of thermal conductivity, 303 - of thermal diffusivity, 304 Coefficients - Fourier, 506-508, 510, 513, 519, 526, 531, 536, 541, 551 , 553, 557
Subject Index - - extremal property, 506 -- Lame, 268, 275 Cohomology group, 356 Compact set, 15 - elementary, 159 -- in a metric space, 16 Companion trihedral, 73 Complete system of vectors, 51 1-516 Completeness of the trigonometric system, 545 Completion of a space, 24-27 Condition - necessary, for convergence, 374 Conditions - Dini, 535, 570 Connected set, 19 Consistent charts, 174, 328 Constant, cyclic, 297, 360 Content of a set (Jordan) , 1 19 Continuity - and passage to the limit, 383 - of an improper integral depending on a parameter, 423-426 - of an integral depending on a parameter, 408-409 Continuous group, 72, 336 Contribution - of a maximum point, 616 Contribution of a point to asymptotics, 615 Convergence - absolute, 375 - - of an improper integral, 420 - in mean, 528, 545 - of a family of functions - - pointwise, 363, 367 - - uniform, 367 - of a series of vectors, 505 - of an improper integral, 153 -- Cauchy principal value, 155 -- of distributions, 463 - of generalized functions, 463 - of linear functionals - - strong (norm) , 58 - of test functions, 463 - uniform, 397 - - Cauchy criterion, 369-370 - weak, 463
663
Convergence set, 363, 367 Convolution, 449-470 - differentiation, 453 - in lRn , 483 - multidimensional, 483-493 - symmetry, 453 - translation-invariance, 453 Coordinate parallelepiped, 107 Coordinates - Cartesian, 259, 261 - curvilinear, 161, 265 - cylindrical, 268-273 - of a tangent vector, 340 - polar, 166 - spherical, 166, 268-273 - triorthogonal, 268-273 Cotangent space to a manifold, 340 Covering - locally finite, 336 - refinement of another covering, 336 Criterion - Cauchy, 422 for uniform convergence, 369-370, 375 for uniform convergence of a series, 378 for uniform convergence of an integral, 418-420 - Darboux, 1 14-1 16, 1 29 - for a field to have a potential, 290 - for compactness in a metric space, 17 - for continuity of a mapping, 31 - Lebesgue, 1 1 1-114, 1 17, 1 19, 138, 145 Critical point, 149 Cube - singular, 357 - - boundary of, 357 Curl, 203, 260, 274 - physical interpretation, 281-282 Current function, 310 Curvature of a curve, 73 Cycle - boundary, 358 Carnot, 226 of dimension p, 358
664
Subject Index
Cycles, homologous, 358 Cyclic constant, 297, 360 Cyclic frequency, 560 Cylinder, 167 D
Darboux integral - lower, 1 14 - upper, 1 14 Darboux sum - lower, 1 14, 396 - upper, 1 14 Deformation (of a closed path) , 293 Degree (order) of a differential form, 195 Delta function (8-function) , 281 , 449, 455, 461 , 464, 469, 473, 485, 496, 559, 591 , 593 - shifted, 485 Derivation - of a ring, 352 Derivative - Lie, 352 - of a mapping, 61, 62 - of order n, 80 - partial, 7Q-71 - second, 80, 83 - with respect to a vector, 81 Derivative mapping, 62 Deviation - mean-square, 529 Diffeomorphism, elementary, 140 Differential - exterior, 342, 349 - exterior, of a form, 201 , 342, 349 - of a mapping, 61 - of order n, 80 - partial, 70, 71 - second, 80, 83 - total, 71 Differential equation with variables separable, 226 Differential form, 197 closed, 296, 352 - exact, 296, 352 - flux, 199 - of class c ( k ) ' 342 - of compact support , 344 -
- of order zero, 201 - on a manifold, 340 - on a smooth surface, 207 - restriction to a submanifold, 350 - work, 198 Differential operator, 486 - adjoint, 486 - self-adjoint, 487 - transpose, 486 Differentiation, 61 - at a point of a manifold, 347 - of a family of functions depending on a parameter, 388-393 - of a Fourier series, 544 - of a generalized function, 465-468 - of a power series, 391 - of a series, 391 - of an integral - - over a liquid volume, 497 - of an integral depending on a parameter, 409-412, 483 - on a manifold, 347, 351 - with respect to a parameter, 426-428 Dimension of a manifold, 320 Dini conditions, 535 Dipole, 299, 496 Dipole moment, 299, 496 Dipole potential, 299 Direct product of metric spaces, 7 Direction of motion along a curve, 176 Dirichlet discontinuous factor, 436 Dirichlet integral, 436, 447, 570, 610 Dirichlet kernel, 532, 557 Discontinuous factor, Dirichlet, 436 Discrete group of transformations, 335 Discrete metric, 2, 8 Disk, n-dimensional, 179, 322 Distribution, 460 - regular, 464 - singular, 464 - tempered, 591 Divergence, 203, 260, 274 - physical interpretation, 278-281 Domain - elementary, 240
Subject Index - fundamental, of a group of automorphisms, 324, 336 - of parameters of a chart, 320 - parameter, 366 - simply connected, 293 Double layer, 496
E Efficiency of a heat engine, 225 Eigenvalue, 518 - of a Sturm-Liouville problem, 518 Eigenvector of an operator, 519 Element of volume, 227 Elementary diffeomorphism, 140 Elliptic integral, 393 - complete - - of first kind, 393, 4 1 1 - - o f second kind, 393, 4 1 1 - modulus, 4 1 1 Embedding - canonical, 349 Entropy, 302 Envelope of a family of curves, 252 Equality - asymptotic, 598 - Parseval's, 520, 529, 547, 569, 592 Equation - Bessel's, 392, 410, 414, 434 - differential, 34, 226 - Euler's - - hydrodynamic, 307 - Euler-Lagrange, 91 - heat, 302, 583 - hypergeometric, 393 - Laplace's, 304, 524 - Mayer's, 224 - of an adiabatic, 224 - of continuity, 304-306 - of state, 222 - Poisson's, 298, 304, 493 - wave, 307-310, 581 - - homogeneous, 309 - - inhomogeneous, 309 Equations - Cauchy-Riemann, 310 - electrostatic, 286 - magnetostatic, 286
665
- Maxwell, 262, 263, 275, 281 , 298, 311 Equicontinuity, 397 Equivalence, asymptotic, 598 Equivalent atlases - with respect to orientation, 175, 328 - with respect to smoothness, 325 Erdelyi's lemma, 635 Error function - asymptotics, 628 Estimate, asymptotic, 597 uniform, 6 1 1 Euler's formula, 446 Euler's hydrodynamic equation, 307 Euler-Gauss formula, 440 Euler-Lagrange equation, 91 Euler-Poisson integral, 433, 442, 567, 580, 610 Eulerian integral, 437-448 Exact form, 352 Exhaustion of a set, 150 Expansion, asymptotic - in the sense of Erdelyi, 610 - uniform, 61 1 - uniqueness, 602 Exponential - of an operator, 71-73 Exponential integral, 600 Exponential system, 502 Exterior algebra, 319 Exterior differential, 201 , 342, 349 of a form, 342, 349 Exterior point, 6, 13 Exterior product , 195, 315, 342 Extremum of a function - necessary condition, 87 - sufficient condition, 87 - with constraint, 106
-
-
F Family of functions, 366 - equicontinuous, 397, 398 - - at a point, 403-404 - separating points, 401 - totally bounded, 397 - uniformly bounded, 397 Fejer kernel, 557 Field
666
Subject Index
- of forms, 257 - of linear forms, 197 - potential, 288 - scalar, 257 - solenoidal, 295 - tensor, 257 - vector, 257, 348 - - smooth, 348 Filter, low-frequency, 563 Flow - planar, 310 - plane-parallel, 310 Flux across a surface, 213-2 18, 233, 278, 302, 496 Force - mass, 306 Form - anti-symmetric, 195 - differential closed, 296, 352 exact, 296, 352 flux, 199 of class c k ) ' 342 of compact support, 344 on a manifold, 340 restriction to a submanifold, 350 work, 198 - Hermitian, 45 - - nondegenerate, 45 - - nonnegative, 45 - on a surface in lRn , 227 - semidefinite, 87 - skew-symmetric, 195, 314-316 - volume in lR k , 227 Formula - asymptotic, 598 - Cauchy-Hadamard, 375 - co-area, 236 - complement, for the gamma function, 441 Euler's, 446 - Euler-Gauss, 440 - for change of variable in an integral, 136 - Fourier inversion, 571, 592 - Frenet, 73 - Gauss' , 447 -
- Gauss-Ostrogradskii, 242-245, 278, 303, 489, 496 - - in vector analysis, 278 - Green's, 496 - homotopy, 360 - Kotel'nikov's, 586 - Kronrod-Federer, 236 - Legendre's, 446 - Leibniz ', 409 - Newton-Leibniz, 237, 278, 408, 573 - Poisson summation, 593 - reduction - - for the beta function, 438 - - for the gamma function, 440 - Stirling 's, 448, 622 - Stokes' , 237, 276, 278, 345-347, 358 general, 248-251 , 345 - - in JR3 , 245-247 - - in vector analysis, 277 - Taylor's, 94, 415 - Wallis', 434, 448, 636 Formulas - Borel's, 581 - differential, of field theory, 263-265 - Green's, 284, 285 - integral, of vector analysis, 278 Fourier coefficients - extremal property, 506 Fourier cosine transform, 564 Fourier integral, 634 - asymptotics, 635, 639 - multiple, 641 Fourier inversion formula, 571, 592 Fourier series, 499-526 - in a general orthogonal system, 499-500 - multiple, 557 - of generalized functions, 558 - partial sum - - integral representation, 531 - pointwise convergence, 527 rate of convergence and smoothness, 540 Fourier sine transform, 564
Subject Index Fourier transform, 561 , 563, 566, 571, 574, 579, 583 - asymptotic properties, 573-576 - frequency shift, 587 - in £ 2 , 589 - inverse, 583 - multidimensional, 577 - normalized, 568 - of a convolution, 569 - of generalized functions, 592 - rate of decrease and smoothness, 573 - time shift, 587 Frame - Frenet, 73 - orienting, 172 Frequencies, natural, 518 Frequency, 560 -- cyclic, 560 - fundamental, 560 - harmonic, 560 Frequency characteristic, 562 Frequency spectrum, 593 Fresnel integral, 436, 447, 610 Function - band-limited, 585 - Bessel, 392, 410 -- - asymptotics, 640 - beta, 437-439 - cardinal sine, 588 - change of coordinates, 321 - current, 310 - delta, 449, 455, 461 , 464, 469, 473 - Dirichlet, 364 - exponential integral, 600 - fundamental, 484 - gamma, 439-442 - - asymptotics, 622, 629 _:_ - incomplete, 610 - generalized, 460 - - differentiation, 465-468 of several variables, 484 - - regular, 464 - - singular, 464 - generating, 471 - Green 's, 470 - harmonic, 286, 304 - -- conjugage, 310
667
- Heaviside, 466, 469, 491 , 497 - limit, 363, 367 linear, 49 locally integrable, 452 multilinear, 49 of compact support, 136, 452 - phase, 634 piecewise continuous, 536 piecewise continuously differentiable, 536 - probability error, 610 -- asymptotics, 628 - rapidly decreasing, 575 - Riemann integrable - - over a set, 1 18 - - over an interval, 109 - sample, 588 sine integral, 610 spectrum of, 560 spherical, 524 support, 451 system, 449, 460 test, 463, 484 - transient pulse, 449 - uniformly continuous, 456 - unit step, 466 - zeta, 447 Functional - linear, 49 - multilinear, 49 Functions, asymptotically equal, 598 Functions, asymptotically equivalent, 598 Fundamental domain of a group of automorphisms, 324, 336 Fundamental frequency, 560 Fundamental sequence, 2 1 Fundamental solution, 469-470, 490 Fundamental tone, 518 G
Galilean transformation, 595 Gamma function, 439-442 - asymptotics, 622, 629 - incomplete, 610 Gauge condition, 3 1 1 Gauss' formula, 447 Gauss' theorem, 496
668
Subject Index
Gauss-Ostrogradskii formula, 237, 242-245, 489, 496 - in vector analysis, 278 Generalized function, 460 - differentiation, 465-468 - of several variables, 484 - regular, 464 - singular, 464 Generating function of a sequence, 471 Gibbs' phenomenon, 544, 556 Graded algebra, 314 Gradient, 203, 260, 274, 282 - physical interpretation, 283 Gram matrix, 185 Grassmann algebra, 319 Green's formula, 496 Green's function, 470 Green's theorem, 237-242 Group - cohomology, 356 continuous, 72, 336 - discrete, of transformations, 335 homology, 298, 356, 358 - - p-dimensional, 358 homotopy, 298 Lie, 72, 336 of automorphisms, 324 one-parameter, 350 - topological, 72, 336 H
Haar system, 525 Hamilton operator (nabla) , 262, 265 Harmonic analysis, 560 Harmonic frequency, 560 Harmonic function, 286, 304 - conjugate, 310 Harmonic polynomials, 524 Hausdorff space, 12, 14 Heat capacity, 223 - molecular, 223 Heat engine, 224 Heat equation, 302, 583 Heaviside function, 469, 491, 497 Hermite polynomials, 524 Hermitian form, 45 - nondegenerate, 45 - nonnegative, 45
Hilbert's fifth problem, 336 Homeomorphism, 31 Homologous cycles, 358 Homology, 358 Homology group, 298, 356 Homotopic paths, 294 Homotopy, 293 Homotopy formula, 360 Homotopy group, 298 Homotopy identity, 352 Hypergeometric equation, 393 Hypergeometric series, 392 I
Identity - approximate, 455 - homotopy, 352 - Jacobi, 72 Improper integral, 151 - depending on a parameter, 407 - with variable singularity, 478-483 Improper integral depending on a parameter - Abel-Dirichlet test, 421 - Cauchy criterion, 422 - continuity, 423-426 - limiting passage, 423-426 - uniform convergence, 431 - Weierstrass' M-test, 431 Induced orientation on the boundary of a surface, 181 Inequality - Bessel ' s, 508-509 - - for trigonometric system, 534 - Brunn-Minkowski, 121 - Cauchy-Bunyakovskii, 46, 452, 505 - Clausius, 225 - Holder's, 126 - isoperimetric, 194, 550-551 - Minkowski 's, 44, 126, 494 - - generalized, 494 - Steklov's, 556 - triangle, 1 - Wirtinger's, 556 Inertia, 306 Inner product, 45-47, 351 - of a field and a form, 351 Instantaneous axis of rotation, 70
Subject Index Integral, 108 - Bernoulli, 309 - canonical - - asymptotics, 633 - Cauchy, 309 - Darboux - - lower, 1 14 - - upper, 1 14 - depending on a parameter - - continuity, 408-409 - - differentiation, 409-412 -- integration, 413 - depending on a paramter, 407-415 - Dirichlet, 286, 436, 447, 570, 610 - double, 109 - elliptic, 393 - - complete, of first kind, 393, 41 1 , 435 - - complete, of second kind, 393, 411 - Euler-Poisson, 433, 442, 567, 580, 610 - Eulerian, 437-448 - - first kind, 437 - - second kind, 437 - Fourier, 561 , 564, 567, 571 , 578, 585, 588, 634 - - asymptotics, 635, 639 - - multiple, 641 - Fresnel, 436, 447, 610 - Gauss' , 253 - improper - - differentiation with respect to a parameter, 426-428 - - integration with respect to a parameter, 429-434 iterated, 127-129 - Laplace, 572, 612, 634 - - asymptotics, 635, 638 - Lebesgue, 395 - line, 212 - multiple, 109 - - depending on a parameter, 476-497 - - with variable singularity, 481 - of a differential form - - over a surface, 215, 218 - of a form on a manifold, 343-345 -
669
- of a function over a surface, 227, 231 - over a chain, 358 - over a set, 1 17 - over a singular cube, 358 - Poisson, 459, 588 - Raabe's, 446 - Riemann, 395 - - over a set, 1 17 - - over an interval, 108 - surface - - of first kind, 231 - - of second kind, 232 - triple, 109 Integral metric, 4 Integral operator, 568 Integral representation of the partial sum of a Fourier series, 531 Integral transform, 568 Integration by parts in a multiple integral, 255 Integration of an integral depending on a parameter, 413 Integration with respect to a parameter, 429-434 Interchange - of differentiation and passage to the limit, 388-393 - of integrals, 127 - - improper, 429-434 - - proper, 127 - of integration and passage to the limit, 386-388 - of limiting passages, 381 - of summation and differentiation of a series, 391 Interior point , 6, 13 Interval in R n , 107 Isobar, 222 Isochore, 222 Isometry of metric spaces, 24 Isomorphism - of normed vector spaces, 60 - of smooth structures, 335 Isoperimetric inequality, 194, 550-551 Isotherm, 222 Iterated integral, 127-129
670
Subject Index
J
Jacobi identity, 72 Jacobian of a coordinate change - cylindrical coordinates, 269 general polar, 166 spherical coordinates, 269 triorthogonal coordinates, 269 Jordan measure, 1 19 K
Kernel - Dirichlet, 532, 557 - Fejer, 557 - Poisson, 472 Klein bottle, 169, 325 Kotel'nikov's formula, 586 L
Lagrange's theorem, 309 Laplace integral, 572, 612, 634 - asymptotics, 635, 638 Laplace transform, 612 Laplace's equation, 304, 524 Laplace's method, 612-615 - multidimensional, 633 Laplacian, 264, 265, 273, 493 Law - Ampere's, 234 - Archimedes', 245 - Biot-Savart , 235 - Coulomb's, 279 - Faraday's, 233 - Gauss', 286 - Newton's, 288, 306, 307 - normal distribution, 471 - of conservation of mass, 279 Layer - double, 496 - single, 493 Legendre polynomials, 504, 516, 623 Legendre's formula, 446 Lemma - Erdelyi's, 635, 640, 641 - exponential estimate, 615, 624, 637 - Hadamard's, 415 - Morse's, 149, 408, 633, 637, 641 - nested ball, 27
- on continuity of the inner product, 505 - on finite c:-grids, 17 - on orthogonal complement, 508 - Poincare's, 296 - Riemann-Lebesgue, 532, 570, 638 - Sard's, 149 - Watson's, 631 , 635 Lie algebra, 72, 348 Lie derivative, 352 Lie group, 72, 336 Limit, 2 1 , 28 - of a family of continuous functions, 383-386 - of a family of functions, 370 - of a mapping, 28 - of a sequence, 21 - of a sequence of functions, 363 Limit function, 363, 367 Limit point, 5, 7, 13 Limiting passage - interchange, 381 - under a differentiation sign, 388-393 - under an integral sign, 386-388 Linear transformation, 49 Local chart, 320 Local maximum, 87 Local minimum, 87 Localization principle - for a Fourier series, 532 - for a Laplace integral, 615 Locally integrable function, 452 Lorentz transformation, 595 Low-frequency filter, 563 M
M-test for convergence, 374 Manifold, 320-325 -- analytic, 325 - compact, 323 - connected, 323 - contractible, 353 embedded in lR n , 161 nonorientable, 328 - of class c< k ) , 325 - orientable, 328 - oriented, 328 - smooth, 325
Subject Index - topological, 325 - with boundary, 322 - without boundary, 322 Mapping - adjoint, 317 - bounded, 29 - continuous, 30-33 - - at a point, 30 - continuously differentiable, 75 contraction, 34 derivative, 62 - - of order n , 80 - differentiable at a point, 61-62 - differentiable on a set, 62 - homeomorphic, 31 - linear, 49 - multilinear, 49 - of class c ( k ) , 325 - partial derivative, 70 - smooth, 325 - tangent, 61, 339, 349 - ultimately bounded, 29 - uniformly continuous, 32 Mass force, 306 Maximum, local, 87 Mean convergence and completeness, 545, 547 Mean value over a period, 560 Mean-square deviation, 3, 529 Measure - of a set (Jordan) , 1 19 - of an interval, 110 Measure zero, 110-1 1 1 , 1 17 Method - asymptotic, 597 - Fourier, 517-520 - Laplace's, 612-615 - - multidimensional, 633 - of Lagrange multipliers, 106 - of least squares (Gauss') , 520 - of separating singularities (Krylov's) , 544 - of tangents (Newton's), 38 - of tangents, modified (Newton-Kantorovich) , 39 - separation of variables, 517-520 - stationary phase, 633, 635 - - multidimensional, 640 -
671
one-dimensional, 638 - Steklov's averaging, 520 Metric, 1 - Chebyshev, 3 - discrete, 2, 8 - integral, 4 - of mean-square deviation, 3, 4 - of uniform convergence, 4, 397, 399 - Riemannian, 267 Metric space, 1 - separable, 15 Metric spaces, direct product, 7 Minimum, local, 87 Mobius band, 168, 175, 180, 184, 327, 330 Modulus - of an elliptic integral, 4 1 1 Moment - dipole, 299, 496 - multipole, 299 - of a function, 404 Morse's lemma, 633, 637, 641 Multilinear transformation, 49 Multiple integral - depending on a parameter, 476-497 - improper, 151 - integration by parts, 255 - iterated, 127-1 29 - with variable singularity, 476, 478 Multiplication of generalized functions, 475 Multipole, 299 Multipole moment, 299 Multipole potential, 299 N
Nabla (Hamilton operator) , 262, 265 Natural frequencies, 518 Natural oscillations, 518 Natural parametrization, 73 Necessary condition for uniform convergence, 374 Neighborhood, 13 - in a metric space, 5, 6, 9 - in a topological space, 1 1 - of a germ of functions, 1 1 Newton-Leibniz formula, 573
672
Subject Index
Norm - in a vector space, 42, 589 - of a transformation, 52 - of a vector, 42, 589 Null-series, Men'shov's, 531 Numbers, Bernoulli, 635 - generating function, 635 0
One-parameter group, 350 Open set - in a metric space, 5 - in a topological space, 9 Operational calculus, 466 Operator - differential, 486 adjoint, 486 - - self-adjoint, 487 - - transpose, 486 - Hamilton (nabla) , 262, 265 - integral, 568 - Laplace, 264, 265, 273, 493 - nilpotent, 71 - of field theory, 260 - - in curvilinear coordinates, 265-274 - symmetric, 519 - translation, 449 - translation-invariant, 449 Orbit of a point, 324, 335 Order (degree) of a differential form, 195 Orientation - induced on the boundary of a manifold, 331 of a domain of space, 172 of a manifold, 328 of a surface, 171-177, 181 of the boundary of a surface, 181 opposite to a given orientation, 172 Orientation class, 328 - of atlases - - of a surface, 175 - of coordinate systems, 171, 172 - of frames, 171 Oriented space, 171 Orienting frame, 172 Orthogonal vectors, 499, 5 19
Orthogonality with a weight, 522 Orthogonalization, 503-504 Oscillation of a mapping, 30 - at a point, 32 Oscillations, natural, 518 Overtones, 518 p
Parallelepiped, coordinate, 107 Parameter domain, 161, 320, 366 Parameter set, 366 Parameters - Lame, 268, 275 Parseval's equality, 520, 529, 569, 592 Partition - locally finite, 189 - of an interval, 108 - - with distinguished points, 108 - of unity, 148, 331-333 - - k-smooth, 332 - - subordinate to a covering, 332 Paths - homotopic, 294 - tangent, 347 Period of an integral, 297 Period, over a cycle, 360 Phase, 560, 634 - stationary, 634 Phase characteristic, 562 Phase function, 634 Piecewise continuous function, 536 Piecewise continuously differentiable function, 536 Planar flow, 310 Plancherel's theorem, 590 Plane - projective, 326 Plane-parallel flow, 310 Point - boundary, 6 - - in a topological space, 13 - boundary (of a manifold) , 321 - boundary (of a surface) , 178 - critical, 149 - exterior, 6 - - in a topological space, 13 interior, 6, 13 - - in a topological space, 13 -·
Subject Index - limit, 5, 7, 13 - of a metric space, 1 Poisson bracket, 348 Poisson integral, 459, 588 Poisson kernel, 472 Poisson's equation, 298, 304, 493 Polar coordinates, 166-167 Polynomial - trigonometric, 526 Polynomials - Bernoulli, 555 - Chebyshev, 525 - Chebyshev-Laguerre, 525 - harmonic, 524 Hermite, 524 Legendre, 504, 516, 623 Positive direction of circuit, 176 Potential - dipole, 299 - multipole, 299 - of a field, 288 - quadrupole, 299 - scalar, 296 - single-layer, 493 - vector, 295, 296 - - of a magnetic field, 296 - velocity, 309 Potential field, 288 Power series, 375 - differentiation of, 391 Principal value (Cauchy) of an integral, 155 Principle - Cavalieri's, 132 - contraction mapping, 36 - d'Alembert 's, 306 - Dirichlet 's, 286 - fixed-point, 34, 38 - localization - - for a Fourier series, 532 - - for a Laplace integral, 615 - Picard-Banach, 34, 38 - stationary phase, 633, 634, 640 - uncertainty, 590 Probability error function, 610 - asymptotics, 628 Problem - asymptotic, 596
673
- brachistochrone, 92, 94 - curve of most rapid descent, 92 - Luzin 's, 531 - Riemann, 530 - shortest-time, 92 - Sturm-Liouville, 523 Process - adiabatic, 223 - quasi-static, 222 Product - exterior, 3 15, 342 - inner, 45-47, 351 - - of a field and a form, 351 -- of functions, 522 - of generalized functions, 4 75 - of manifolds, 320 - of topological spaces, 13 - tensor, 313 Projective line, 326 Projective plane, 326 - real, 326 Pulse - rectangular, 587 - triangular, 588 Q
Quadrupole, 299 Quadrupole potential, 299 Quantities of the same order, 598 R
Raabe's integral, 446 Rademacher system, 525 Range of a chart, 161, 320 Rapidly decreasing function, 575 Rectangular pulse, 587 Restriction of a form to a submanifold, 207, 350 Riemann sum, 108 Riemann-Lebesgue lemma, 532, 638 Riemannian metric, 267 Rule, Leibniz, 134, 409, 453 s
Sample function, 588 Sard's lemma, 149, 150 Sard's theorem, 236 Scalar potential, 296 Schwartz space, 592
674
Subject Index
Schwarz boot, 194 Separable metric space, 15 Separation of points by functions, 401 Separation of variables, 517-520 Sequence - asymptotic, 601 - Cauchy, 21 - convergent, 21 - - uniformly, 368 - convergent at a point, 363 - convergent on a set, 363 - fundamental, 21 - monotonic, of functions, 376 - nondecreasing, of functions, 376 - nonincreasing, of functions, 376 Series - asymptotic, 601 , 602 general, 601 in the sense of Erdelyi, 610 in the sense of Poincare, 602 power, 606-609 - continuity of sum, 384 - Dirichlet, 380 - Fourier, 499-526 in a general orthogonal system, 499-500 multiple, 557 of generalized functions, 558 partial sum, 531 pointwise convergence, 527 rate of convergence and smoothness, 540 - hypergeometric, 392 - of functions, 366 - power, 375 - Stirling ' s, 636 - trigonometric, 526-560 - uniformly convergent - - Cauchy criterion, 378 Set - admissible, 1 1 7 - bicompact, 15 - Cantor, 23 - closed - - in a metric space, 5, 6 - - in a topological space, 13 - compact, 15
-
conditionally compact, 18 everywhere dense, 12 Jordan measurable, 1 19 nowhere dense, 28 of area zero, 190 of content zero, 120 of convergence, 363, 367 of first category, 28 of measure zero (Jordan) , 120 of measure zero (Lebesgue) , 1 17, 120, 121 - of second category, 28 - of volume zero, 120 - open - - in a metric space, 5 - - in a topological space, 9 -- parameter, 366 - relatively compact, 18 - totally bounded, 18 Signal - band-limited, 585 - spectrum of, 560 Simply connected domain, 293 Sine integral, 610 Sine, cardinal, 588 Single layer, 482 Single-layer potential, 493 Singular cube, 357 - boundary of, 357 Smooth structure, 326 Smooth structures, isomorphic, 335 Solenoidal field, 295 Solution - fundamental, 469-470, 490 - - of the Laplacian, 490 Space - Banach, 43 - complete, 21 - connected, 19 - cotangent to a manifold, :340 - Euclidean, 4 7 - Hausdorff, 12, 14 - Hermitian (unitary) , 47 - Hilbert, 47 - locally compact, 18 - locally connected, 20 - metric, 1 - - separable, 15
Subject Index - normed affine, 62 - normed vector, 42 - - complete, 43 - of distributions, 463 of fundamental functions, 484 of generalized functions, 463, 484 of tempered distributions, 591 of test functions, 463, 484 path connected, 20, 34 pre-Hilbert, 47 Schwartz, 591 , 592 Sobolev-Schwartz, 464 - tangent, 61 - tangent to JR n , 337 - tangent to a manifold, 337, 338 - topological, 9, 10 in the strong sense, 14 - - Tl , 14 - - T2 , 14 Spectral characteristic, 562 Spectrum - bounded, 585 - discrete, 560 - frequency, 593 Sphere, Alexander horned, 162 Spherical functions, 524 Stabilizer of a point, 335 Stationary phase method, 633, 635 - multidimensional, 640 - one-dimensional, 638 Stationary phase principle, 634 Steklov's inequality, 556 Stirling's formula, 448, 622 Stirling's series, 636 Stokes' formula, 237, 345-347, 358 - general, 248-251 , 345 -- in JR 3 , 245-247 Structure - smooth, 326 Sturm-Liouville problem, 523 Submanifold, 335 Subset, everywhere dense in c([a, bJ) , 400 Subspace - of a metric space, 7 - of a topological space, 13 Sum - Darboux
675
- - lower, 1 1 4 - - upper, 1 14 - Riemann, 108 Summation method - Abel, 384-385, 394 - Cesaro, 394 Support - of a differential form, 344 - of a function, 136, 451 Surface - elementary, 162 - k-dimensional, 161 - - smooth, 1 73 - nonorientable, 1 74 - of dimension k, 178 - of measure (area) zero, 190 - one-sided, 176 - orientable, 1 75 - oriented, 175 - piecewise smooth, 183 - - orientable, 184 two--sided, 176 - with boundary, 178 - without boundary, 1 78 - zero-dimensional, 183 Surface integral - of first kind, 231 - of second kind, 232 Symmetric operator, 519 System - exponential, 502 - Haar, 525 - Rademacher, 525 - trigonometric, 501 - - completeness, 545 - - in complex notation, 502 System function, 449, 460 System of sets - locally finite, 336 - refinement of another system, 336 System of vectors - complete, 51 1-516 - condition for completeness, 512 - linearly independent, 500 - orthogonal, 499-506 - orthonormal, 500 - orthonormalized, 500
676 T
Subject Index
Tangent mapping, 339, 349 Tangent paths, 347 Tangent space - to JR. n at a point, 337 - to a manifold, 337, 338 Tangent vector to a manifold, 338, 339, 349 Tempered distribution, 591 Tensor product, 313 Test - Abel-Dirichlet , 376 - Weierstrass', 374-376 - - for integrals, 425 Test function, 463 Theorem - Abel's, 379 - Arzela-Ascoli, 397-399 - Brouwer - - fixed-point , 242 - - invariance of domain, 321 Carnot 's, 225 - Cauchy's multidimensional mean-value, 287 - curl, 283 - Darboux', 1 15 - de Rham's, 359 - Dini's, 386, 430 divergence, 283 Earnshaw's, 286 Fejer's, 538, 557 finite-increment, 73-79 - Fubini's, 127-129, 580 Gauss', 285, 496 gradient , 283 - Green's, 237-242 - Hardy's, 395 Helmholtz', 302 implicit function, 96-106 inverse function, 105 - Kotel'nikov's, 585 - Lagrange's, 309 - Lebesgue's - - dominated convergence, 395 - - monotone convergence, 396 - mean-value for harmonic functions, 287 - - for the integral, 125
- on asymptotics of a Laplace integral, 621 - on existence of solutions of differential equations, 36-37 - on interchange of limiting passages, 382 - on the principal term of asymptotics of an integral, 621 - on uniform continuity, 32 - Plancherel 's, 590 - Poincare's, 353, 356 - Pythagorean, 505 - - for measures of arbitrary dimension, 193 - sampling, 585 - Sard's, 236 - Stone's, 40Q--403 - - for complex algebras, 405 - Tauberian, 395 - translation, 587 - Weierstrass' on approximation by polynomials, 400, 472, 494 on approximation by trigonometric polynomials, 484, 540 - Whitney 's, 169, 334 Whittaker-Shannon, 585 Timbre, 518 Tone, fundamental, 518 Topological group, 72, 336 Topological space, 9, 10 - base of, 10 - in the strong sense, 14 - weight of, 1 1 Topological spaces, product of, 13 Topology - base of, 10 - induced on a subspace, 13 - on a set, 9 - stronger, 14 Torsion of a curve, 73 Torus, 167 Total boundedness, 397 Transform - Fourier asymptotic properties, 573-576 - frequency shift, 587
-
Subject Index - - in L2 , 589 multidimensional, 577 of a convolution, 569 of generalized functions, 592 rate of decrease and smoothness, 573 time shift, 587 - Fourier cosine, 564 - Fourier sine, 564 - integral, 568 - Laplace, 612 Transformation - Abel's, 376 - bilinear, 49 - bounded, 53 - continuous - - multilinear, 56 - Galilean, 595 - linear, 49 - Lorentz, 595 - multilinear, 49, 52, 56 - trilinear, 49 Transient pulse function, 449 Translation operator, 449 Translation theorem, 587 Translation-invariant operator, 449 Triangle inequality, 1 Triangular pulse, 588 Trigonometric polynomial, 526 Trigonometric series, 526-560 Trigonometric system, 501 - completeness, 545 - in complex notation, 502 Trihedral, companion, 73 Triorthorgonal coordinates, 268 u
Uncertainty principle, 590 Uniform boundedness, 371, 397 - of a family of functions, 376 Uniform convergence, 397 - Cauchy criterion, 369-370
677
Uniformly convergent series - Cauchy criterion, 378 Unit step, 466 v
Value (Cauchy principal) of an improper integral, 155 Vector - angular velocity, 70 - tangent to a manifold, 338, 339, 349 Vector field - central, 213 - on a manifold, 348 - - smooth, 348 Vector potential, 295, 296 of a magnetic field, 296 Vector space, tangent to a manifold, 337, 338 Vectors - orthogonal, 499, 519 Volume - of a ball in R n , 192, 444 - of a set (Jordan), 1 19, 245 - of an interval, 107 Volume element, 227 -
w
Wallis' formula, 434, 448, 636 Watson's lemma, 63 1 , 635 Wave equation, 307-310, 581 - homogeneous, 309 - inhomogeneous, 309 Weak convergence, 463 Weierstrass M-test, 374 Weight of a topological space, 11 Wirtinger's inequality, 556 Work of a field, 2 1 1 z
Zeta function, 447
Name Index
A Abel, N. , 376, 377, 379, 384, 42 1 Alexander, J., 162 Ampere, A., 234 Archimedes ( ' ApXLJlifJOTJ') , 245 Arzela, C., 398, 404 Ascoli, G . , 398, 404 B
Banach, S., 34 Bernoulli, D., 309 Bernoulli, Jacob, 555, 635 Bernoulli, Johann, 92 Bessel, F. , 392, 410, 414, 434, 508 Biot , .J . , 235 Borel, E. , 581 Brouwer, 1 . , 178, 242 , 321 Brunn, H., 121 Bunyakovskii, V. , 46, 452 , 505 c
Cantor, G., 23, 32, 1 1 1 Carleson, 1 . , 531 Carnot, S., 226 Cart an, E. , 250 Cauchy, A. , 2 1 , 29, 46, 155, 287, 309, 310, 369, 375, 378, 422, 452, 505 Cavalieri, B., 132 Cesaro, E., 394 Chebyshev, P. , 3, 525 Clapeyron, E., 222 Clausius, R. , 225 Coulomb, Ch. , 279, 288 D
d'Alembert , J., 306 Darboux, G., 1 14- 1 16, 120, 129, 396
de Rham, G . , 359 Dini, U., 386, 430, 535, 570 Dirac, P. , 237, 281 , 464 Dirichlet , P. , 286, 364, 377, 379, 380, 42 1 , 436, 447, 520, 532, 557, 570, 610
E Earnshaw, S . , 286 Erdelyi, A., 610, 635 Euler, 1 . , 91, 307, 433, 437, 440, 442, 446, 530, 567, 580, 610 F Faraday, M . , 233, 237 Federer, H., 236 Fejer, 1 . , 538 Feynman, R. , 262 Fourier, J., 499-593, 634, 635 Frechet , M . , 1 1 Frenet , J . , 73 Fresnel, A., 436, 447, 610 Fubini, G., 127, 580 G
Galilei, Galileo, 595 Gauss, C . , 237, 242 , 249, 278, 285, 392, 440, 447, 471, 489, 496 Gibbs, J . , 544, 556 Gram, J . , 185, 503, 5 1 1 Grassmann, H . , 319 Green, G . , 237, 284, 470, 496 H
Haar, A., 525 Hadamard, J. , 375, 415 Hamilton, W. , 262 , 265 Hardy, G . , 395 Hausdorff, F., 1 1 , 13
680
Name Index
Heaviside, 0 . , 466, 469, 491 , 497 Heisenberg, W., 573 Helmholtz, H., 302 Hermite, Ch. , 45, 524 Hilbert, D., 47, 336 Holder, 0 . , 126 Hurwitz, A., 550 J
Jacobi, C., 72 Jordan, C . , 1 19-12 1 , 126, 130-132, 137, 144 Joule, G., 224 K
Kantorovich, L., 39 Kelvin, Lord (W. Thomson ) , 224, 237, 250 Klein, F . , 169, 325, 550 Kolmogorov, A., 530 Kotel'nikov, V., 585 Kronrod, A., 236 Krylov, A . , 544 L
Lagrange, J . , 39, 91, 106, 287 Laguerre, E., 525 Lame, G . , 268 Laplace, P. , 264, 265, 273, 304, 310, 493, 524, 572, 612, 634, 635, 638 Lebesgue, H., 23, 1 1 1-1 14, 1 17, 120, 1 2 1 , 125, 129, 136, 145, 146, 149, 395, 532, 570, 638 Legendre, A., 414, 446, 504, 516, 623 Leibniz, G., 134, 237, 278, 408, 453, 573 Lie, S . , 72, 336, 348 Liouville, J., 523 Lorentz, H., 595 Luzin, N., 531 Lyapunov, A., 520 M
Maxwell, J . , 237, 262, 263, 275, 281, 286, 298, 311 Mayer, J., 224 Men'shov, D., 531
Milnor, J . , 335 Minkowski, H., 121, 126, 194, 494 Mobius, A., 168, 327, 330 Morse, A., 149 Morse, M., 633, 637, 641 N
Newton, I., 38, 237, 278, 288, 306, 307, 408, 573 0
Ostrogradskii, M., 237, 242, 249, 278, 286, 489, 496 p
Parseval, M., 520, 529, 547, 569, 592 Picard, E . , 34, 38 Plancherel, M., 590 Poincare, H., 250, 296, 353, 356, 602, 610 Poisson, S., 224, 298, 304, 348, 433, 442, 459, 472, 493, 567, 580, 588, 593, 610 Pythagoras (IIv8o:'"'(6po:c;) , 193, 505 R
Raabe, J . , 446 Rademacher, H., 525 Riemann, B., 23, 107-109, 310, 395, 447, 530, 532, 638 s
Sard, A., 149, 236 Savart, F., 235 Schmidt, E., 503 Schwartz, L., 464, 591 , 592 Schwarz, H., 193 Shannon, C., 585 Sobolev, S . , 464 Steklov, V., 520, 556 Stirling, J., 448, 622, 636 Stokes, G., 237, 249, 345-347, 358 Stone, M., 400 Sturm, Ch. , 523 T
Tauber, A., 395
Name Index Taylor, B., 94, 415, 517 Thomson, W. (Lord Kelvin) , 224, 237, 250 w
Wallis, J . , 434, 448, 636
681
Watson, J., 631 , 635 Weierstrass, K., 374, 393, 400, 425, 431 , 472, 484, 494, 540 Whitney, H., 149, 169, 334 Whittaker, J . , 585 Wirtinger, W., 556
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Vladimir A. Zorich is professor of mathematics at Moscow State University. His areas of specialization are analysis, conformal geometry, quasiconformal mappings, and, most recently, mathe matical aspects of thermodynamics. He solved the problem of global homeomorphism for space quasiconformal mappings, for whicH he was awarded the National Prize for Young Mathe maticians. He also holds a patent in the technology of mechanical engineering. Vladimir A. Zorich was one of the organizers of advanced experi mental courses at the Department of Mechanics and Mathematics of Moscow State University. He has designed and taught modern analysis courses in several universities and is the author of numerous research mathematical publications.
Mathematica l Ana lysis I I This two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, integral transforms, and distributions. Especially notable in the course is the clearly expressed orientation toward the natural sciences and its informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of in structive exercises, problems and fresh applications to areas seldom touched on in real analysis books. This second volume expounds classical analysis as it is today, as a part of uni fied mathematics, and its interactions with other modern mathematical courses such as algebra, differential geometry, differential equations, complex and func tional analysis. The book provides a firm foundation for advanced work in any of these directions. " . . . The textbook of Zorich seems to me the most successful of the available comprehensive textbooks of analysis for mathematicians and physicists. It differs from the traditional expo sition in two major ways: on the one hand in its closer relation to natural-science applications (primarily to physics and mechanics) and on the other hand in a greater-than-usual use of the ideas and methods of modern mathematics, that is, algebra, geometry, and topology. The course is unusually rich in ideas and shows clearly the power of the ideas and methods of modern mathematics in the study of particular problems. Especially unusual is the second volume, which includes vector analysis, the theory of differential forms on manifolds, an introduction to the theory of generalized functions and potential theory, Fourier series and the Fourier transform, and the elements of the theory of asymptotic expansions. At present such a way of structuring the course must be considered innovative. It was normal in the time of Goursat, but the tendency toward specialized courses, noticeable over the past half century, has emasculated the course of analysis, almost reducing it to mere logical justifications. The need to return to more substantive courses of analysis now seems obvious, especially in connection with the applied character of the future activity of the majority of students. . . . In my opinion, this course is the best of the existing modern courses of analysis." From a review by Vladimir 1. Arnold
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