Handbook of vector and polyadic analysis

Handbook of

VECTOR and POLYADIC ANALYSIS

THOMAS B. DREW Professor of Chemical Engineering Columbia University New York, N. Y.

REINHOLD YUl3LISHlIl6:-47KYOKA"1 ItliN

NEW YORK, N. Y. CHAPMAN & HALL, LTD., LONDON

Copyright Q 1961 by

REINHOLD PUBLISHING CORP. All rights reserved

Library of Congress Catalog Card Number: 61-18276

Printed in the United States of America

Preface This book results from occasional sets of notes prepared over the past twenty years for the use of engineering graduate students in fluid mechanics, heat transmission, and the theory of diffusional processes. The primary service of mathematical analysis in teaching such subjects is, in my view, linpial rather than computational. The use of tensor analysis in the form of Gibbsian polyadics clears away the fog of multiple scalar partial differential equations in which the student otherwise finds himself without requiring him to learn a secret code. the symbols and ideas are only slightly different from those met in undergraduate

calculus and the mathematical models mirror the physical models clearly. The apace given the several topics reflects my own teaching experience; there could be no pretense of completeness in a book of this length. The admission of complex vectors and polyadice has proved useful in developing the notion of function-vectors in Fourier analysis and in the treatment of integral equations. I wish to acknowledge my debt to H. B. Phillips who first led me to appreciate the value of vector analysis using electromagnetic theory as the vehicle. Thanks are due my daughter Sarah who typed much of this manuscript and the staff of the Waverly Press, Inc., for their help in devising printable symbols that suggest the intended meaning. Thomas B. Drew Temple, New Hampshire September, 1961

Contents 1. INTRODUCTION Object & Scheme Scalars vs. Vectors Notation

3

II. VECTOR ALGEBRA Definitions and Elementary Operations Scalar Product Vector Product

5 7 9

1

1

Position Vectors Linear Vector Functions of Vectors Dyadics, Tensors, and Linear Vector Operators Dyadic Representation of Linear Vector Operators Multiple Scalar Products: Higher Linear Operators Definitions Polydot Products of Polyads Unit Polyads Polyadics Transposes and Adjoints of Polyadics Idemfactors, Deviation Factors, Transposers, and related functions Linear Polyadic Functions of Polyadics Use of Linear Operators with Arguments of Various Ranks Polycross Products Versors: Rotation Operators Invariants of Vectors and Polyadics

III. VECTOR and POLYADIC CALCULUS Differentiation and Integration with respect to Scalars

10 11

14 17 21 21 21 22

24 27 31 45 52 57 58 64

Differentiation with respect to Vectors and Polyadics The Divergence and Gauss's Theorem The Circulation, the Curl, Stokes Theorem

69 69 74 75

Operations with V and dW

78

Mixed Derivative Operators Maclaurin's and Taylor's Series

82 83 vu

viii

CONTENTS

APPENDICES Appendix A: Summary of Formulas and Notation Appendix B : Ternary Numeration Appendix C : Comparison of Notation Appendix D: Miscellaneous Exercises References on Vector Analysis

I. Introduction 1.1 Object and Scheme

This book constitutes a brief exposition of the vector and tensor concepts and methods convenient in studying the theory of diffusion, fluid dynamics, and related topics. There is no intent to provide a text or treatise but rather to give the reader with a knowledge of calculus a handbook sufficiently interlineated to be intelligible to one familiar with the elementary vector usage in college physics and engineering mechanics. No prior formal study of vector or tensor analysis is assumed. Willard Gibbs' system of vector analysis is used. It has the advantage

for the uninitiated that many of the notations of scalar calculus carry over with analogous meanings into the calculus of vectors and tensors; moreover, the modifications required to include complex vectors (the "bi-vectors" of Gibbs) parallel those required in scalar analysis to admit complex scalars. Because diffusion theory involves tensors of rank higher than two, Gibbs' theory of dyadics has been extended to that of "polyadics": linear vector functions of variable vectors are thus extended to linear polyadic functions of polyadics. The presentation is largely in terms of three-dimensional real vectors but in general the formulae are so written that they are valid for complex vectors, and any changes necessary for application to N-dimensions are pointed out. In the first few sections, the geometric (or vector) method and the alge-

braic (or matrix) method are used in parallel so that the essential equivalence of vector and matrix analysis may be seen and, further, so that matrix interpretation may be available for numerical computation

when convenient. Polyadic analysis is tensor analysis restricted to "Cartesian tensors" but written in terms of the usual symbols of calculus rather than in a shorthand wholly new to the student. 1.2 Scalars vs. Vectors

The size of a physical quantity is commonly specified with reference to an agreed scale by a real number. For this reason properties such as mass, volume, and temperature, of which size on an agreed scale is the primary analytical characteristic, are called scalar properties or, more briefly, scalars. The numbers used to express size, and by extension all algebraic numbers, are themselves often called scalars. This is the usage 1

2

VECTOR AND POLYADIC ANALYSIS

herein. Some writers restrict the word scalar to mean what are here termed scalar invariants (v. infra). Certain physical quantities, force and velocity for example, require for their descriptions the specification of their directions as well as their

sizes; these are vector quantities. The fundamental idea of vector analysis is that the notions of magnitude and direction may be regarded as a composite property and represented jointly by a single letter symbol.

The dual entity is a vector. In the same sense in which the abstract number 3 belongs to all classes of threes so do all quantities of a given size and direction possess the same vector. The geometrical representation of a real three-dimensional vector as an arrow is familiar from elementary mechanics. The arrow flies with the vector and its length measures the magnitude. All parallel arrows of equal length, wherever located in space, are equally good representations of the same vector. The notion of position is not part of the vector concept. To state the numerical value of a vector, select any convenient system of non-coplanar rectilinear axes and consider the particular representative arrow which rises from the origin. The coordinates of the arrow's tip, stated in x, y, z order, fully determine the vector if the system of axes is known; when reference to the vector is intended they are called

the coordinates of the vector with respect to that system of axes. A system of rectilinear axes may be chosen in infinitely many orientations, hence there belong to each vector infinitely many ordered sets of three

numbers, each associated with a particular frame of reference. Algebraically speaking, it is usual to take the collection of such sets associated with right-handed rectangular Cartesian frames of reference to be the vector. Every set of this restricted collection has the property that the sum of the squares of its members is the same number: the square of the magnitude of the vector. An ordered set of quantities is a matrix;* algebraically speaking, therefore, a vector is a restricted colTypically, a matrix is a set ordered in two senses; one sense is represented by position in rows of a rectangular array; the other, by position in columns. Thus the matrix a,

b,

c,

a,

b,

c,

as

b,

cc,

a

b+

.

might be a tabulation in order of the coefficients of x, y, and z in the first, second, third, and fourth of a set of linear equations. This is a "4 X 3 matrix"; that

INTRODUCTION

3

lection of matrices. Thus, following Murnaghan ("Applied Mathematics", Wiley, 1948) :

A three-dimensional vector is a collection of 3 X 1 matrices such that (a) the sum of the squares of the absolute values of the elements (coordinates) is a given number, a scalar invariant of the collection, called the square of the magnitude of the vector, and (b) the corresponding coordinates of any pair of the matrices, say Y

and

11

z11

are related by a change of variables of the type

x' = 1,x + m,y + n,z y' = 12x + m2y + n2z

z' = lax + may + n3z where the determinant 1,

m,

l2

m2 ma

l3

nx

%=+1 n3

This definition is valid for vectors of any number of dimensions with real or complex coordinates; in N-dimensions, N X I matrices replace the 3 X 1 matrices and so on. 1.3

Notation

Vectors are denoted by symbols in bold face type; scalars, by letters in ordinary italics. When in the same discussion symbols are needed

both for a vector and for its magnitude (a scalar), the same letter is used when the typeface makes the interpretation clear. When necessary for clarity J A I is used to denote the magnitude of the vector A; there will be no confusion if I a I is used for the absolute value (or modu-

lus) of the scalar a. The common practice of indicating the complex conjugate of a complex number by an overscore is extended to complex vectors; thus, if, is, it contains four rows and three columns. If a set is ordered in only one sense as is true of the set of coordinates of a vector, it is simply a matrix of one column or, if we prefer, of one row.

4

VECTOR AND POLYADIC ANALYSIS

with x and y real, w = x + iy, then iV = x - iy whether w, x, y represent scalars, vectors or polyadics.

The coordinates of a vector A with respect to the set of rectangular axes x, y, z in use will often be denoted by A., A , A.. When a single letter is used to designate a matrix, it will be enclosed by double bars; thus, 11 A 11. A curled overscore will be used to indicate the transposed matrix. For example, if als 1

an

au an an

all

a21

a0i

an an

an

an an

all IIA11 a-

an

azs

an

then

11 1111

ass

that is, the columns of (I A II are the rows of II 111. Tensors and polyadics (cf. ¢2.5, 2.8) will be denoted by Gothic letters, e.g., P.

II. Vector Algebra 2.1

Definitions and Elementary Operations

The zero vector 0, and no other vector, has zero magnitude. All its coordinates are zero in every frame of reference. The magnitude of every vector other than the zero vector is a positive real scalar invariant, i.e., its value is independent of the frame of reference.

The negative - V of a vector V has the same magnitude V as does V but extends in the opposite direction; its coordinates in each reference frame are opposite in sign of those of V. Formally, addition and subtraction of vectors obey the laws of scalar algebra. The sum of a number of vectors

s=a+b+c

(2.1-1)

means the resultant of the vectors a, b, c in the same sense that the word is used in statics if the three vectors are the values of three forces acting at the same point. The force polygon of statics illustrates the geometrical construction of s from a, b, c. In each reference frame the coordinates of the sum are the algebraic sums of the corresponding coordinates, in that reference frame, of the parts. To subtract a from b reverse the sign of a and add.

Figure 2.1-1. The sum a of at b, and c.

Two vectors are equal when, and only when, their difference is the zero vector.

Multiplication and division of vectors by scalars also obey the laws of scalar algebra. The product, cV - Vc, of a vector V by a scalar c is a 5

VECTOR AND POLYADIC ANALYSIS

6

vector in the direction of V or in the opposite direction according as c is positive or negative, and in magnitude that of V multiplied by the absolute value of c: (2.1-2) CV 1 = jcj I V j In every reference frame the coordinates of c V are the algebraic products of those of V by c.

A vector of magnitude unity is called a unit vector. The symbols i, j, and k are customarily reserved for unit vectors in the directions of positive extension of the real x-, y-, and z-axes, respectively; n, for a unit vector normal (i.e., perpendicular) to a specified direction or surface. A unit vector in the direction of an arbitrary vector V may always be found by dividing V by its magnitude. Every vector may therefore be written as a unit vector with a scalar coefficient equal to the magnitude. In the complex field there is a multiplicity of unit vectors in any given direction; any unit vector remains a unit vector when multiplied by the complex scalar Exp (i¢), where ¢ is real, because for all real values of -0,

1 Exp (W 1 = 1.

If A, B, C are the coordinates of a 3-dimensional vector V with respect

to any set of non-coplanar rectilinear axes along which the respective unit vectors are a, Ia, c, then from the definition of coordinates (11.2) and of vector addition it is geometrically evident that

V=Aa+Bb+Cc

(2.1-3)

The set of vectors (Aa, Bb, Cc) of which the sum is V are called the components of V resolved along the given axes. In keeping with the customary usage of referring coordinates to rectangular axes, the unmodified term "components of V" is understood to mean the rectangular components found by the perpendicular projection of the representative arrow upon the set of mutually perpendicular axes in use. Hence, the `component of V in the direction of W" has come to mean the perpendicular projection of the arrow on a line parallel to W. When the unit vectors a, b, c are mutually perpendicular, the expression on the right of equation 2.1-3 is in essence merely an alternative way of writing the matrix A

B 11 C 11

The a, h, and c amount to mnemonic tags which serve both to identify the reference frame and to avoid the hazard of taking the coordinates in a wrong order.

VECTOR ALGEBRA

7

As is true of complex scalars, the conjugate V of a complex vector V differs from V only in that its imaginary part is oppositely signed; its magnitude is the same as that of V; and the conjugates of the coordinates of V are those of P in the conjugate reference frame.

Scalar Product The scalar or dot product of two real vectors a and b is the scalar

2.2

quantity

ab

(2.2-1)

cos B

where 0 is the angle between the two vectors. From the generalized Pythagorean theorem, an equivalent definition is

J[Ia+b12- IaI'- Ib1']

(2.2-2)

These formulas are valid for any number of dimensions; in fact the "angle" between two vectors in real N-dimensional space is, by definition,

MMCIria+b12-Ia12-Ib1'1 ll

2ab

(2.2-3)

If a and b are complex, a-b is in general a complex number. Its real part is

J[ I a + b Is - 15 I' - I b I2]

(2.2-4)

which reduces to 2.2-2 when the vectors are real. Its imaginary part is 2Z[ia+ibI2

- IaI' _ Ibll

(2.2-5)

Then

If either factor of a scalar product is zero, the product is zero. The inverse is, however, not necessarily true. The necessary and sufficient condition for the perpendicularity of two vectors is 0 or

0

one expression being equivalent to the other. Further, the dot-product of a complex vector by itself will vanish if the real and imaginary parts are perpendicular and of equal magnitude. In other respects the laws of scalar algebra apply to scalar products:

VECTOR AND POLYADIC ANALYSIS

8

a (b + c) =

(2.2-6b)

(ma)-b = a-(mb) -

(2.2-6e)

(2.2-6d)*

a I'

(2.2-6e)*

If the coordinates of a and b in the same real rectangular reference frame are respectively as , a,,, a. and b., b , b., substitution in the formula 2.2-2 or in 2.2-4 and 2.2-5, as appropriate, will show that

a-b = a,bx + ayb, + a,b.

(2.2-7)

which serves conveniently to compute the value of the scalar product when the coordinates are known. If the directions of the axes are complex,

in applying 2.2-7 the coordinates of one of the vectors are taken with respect to the conjugate system of axes. The operation in matrix algebra corresponding to the dot product of erectors is found by considering how the rules of matrix multiplication must be applied to obtain 2.2-7 when the vectors a and b are written as 3 X 1 matrices : ax

b,

a

Ila11 m

II b II =

b b.

as

By definition, the product II c 11 11 d I) of twc matrices cu

eat

ela

C21

Cr22

022

eal

ea2

Cas

I

and

do

du

dia

d2i

d22

d2a

dar

da

da

is

(Clldll + cud21 + ciadai)

(ciid,2 + cud22 + ciada2) ...

(endu + ca do + c sdai) (caidu + e8d2i + caadn)

(c2idi2 + c29d22 + csads2) . , . (catdi2 + ca2d22 + caada2) ...

that is: to obtain the element in the i'th row and j'th column of the product one multiplies the elements of the i'th 'row of the prefactor by the corresponding elements of the j'th column of the po8tfactor and adds For real vectors, these equations may be read without the overscores; a real number or real vector is identical with its conjugate.

VECTOR ALGEBRA

9

the products. If we regard the 3 X 1 matrices II a II and II b II as 3 X 3 matrices with the last two columns filled by zeros, trial will show that one must assign the following meaning to a-b when the frame of reference is real: i1611 IIbII =

(2.2-8)

IIa1I

116 11

if 2.2-7 is to be the result. For complex frames of reference, the coordinates of one of the vectors must be those in the conjugate frame as in the case of equation 2.2-7. From the rule for multiplication of matrices it will be seen that, in general II c II

11 d 11 3-6 11 d 11

II c 11

For a mutually perpendicular set of unit vectors, e.g. U, v, w: 1

i

(2.2-9)

0

The component of a vector V in the direction of a unit vector w is (w V) w. The quantity in parenthesis is the coordinate of V with respect to an axis in the direction of w. 2.3

Vector Product

The vector or cross product a x b of two real 3-dimensional vectors is a vector of magnitude ab sin 0, where 8 is the smaller of the angles be-

tween a and b. It extends, perpendicular to each of its factors, in the direction it would advance were it a right handed screw so turning that a

swings through the smaller angle 0 toward b. If the coordinates of a and b in the same reference frame are respectively (a= , a , a,) and (bx , b,, , b,), those of a x b are:

(arb, - a,b,,)

(a,b, - a:b,) (axb, - arb.) The following relations apply both to real and to complex vectors in three dimensions:

axb=-bxa ax (b+c)axb±axc

(2.3-1)

(2.3-2)

a x mb = ma x b = m(a x b)

(2.3-3)

axb -axb

a x (b x c) =

(a x b) x c =

(2.3-4) bc)

c (ab - ba)l

(2.3-5)

VECTOR AND POLYADIC ANALYSIS

10

A

a x b

b x a

Figure 2.3-1. The vector products of a and b.

Various special relations for complex vectors may be derived by writing them in the form r, + ire , where rl and r2 are real, and using the formulae of this paragraph. Notice that (1) changing the order of the factors in the vector product changes the sign of the result (Eq. 2.3-1) and (2) the associative law of multiplication does not apply to the vector product (Eq. 2.3-5) so that the parentheses must be retained in expressions like those on the left of equations 2.3-5.

For the real right-handed three dimensional set of mutually perpendicular unit vectors i, j, k:

i x i = j x j= k x k= Ol ixj =-j xi=k

J x k=- k x j= i kxi

2.4

(2.3-6)

i x k= j

Position Vectors

The position of a point P relative to an agreed origin 0 may be specified by stating the vector R of the arrow drawn from 0 to P. In vector analysis the "position vector" R plays much the same part as does the set of scalars (x, y, z) in scalar analysis. Any quantity, scalar or vector, which varies with the position in space of the point under consideration may be thought of as a function of R just as it may be thought of as a function of (x, y, z). Examples of such quantities are the local temperatures in a body conducting heat and the local velocities in a mass of fluid. When a quantity Q depends upon the value of a vector V, the

VECTOR ALGEBRA

11

vector may be imagined to issue from a fixed origin so that Q is visualized as a "function of position" and V as the position vector. Although real graphical representation is possible only in three dimensions, the ideas of the "position vector" and of a "function of position" are easily extended to N-dimensions.

Linear Vector Functions of Vectors Whatever be the natures of the independent variable X and of the dependent variable Y, if Y = f(X) is such a function that 2.5

f(A) + f(B) = f(A + B)

(2.5-1)

when A and B are any arbitrary pair of values of X within the range of definition of the function, then f(X) is a linear function of X. If Y is a vector, f(X) is a linear vector function of X. In this section, X will also be supposed a vector. Linear tensor functions of vectors and tensors are discussed in Section 2.8.

It is usual in vector analysis to follow Gibbs in writing Y = instead of Y = f (X) and to describe K as an "operator which operates on X to give Y." In Gibbs' notation equation 2.5-1 is written B)

(2.5-1a)

From equation 2.5-1a and vector algebra 0

(2.5-2)

and, if c is a scalar, (2.5-3)

Two linear vector operators K and L are equal if, and only if,

for every arbitrary V. An operator S which satisfies (2.5-4)

where K and L are linear vector functions and V is an arbitrary vector, is a linear vector fanction called the sum of K and L. An operator N which satisfies

(2.5-5)

function called the dot product of where V is arbitrary, is a linear K and L. The order of addition is immaterial in forming a sum but the

12

VECTOR AND POLYADIC ANALYSIS

order of the factors in a product generally affects its value; thus, in general,

0 If, and only if, for every arbitrary vector V

is, by definition, the zero function; it is written 0. When equation 2.5-6 is true for some values of V other than zero, but not for all, K is K

said to be singular. If 0 is a factor in a dot product of linear vector operators the product is zero, but its presence as a factor cannot be inferred if a product vanishes. A linear vector function K such that, for all values of Y, V

(2.5-6a)

is called an idemfactor; it is written 1. Dot multiplying an arbitrary linear vector function by I in either order reproduces the function: (2.5-8b) L 1L= Idemfactors of higher rank are discussed in Section 2.8-4.

The vectorial transpose 9 of any linear vector operator K is a linear vector operator such that (2.5-7)

for every arbitrary choice of the vectors U and V. The relation between K and 9 is reciprocal; either is the transpose of the other. Usually the expressions in equation 2.5-7 are written without parentheses: (2.5-7a)

Formally, this assigns a meaning to an expression like U- K in which the operator follows instead of preceding the argument U: U.K

(2.5-8) V

If we operate on the vector K. V with the operator L, the result is Thus or by 2.5-8, m

(2.8-9)

which shows that the vectorial transpose of a product of a number of linear vector operators is the product of their transposes in the reverse

VECTOR ALGEBRA

13

order. The transpose of the sum or differences of two operators is the sum or difference of their transposes. When R K, the operator is called vectorially symmetric; when

w -K, it is called vectorially antisymnletric. Every linear vector operator may be written as the sum of a symmetric part

K=J(K+R)

(2.5-10)

.and an antisymmetric, or skew, part

K=+}(K-R)

(2.5-11)

The idemfactor I is symmetric. Sums of symmetric linear vector operators are symmetric. The product in either order of any linear vector operator by its transpose is symmetric. In the complex domain, there is to be distinguished from the transpose of K another linear vector function, the adjoins of K, which will et,

be denoted by K. The adjoint of K is defined by the relation in

- of

(2.5-12)

or, equivalently,

U.K an K-U

(2.5-12a) N

where U and V are arbitrary vectors. The relation between K and K is reciprocal. If K and its adjoint are identical so that U is arbitrary, K is said to be vectorially Hermitean, self-adjoint, or self-conjugate. A linear vector operator is Hermitean when, V is real, and thus equal to its conjugate, where V and only when, is an arbitrary vector, real or complex. The sum of any number of Hermitean linear vector operators is Hermitean. The product in either order of any linear vector operator by its adjoint is Hermitean. In the domain of reals the adjoint and the transpose are the same; a real Hermitean linear vector operator is symmetric. Linear vector operators that commute with their adjoints in dot products are said to be normal. All Hermitean linear vector operators

are normal.

For every linear vector operator K other than zero, there exists at least one characteristic unit vector v such that av

(2.5-13)

VECTOR AND POLYADIC ANALYSIS

14

where A is a scalar, associated with K and v, called a characteristic number or eigenvalue of K. From equation 2.5-3 all scalar multiples of v satisfy 2.5-13. The formula A

(2.5-14)

gives the characteristic number of K associated with the characteristic unit vector v. If K is Hermitean, the expression on the left of equation 2.5-14 is real; in consequence all characteristic numbers of Hermitean linear vector operators are real. Every normal linear vector operator has

a set of mutually perpendicular characteristic unit vectors equal in number to the number of dimensions.

Dyadics, Tensors, and Linear Vector Operators Considered as mathematical entities, linear vector operators share

2.6

with vectors the property of being independent of any specific reference

frame although they possess representations in each of the infinitely many frames appropriate for representations of vectors (cf. §1.2). By reason of this feature they prove useful as mathematical measures of certain kinds of physical quantities which, like the vector quantities measured by vectors, are visualized as independent of the particular frame of reference that an observer may have set up. Vectors and linear vector operators are members of a class of mathematical entities called tensors; vectors are tensors of rank one, linear vector operators, of rank two. Tensors of higher rank are considered in Section 2.8. Tensor quantities more complex than vectors arise naturally in flow problems. Suppose J is the local mass velocity of a fluid of mass density -y and let n be the unit normal to a stationary plane element of area 58

within the fluid. The time rate of flow of mass (a scalar) across bS is then (2.6-1)

Now the momentum (a vector) of each unit mass is J/gy if the mass velocity is J. Consequently the time rate at which momentum is carried across SS by the motion of the fluid is the vector (2.6-2)

From one point of view, the right hand expression is merely a condensed

style of writing that on the left; no confusion can arise if it is agreed that the two vectors immediately adjacent to the dot are to be read as combined in a scalar product. On the other hand, if 2.6-2 is compared

VECTOR ALGEBRA

15

with 2.6-1, it appears that JJ/g-y enters the former in the same way is the vector expression for the magnitude that J enters the latter: of the component parallel to n (and so perpendicular to SS) of the is the expression for the "vector mass current per unit area, magnitude and direction" of the component parallel to n of the total momentum current per unit area. Physically JJ/gy represents the flux density of momentum within the fluid in the same sense that J represents the flux density of mass. An expression such as JJ is called a dyad. Formally, as used in equa-

tion 2.6-2, JJ is a linear vector operator because when dot multiplied by a vector n, the result is a new vector and if n were a sum of vectors, equation 2.5-1 would be satisfied. It is a singular operator (cf. §2.5) because n, being a unit vector, is never zero yet whenever n is perpendicular to J the result 2.6-2 is zero. Further, in accord with our agreement (below 2.6-2) as to the interpretation of dot products of dyads and vectors,

JJ is a symmetric operator. It appears, therefore, that flux densities of vector quantities are instances of tensor quantities of the type representable by linear vector operators of at least one kind, namely, dyads.

The following laws of direct multiplication apply in forming dyads. They may be verified by showing that dot multiplication of the right and left hand members of each equation by an arbitrary vector leads to an equality.

(a + b)(c + d) = ac + ad + be + bd mab = (ma)b = a(mb)

ab = 00 (T,) b = ba

(2.6-3) (2.6-4) (2.6-5) (2.6-6) (2.6-7)

Note that ab ;-6 ba unless a and b are parallel. A sum of dyads with or without scalar coefficients is termed a dyadic. In matrix form, if the coordinates with respect to a given reference frame of a and b are ax , a, , as and b., , b, , b. respectively, the dyad ah may be interpreted as the matrix

16

VECTOR AND POLYAD.IC ANALYSIS aab=

a=by

a.b=

a,b,

aba

I

II ab II

a fl lI b II

(2.6_8)*

I

if the operation ab II li c II. In this form the operation c,ab must be replaced by its equal etc. are called the coordinates ba 11 c II. The nine numbers a=b, , of the dyad in the given reference frame. Just as in the case of vectors 11

(cf. §1.2), there are infinitely many sets of coordinates (matrices) belonging to the same dyad, each set associated with a particular set of axes. The law for transforming the coordinates of a dyad from one set

of admissible axes to another may be readily deduced from that applicable to vectors. Thus, given the transformation T for vectors on page 3, which is written in matrix algebra as

II'll=.=IITII IIVIla &..a,

(2.6-9)

where 11

IIT11w

Vh

n1

M2

n2

MS

n,

(2.6-10)

one has

IIaII='=IITII IIaII' IIbll

IITIIIfbfl Na.

Ilabll=-= IIaII=IIbII=, =IITII IIaII=IIbII=llfill = IITII If a and b are written in terms of the unit vectors, i, j, k along the set of rectangular axes in use and then multiplied together: ab = a=b=ii + aabjj + a,bAk +

j+abjk+

(2.6-12)

a,b=ki + a,bkj + a b=kk

It is seen that the elements of the matrix in 2.6-5 are the coefficients of 0 Note difference from equation 22-8.

VECTOR ALGEBRA

17

the nine unit dyads that can be formed from i, j, and k; in fact the expanded dyadic 2.6-12 is essentially a mode of writing the matrix with its elements so tagged that they cannot be mis-ordered.

2.7 Dyadic Representation of Linear Vector Operators A fully defined linear vector function is expressible as a sum of dyads. A function is fully defined if one is able to find its value for all allowable values of its argument. Inasmuch as an N-dimensional vector can always

be written as the sum of its components along N independent vectors, we can, in the light of 2.5-1, find the value of a linear vector function of

an arbitrary vector r if we know the results when it operates on N independent unit vectors. To illustrate in three dimensions, suppose that when K operates on the non-coplanar vectors a, b, c, the results are:

g= h = K-c

(2.7-1)

Let us resolve the vector r into its components Aa, Bb, and Cc along a, b, and c respectively. Then

r = Aa+Bb+Cc and from 2.5-1 and 2.5-3,

Kr = AKa + BKb + CKc

= Af+Bg+Ch

(2.7-2)

which, because f, g, and h are known vectors, furnishes the vector resulting when K operates on r. There is no loss in generality if a, b, c are now supposed a set of mutually perpendicular unit vectors because, if this is not the case, the results of allowing K to operate on such a set may be found by the method just given. Granted this supposition:

A = ra` = a"r

B = rb = b-r

(2.7-3)

C=rc=c'r where the overscores are needed only if the directions of the unit vectors are not real. Substitution of equations 2.7-3 in 2.7-2 gives

Kr = far + gbr + her + gb + - (fa hc)r

(2.7-4)

VECTOR AND POLYADIC ANALYSIS

18

or alternatively

r (af + bg +eh)

(2.7-5)

which show K and its transpose to be sums of dyads, viz.:

K=fa+gb+hc

(2.7-6a)

K = of + bg + ch

(2.7-6b)

K = of + bg + ch

(2.7-6c)

Analogously one finds

Thus, a linear vector operator is equivalent to a dyadic which, once known, permits its transpose and adjoint to be written in like form by inspection.

By appropriate algebraic manipulation an N-dimensional dyadic may

be reduced to a sum of not more than N dyads in which the N first factors or the N final factors (but not in general both at once) constitute an arbitrary set of N independent vectors. These N vectors need not be unit vectors, nor need they be mutually perpendicular. Whichever set of factors is chosen to be taken from the set of independent vectors, the members of the set of vectors constituted by the other factors may each be expressed as a linear combination of the N independent vectors. If these expressions are substituted in the N-term dyadic, the resultant expression is a linear combination of N2 dyads each factor of which is

one of the N-independent vectors (or its conjugate). Thus, if f, g, and h are expressed in terms of their components along the unit vectors as b, c, e.g., f = faa + fbb + fee

and the resulting expressions are substituted in equation 2.7-6a, one obtains K in the form

K = aaaaa + aabab + aacae. + ababii + abbbb + ab,be + acaca + a,bcb + a,cec

(2.7-7)

This is often taken as the standard (or "canonical") form of a linear vector operator. If K is Hermitean, a.., abb , a., will be real and the .'other coefficients will satisfy the relations: aab = aba aac = aea

abc = acb

19

VECTOR ALGEBRA

In most applications it is convenient to choose a set of real mutually perpendicular unit vectors, e.g. i, j, k, as the basis. This is always possible because a complex vector may be expressed in terms of such real unit vectors with complex coefficients. Typically the representation of K relative to such a real basis is

K = A,,ii + Axvij + A=,ik + A,ji + A, ,jj + A v,jk +

(2.7-8)

AsZkI + A,,,kj 4 A.,kk where, in general, the A's are complex numbers. If a, b, c in 2.7-7 are real, equation 2.7-7 is itself in the form of 2.7-8. When a linear vector operator is expressed in the form of 2.7-8, it is necessary and sufficient that: 1. If symmetric,

As, = A,, ;

A,v = A.vs ;

Av,

=

A,,,

2. If antisymmetric,

Axv = -Ay: ;

A,,

-A,x ;

AY, _ -A,v

3. If Hermitean, A,,=

;

Aa = A,v

and A.., Avv , A =, are each real.

If the A's are complex, clearly K may be written as the sum of two expressions like 2.7-8, one with real coefficients, the other with imaginary coefficients. If K is Hermitean the real part will be symmetric and the imaginary part, antisymmetric. The idemfactor in terms of any set of mutually perpendicular unit vectors is

1 = aA+bfi+cL

(2.7-9)

which reduces to the form

1 = ii + jj + kk

(2.7-9a)

if i, j, k are any set of real mutually perpendicular unC N®dt

9-

dyadic idemfactors are equal. The laws of direct multiplication of vectors (Eqs.

transforming dyadics from one form to another, ar4beyed by dot products of vectors. If, therefore, a dot is inserted betaden the seatora

20

VECTOR AND POLYADIC ANALYSIS

of every dyad in a given dyadic, the scalar value of the resulting expression is independent of the way in which the dyadic may have been ex-

pressed. This invariant quantity is called the scalar of the dyadic; it will be denoted by adding the subscript "S" to the symbol of the dyadic. If K in 2.7-7 and 2.7-8 is the same operator,

Ks = A. -}- A., + A. - a., + aaa + a.,

(2.7-10)

because the dot products arising from the other terms are all zero. The scalar of an antisynunetric dyadic is zero but not all dyadics with zero scalars are antisymmetric. If K is any N-dimensional dyadic, its deviational part, or "deviator", O

1V

(2.7-11)

Ks1

has zero scalar. Evidently

= Ks

(2.7-12a)

Ks = Ks

(2.7-12b)

Is = 3 (or N, if N-dimensional)

(2.7-12c)

Ks

If a, b, c are any set of mutually perpendicular unit vectors a general formula for the scalar of K is Ks =

(2.7-13)

as may be seen by supposing K to be in form 2.7-7 and performing the indicated operations. Indeed, the formulae for the coefficients of the several dyads in 2.7-7 are A-K-a;

r)-K-,;

etc.

Inasmuch as the real parts of all the diagonal coefficients of an antiselfadjoint (if real, antisymmetric) dyadic are zero, the real part of

F-K-r is zero for such a dyadic for all values of r. This follows because one need only suppose r to be a scalar multiple of one of the mutually perpendicu-

lar unit vectors in terms of which the dyadic is expressed. When the vectors involved in these formulae are real the oversoores naturally may be omitted. The extension to N dimensions should be obvious. The scalar of a dyadic is the sum of its characteristic numbers. In the case of real three dimensional systems, a vector invariant of a

VECTOR ALGEBRA

21

dyadic results if a cross is placed between the vectors of each dyad. This

is called the vector of the dyadic and indicated by a subscript "V". For a dyadic to be symmetric it is necessary and sufficient that its vector vanish. If K is antisymmetric and r is any real vector

r 2.8

(2.7-14)

Multiple Scalar Products; Higher linear Operators

2.8-0 Definitions

An expression such as abed... is a "polyad"; an "n-ad" is a polyad with n direct vector factors. A "polyadic" is a linear combination of polyads of the same rank with scalar coefficients. The rank or valence of a polyadic is the common rank of the polyads that constitute its terms. A dyad is a polyad of rank 2, a vector is a polyad of rank 1. Polyadics are tensors. A polyadic will be represented a Gothic letter with a superscript preceding it to indicate its rank; thus `K will denote a pentadic. In the case of dyadics the superscript will be omitted unless clarity requires its presence. Polyadics are required to represent certain physical quantities: thermal conductivity is, in general, at least a dyadic; viscosity, at least a tetradic;

the derivatives with respect to their vector arguments of functions of vectors are polyadics. 2.8-1 Polydot Products of Polyads The n-dot product of two n-ads is the scalar formed by the rule

abed... ®fghm = = fghm ® abed...

(2.8-1)

The squared magnitude of an n-ad is its n-dot product with its complex conjugate:

abed...®abcd

aI'IbI'I cd1=

(2.8-2)

= f abed... I' The several formulae of Section 2.2 (viz.: 2.2-1 to 2.2-6e inclusive) are true of n-ads of every rank if, as is true by definition, they are true of monads (i.e. of vectors). For example, equation 2.2-2 is a correct and consistent definition of the triple-dot product of two real triads:

abc; def = M I abc + def 1' -- I abc I' -- I def 11 (2.8-3)

22

VECTOR AND POLYADIC ANALYSIS

For n = 2, the n-dot product as here defined is the double-dot multipli-

cation which Gibbs denoted by a colon (:) (E. B. Wilson, "Vector Analysis--Gibbs", Yale University Press, 1901). Herein the notation is used when n exceeds 3 or 4; for n = 1, 2, 3 respectively the Gibbs'

notations* , :, 1 are usually employed. The two factors in a simple polydot product each must be of at least the rank of the multiplication sign. If one or both of the factors exceeds the rank of the sign, the expression is read as though parentheses were so inserted that the two factors actually to be multiplied agree in rank with the sign, thus

abcdeshjks = ab(cde:jhk)s

(2.8-4)

Polydot products of polyads may be factored into products of lower rank. For example,

abcdef ® kjmpq = a(bc: kj)(mpg;def) =

(2.8-5)

= a(be:kj)(def=mpg)

= (kj: because all these expressions reduce to the vector Study of the example in the light of equation 2.8-1 should make the rules of factoring clear. In the examples of equation 2.8-5 no ambiguity results

if the parentheses are omitted. On the other hand the parentheses may

not be omitted from the two expressions abc(de: uv)w:pgstx and abcde: (uvwipgs)tx without confusion; the first reduces to (de: uv) (bcwipgs)atx and the second, to (uvw;pgs)(de:tx)abc. Parentheses must therefore be kept if their removal would result in enclosing a polyad

with multiplication signs whose combined rank exceeds that of the polyad.

2.8-2 Unit Polyads

A polyad composed entirely of the unit vectors of a mutually perpendicular set, such as the pentad ikijk which is composed of the real * Chapman and Cowling and some other writers who in other respects use Gibbs' notation use ab:cd to mean what Gibbs wrote as ab:dc. Especially when an extension is made to polyadics of higher rank Gibbs' original meaning seems preferable. For symmetric dyads and dyadics the difference in meaning disappears.

VECTOR ALGEBRA

23

set i, j, k, has unit magnitude and is normal with respect to polydot multiplication of its own rank to every other polyad of the same rank formed from the same set of unit vectors:

ikijk ® ikijk = 1 ikjjk ® ikijk = 0

(2.8-6)

Such polyads may be called unit-polyads. From the rules of direct multiplication of vectors and the fact that any vector may be expressed as a

linear combination of a set of mutually perpendicular unit vectors it follows that any polyad may be expressed as a polyadic in terms of unit polyads of its own rank. Each unit polya.d is independent of every other of the same rank and basis. The number of independent unit n-ads is 3 in a three dimensional system: there are nine independent unit dyads, 27 independent unit triads, and so on. It will be convenient to consider the unit n-ads to be ordered just as i, j, k are considered to be ordered. A simple way of ordering results if

the first three unit (n + 1)-ads of the i, j, k basis are, in order, those obtained by adding i, j, k respectively as a terminal direct factor to the first unit n-ad; the second three, those obtained by adding i, j, k respectively to the second unit n-ad; and so on. With this scheme, if we begin numbering with zero, the number of a given polyad is, for the three dimensions, obtained in the ternary system of numeration (see Appendix B) by replacing i, j, and k by their numbers, viz.: 0, 1, 2. Thus

the pentad ikijk is the 02012nd unit pentad in the ternary system of numeration or the

0(3') + 2(31) + 0(3') + 1(31) + 2(3°) - 59th in the decimal system of numeration. For brevity and ease of identification when n is large, the m-th n-ad will be written , with both n and m in decimal numeration; thus ikijk = <5/59>. The following rules result for three dimensions:

1. A unit n-ad has the zero'th unit q-ad as a direct prefactor if

0<m<3"-°; the unit q-ad number one as a direct prefactor if 3'

5 m < 2(3)*

;

and so on. R. In direct multiplication

- <(a + b)/(m`3' + »n)>

24

VECTOR AND POLYADIC ANALYSIS

3. The (2p) -ads numbered m = 8 (1 + 3') where 8 is zero or an integer less than 3' are given by

. These occur in the diagonal terms when a (2p-)adic is written in a square array analogous to equation 2.7-8.

Polyadics An n-adic is, from its definition and the results of 2.8-2, always ex-

2.8-3

pressible in the form

"K = Ao + Al + A2

+

A

1)>

(2.8-7)

with not more than 3" terms. If, and only if, its squared magnitude I"K(:= EIA..12-0 "K®"K= "K is the zero n-adic; then every A. is zero, but otherwise at least one is not zero. The A's are the coordinates of the n-adic with respect to the basis (e.g. i, j, k) in which the unit n-ads are expressed; the A's are scalars and may be complex. Herein, unless otherwise stated, the basis will be supposed real and three-dimensional. Figure 2.8-1 shows the array of a tetradic in square form.

An n-adic may be written as a vector in two different ways "K = "- "1Ao i "-"lA "-rilA3 k 1j

= i "nBo + j " nB1 + k "nB2

(2.8-8)

wherein the (n -- 1)-adic coefficients in the first form may be called the

(n - 1)-adic pre-coordinates of "K vectorially written and those of the second form, the (n - 1)-adic post-coordinates of "K vectorially written. Similarly, "K may be "dyadically written" in two ways

nK = n.2A0 ii + n.2A1 ij + ri ,A2 ik + A3 ji + " nA4 jj + "-2A5 j k + nA6 ki + n -.2A., kj + kk

(2.8-9)

II" "B.+ ij ",,Bl+ ik" "B2+ iI""B6+ wherein the (n - 2)-adic coefficients of the first and second form may

be called the (n - 2)-adic pre- and post-coordinates of "K dyadically written. Similarly "K may be written in two ways as a polyadic of any rank m less than n; the coordinates will be of rank n - m.

VECTOR ALGEBRA

-ii ii-

ij -

ik-

-ij

-kk

As

A7

A,

iiki

iikj

iikk

As

iiik

As ,..iii

A4

iiij

lijj

iijk

<.t

Aio

A»

Au

Ai6

A16

A17

ijij

AIf

A12

ijii

ijjk

ijki

ijkj

ijkk

Ais

A22

Ais

A24

Ash

A,.s

ikjk

ikki

ikkj

ikkk

As,.

A64

jiki

jikj A43

ijik

ijji

Aft

Aso

Ail

ikij

ikik

ikji

ikjj

Ate

Alt

Aso

Aai

A32

Ass

Alt

jjji jjjj jjjk jjki jjkj

A4o

A41

A4,

jjkk

As

jiii

jiij

Ass

A67

jiik

jjik

ijjj

jiji

jiii

jijk

Aas

jikk A44

jkji

jkjj

jkjk jkki

Aso

AsI

jkkj jkkk

A52

Ass

Ass

As7

Ass

Act

kiji

kijj

kijk

Aso

As,

A62

kiik

kikj

kikk

A64

Ass

A6s

kjji

Ash

As*

kjik

A67

kjjk

kjki

A7o

A71

kjij

kjkj

kjkk

A72

A73

A7,

An

A76

A77

A7,

kkij

kkik

kkjj

kkjk kkki

A,, kkkj

Aso

kkii

A46

A47

jkii

jkij

jkik

A64

Ass

kiii

kiij

A6a

-A45

kjii kk-

-kj

Ai

jjii jjij

kj-

As

-ki

iiii

A27

ki-

-jk

-jj

-j i

Ao

ikii

jk-

-ik

25

A40

kjjj

kkji

kiki

kkkk

Figure 2.8-1. The Array of a 3-dimensional Tetradic expressed in terms of a set of mutually perpendicular unit vectors i, j, k: Aoiiii + Aiiiij + A,.iiik + etc.

In the above formulae the subscripts of the A's and B's are the numbers of the unit polyads for which they are the coefficients. The A's and B's are interrelated by formulas typified by: ,,-1

nA, =

,A0 i + n-2nA, j

n-2

+

n-2

,Aa k

nAm = Am 1 + Am+(a)n-1 j + n-nAm = n-q-iAA,. 1 + n-Q -nAm-at j nnBO

Am+2(2)A-i k

+ = 1nnB0+JnnBi+k*-2B2

n-q-lnAm+2(,)t

A

n-q

nBm = i

n-q-i n--1,Bam+2 n-9-1 Ba+A+i + k nBam + j

Bm = i Aam + j Aam+i + k Asm+2

They are found in terms of nK by the formulae

k

(2.8-10)

26

VECTOR AND POLYADIC ANALYSIS "

nA," =

"K



nB," _ Q -K Figure 2.8-2 gives the arrays of some of the A's and B's of the tetradic of Figure 2.8-1. To find the q-dot product of "K and 'Q write each q-adically and sum An

Ae

A.

As

iii

iij

ilk

iii

As

A

A

Aso

iji

iij

ijk

iji

A

A

A,,

A21

iki

ikj

ikk

An

Aso

Ass

jii

jij

Ass

As,

jji

A,s

ik

A s,

A,2

A s,

iii

ijk

A s4

ikj

A 6,

Au

ikk

ki

kj

A kk

A,,

A

A,o

A

a "column dyadic".

jji

A,,

jkk

jki

A so

jjj A jkj

A A kij

Ass

Age

A,,

Aso

kjj

kjk

kji

kkj

Ass

The Dyadic Prc-coordinate

kik

Avs

jk

jik

kij

A,' kki

A47

ii

An

Ass

kii kji

A

ji

An

A6,

As,

A so

ij

iii

A,,

A 6,

A ix

ii

jii

jjk

jkj

As

iik

jik

jjj

jki

iki

Ass

iij

A,o

Ar,

kkk

The Triadic Pre-coordinate

kii

A6,

kki

jjk A,,

jkk As,

A,e

ij

kik

A,, kjk

A:,

kjj

Ass

Ass

A

A s,

kkj

kkk

The Triadic Post-coordinate

iAe

i5,

a "column triadic"

a "row triadic"

MA9

*A,

A 22

jj

ji

A,, kj

ki

A ^o

ik

A jk

A2,

kk

The Dyadic Post-coordinate

is,

a "row dyadic"

- Aol + A n j + Aukl Vector Pre-coordinates

iA,f - A ,si + A,s.j + A,.k 14511

4B,s

sAi

A,j + A k

- A,,i + A,,j + A sok

Vector Post-coordinates

Figure 2.8-2. Examples of the Pre- and Post-coordinates of the tetradic of Fig. 2.8-1.

VECTOR ALGEBRA

27

the direct products of the pre-coordinates of the prefactor by the corresponding post-coordinates of the post-factor. Thus, if

'K = E " A," = E n n8m w a 'Q =.

l

then

"K©'Q= PQ ®"K

(2.8-12)

E"nA.nDybm In

- E n-va ,. P

(2.8-13)

»

provided n, p > q > 0. If p = n = q = 1, this reduces to equation 2.2-7 (p. 8). If q = 0, it reduces to the direct product. The squared magnitude of an n-adic is a special case:

"K®"Ka l"KI' - wEIA.I -EI"nAml'= EI`,9,B. It " w

(2.8-14)

It is the sum of the squared magnitudes of the coordinates however "K may be written and is a scalar invariant of the n-adic. From 2.8-13, it is seen that the rank of the q-dot product of an n-adic and a p-adic is n + p --- 2q. In general, the polydot product of two polyadics is commutative only if the result is a scalar, i.e., if the rank of the product is zero.

2.8-31 Transposes and Adjoints of Polyadics

nn n»

The forward n-adic transpose 'K of a polyadic 'K satisfies 'K ®"L = "L ®'K

' "Y

(2.8-I5a)

(n < p) and, equivalently,

' "M P-

1-1n

'K 0 "L - "L ®'K p-N P -"M = S

(2.8-15b)

where "L and "M are aribtrary polyadics of the ranks indicated. The "(l

backward n-adic transpose 'K, analogously, satisfies

nn

"L ®v 'K = 'K ® "L = '"X

(2.8-16a)

and

% ®sK P--N y-"M = 9-MM

ISM

0 "L - T (2.8-16b)

28

VECTOR AND POLYADIC ANALYSIS

In general 'nY 0 "X and S p T when the some choices of nt and '-'"M are made in equations 2.8-15a, b and 2.8-16a, b. The index written on the bent arrow, n in equations 2.8-15 and 2.8-16, will be called the

order of transpose. When a polyadic 'K is expressed as a sum of unit p-ads, the forward

nn n-adic transpose 'K is formed by transposing the terminal set of n vector factors of each polyad from the rear to the front without changing their nF--j

order in the set. The backward n-adic transpose 'K is formed by transposing to the rear the initial set of n vector factors of each polyad. From these rules the following formulae follow:

f7m

nn

rln I -jm+n 'K = 'K = 'K f--Jm

(2.8-17)

MM

j--In m-nEl

11n-m

'K = 'K apn nap "K = 'K = 'K, if a is an integer or zero nn np--n nn p-nl-i 'K = 'K; 'K = 'K 'K

(2.8-18)

(2.8-19) (2.8-20)

By successive application of the operation of transposition we can construct formally a transpose of order higher than the rank of the polyadic. A transpose of order ap + n, where a is integral and n < p, is identical with the transpose of order n in the same direction. Backward and forward transpositions of the same order are inverse operations; correct formal results follow if a transpose of negative order is interpreted

as a transpose of the same absolute order in the opposite direction:

-nn. Fn

'K =- 'K. The operation of transposition is linear; in fact

8n-1 ++(

z'Trn

0

8P-1

t-0

(2.8-21)

and an-1

j -Jn

,E

M-0

2n-1

nF---l

(2.8-22)

t-0

are transposing operators for p-adics if applied with p-dot multiplication :

29

VECTOR ALGEBRA

n( l

'K = 2'Tr, ® 'K = 'K ®2'Trr

(2.8-23)

I-I n 'K = zpTr; ® 'K = 'K ® 2'Tr,,

(2.8-24)

These operators are the p-adic transposes of each other. They are functions of idemfactors and are discussed in Section 2.8-4. nF-l

r--j n

If it happens that 'K = 'K, then also 'K m 'K and 'K is said to be

nrl

n-adically symmetric; if 'K

nn

-'K, then also 'K as -'K and 'K is

nF1 r7-

said to be n-adically antisymmetric. If 'K = 'K, the polyadic may be subdivided into n-adically symmetric and antisymmetric parts: n-

nT--j

'K = I('K + 'K) nX

(2.8-25)

"r--j

'K = J('K - 'K)

(2.8-26)

ftF__j

f--J"

Such polyadics may be called separable; those for which 'K P'- "K will be called n-adically asymmetric. Antisymmetric polyadics are always of even rank; in consequence so also are separable polyadics. An n-adically symmetric polyadic is symmetric also for all integral multiples of n. If p is its rank and mp is the least common multiple of n and p, then mp/n of the p possible transposes will be identical with the original p-adic. The polyadic therefore has symmetry of orders n/m and its integral multiples. Here n/m will always be integral. An n-adically antisymmetric polyadic is also antisymmetric for odd multiples of n but symmetric for even multiples of n. If mp is the least common multiple of n and the rank p, and mp/n is even, then mp/2n of the possible p transposes will be identical with the original polyadic and mp/2n of them will be identically the negative thereof. The polyadic therefore possesses antisymmetry of order n/m. Here p, and therefore

mp, are always even; if, therefore, n is odd, mp/n is even but, if n is even, mp/n is not necessarily even. Actually no antisymmetric p-adic exists of such order n that mp/n is odd: mp/n is the number of successive times a p-adic 'K must be n-adically transposed to obtain the transpose of order mp which, by 2.8-19, is always 'K, the original p-adic. By definition, an odd number of such transpositions results in -'K if the p-adic be n-adically antisymmetric.

VECTOR AND POLYADIC ANALYSIS

30

But 'K # -'K. Therefore n-adically antisymmetric p-adics such that mp/n is odd do not exist. Examples: a)

IF_-I ijjkjj - jkjjij $=X

b) FK &;78Z 'Z 'K 3T 2. Any direct power of an n-adic is n-adically symmetric and also is symmetric for multiples of n up to the value of the exponent: a) VVV = Ya is vectorially, dyadically, and triadically symmetric.

b)

"M "M "M "M - "M' is n-adically, 2n-adically, 3n-adioally, and 4n-adically symmetric.

3. (ijij -- jiji) is vectorially and triadically antisymmetric; and dyadically and tetradically symmetric.

am M-3 -

4. If'K=Ajiji+Bijij,`Kw`Kfora= 1,2,3,4. If A - B, it is symmetric for all these values of a; if A = -B, we have example 3; if A 0 ±B, it has both symmetric and antisymmetric parts for a - 1 or 3: I-

1-

4K = 4K = J(A + B)(ijij + jiji) Ix.

ix

`K= 'K=J(A -B)(jiji-ijij)

It is necessary and sufficient for n-adic symmetry of a polyadic that, when it is n-adically written, the corresponding pre- and post-coordinates

be equal; for n-adic antisymmetry, that they be the negatives of each other.

The scalar p-dot product of an n-adically symmetric p-adic by an n-adically antisymmetrie p-adic is zero.

The forward n-adic adjoint pK of 'K is defined by 'K®WE

and the backward

"L®aK

(2.8-27)

adjoint Kn by w("`l

WE ®'K = ik ®"L

(2.8-28)

am

where "L is an arbitrary n-adic. If it happens that 'IC m 'K, then so also

VBCTOR ALGEBRA

n

31

'K and the polyadic is n-adically self-adjoins. For real polyadics, the adjoints are identical with the corresponding transposes. A, complex polyadic may always be written as the sum of a real and an imaginary part: 'K = 'K, -1- i'Kj is aK

For an n-adically self-adjoint polyadic, the real part 'K, is n-adically symmetric and the imaginary part 'Ki is n-adirally antisymmetric. The n-dot product of a p-adie as prefactor and its forward n-adic adjoint is (p -- n)-adically self adjoint as is also its n-dot where i

product as post-factor with its backward n-adic adjoint:

(2.8-29)

The (p - n)-dot products taken in the reverse order give n-adically self-adjoint polyadics (2.8-30)

The A's and B's are the pre- and post-coordinates as defined on page 24.

2.8-4 Idemfactors, Deviation Factors, Tranaposers, and Related

Functions An idemfactor for polyadics of rank m is a linear operator that when used either as a pre-factor or a post-factor in polydot multiplication with every arbitrary 'Y simply reproduces "Y. The idemfactors thus play the part of unity in polydot multiplication. The 2n-adic

"1

F cn/s>cn/a>

(2.8-33)

$*-1

E <2n/a(1 + 3k)> satisfies the definition

2"1 ®"`Y - 'hl ®'"1 - "Y

(2.8-34)

32

VECTOR AND POLYADIC ANALY E'rS

provided m > n. For n = 1, it reduces to the dyadic idemfactor 1 = ii + jj + kk of page 19. It is customary to call this 2n-adic itself the idemfactor of rank 2n even though it properly constitutes an idemfactor operator only when used with a multiplication sign of rank n. Idemfactors and functional operators constructed exclusively of idemfactors are peculiar among polyadics in having the same coordinates in every real Cartesian frame of reference. This section discusses briefly the structure of idemfactors, their properties, and those of some func-

ttons derived from them. In general only the 3-dimensional case is treated; the N-dimensional case will be seen to be obtained in general by substituting N for 3 where powers of 3 occur in the formulae. Whatever complete set of mutually perpendicular real unit vectors

is adopted, 2"1 is the sum of the full set of centrally symmetric unit 2n-ads. Unity is the idemfactor of zero rank. The next two in rank are, in terms of i, j, k,

21 = 1 = ii + ii + kk 41

(2.8-35)

= iiii + ijij + ikik + jiji + jjjj + jkjk + kiki + kjkj + kkkk

(2.8-35a)

A centrally symmetric unit 2n-ad is factorable into a unit q-ad and a unit (2n - q) -ad in two ways: (2.8-36) <2n/s(1 + 3")> = <(2n - q)/t> (2.8-36a) <2n/s(1 + 3")> = <(2n - q)/t'> wherein (cf. p. 23) the pairs of positive integers (p, t) and (p', e)

satisfy (2.8-37) +t 8(1 + 3") = (2.8-37a) s(1 + 3") = t'3° + p' Because p and t are restricted to be positive integers equation 2.8-37 has only one solution (p, t)., for each integral value 81 of 8 (0 < s, < 3" -- 1). Similarly, equation 2.8-37a has only one solution (p', t')., for p32"_Q

each s, . The 3" different pairs (p, t). derived from 2.8-37 prove, however, to be the same set as the 3" pairs (p', t'), derived from 2.8-37a but for a different sequence of values of s; that is (2.8-38) (p, t),, = (p', t')., (2.8-38a) (p, t)., = (p', t')., where, in general, s, , 4, and ss are not equal. By substituting 2.8-38

VECTOR ALGEBRA

33

in 2.8-37a, one finds

s,=8130 -pl(3"-1) 2

= [8, + t1(3" - 1)]/3

(2.8-39)

"-

and, from 2.8-38a and 2.8-37,

8, = [Si + pi(3" - 1)]/3" (2 . 8-39a ) = 8132"-Q - tit (3" - 1) where pi and tl constitute the pair (p, t)., and pi and ti pair (p', , the t'), . By considering equations 2.8-36 and 36a in the light of these results we can see that

i

QI

<2n/sl(1 + 3")> = <(2n - q)/tl>

(2.8-40)

_ <2n/82(1 + 3")> 1

41

<2n/al(1 + 3')> = = <2n/s,(1 + 3")>

(2.8-40a)

so that the full set of centrally symmetric unit 2n-ads contains within itself the q-adic transposes of each member. Three of these 2n-ads are direct powers of single base vectors (e.g., iiii , jjjj , kkkk. . .) and are therefore individually q-adically symmetric. The number remaining, (3" - 3), is an even number of which one half may be so selected that the second half consists entirely of the forward q-adic transposes, or entirely of the backward q-adic transposes, of the first half. The sum of all 3" of the centrally symmetric unit 2n-ads, i.e. 2"1, is therefore qadically symmetric for all q < 2n. The matrix of the coordinates of 2"1 is the unit 3" X 3" matrix; the determinant of 2"1, which is that of the matrix, is unity. Its "scalar"the sum of the diagonal coordinates- is 3" and so also is its squared magnitude 2"1 2h '"1. With respect to n-dot multiplication every unit rn-ad (m > n) is a characteristic m-ad of 2'1 and all the corresponding characteristic numbers are unity (cf., p. 13, 68).

Example 2.8-1. Tetradics: n = 2. Let q = 3; 2n - q = 4 - 3 = 1. The centrally symmetric tetrads are <2n/s(1 + 3")> _ <4/108> _ <2/8X2/s>

= c3/p> = cl/t'>c3/p'> (cf. equations 2.8-33, 33a, 36, 36a). Here a ranges over 0, 1, 2,

, 8.

34

VECTOR AND POLYADIC ANALYSIS

To solve eqs. 2.8-37, 37a note that p is the integral part of 8(3' ±

3") Q

=

1

s (1

. and t is the remainder. Similarly t' is the integral part of

3) w

3

, 27 and p' is the remainder.

From eqs. 2.8-39 and 2.8-39a if p1 and pi are the values of p and p' determined as above for e = a, , then

as27s,--8p, sa = (at + 8pi)/27 and IF --I

<4/1081> = <4/10s,> <4/1081> _ <4/10ss>

Tabulated triadic and vector factors of the centrally symmetric tetrads are: a(or t1)

0

<4/100

( p,

(p'

2

1

4

a

6

7

8

<4/0> <4/10> <4/20> <4/30 <4/40> <4/50 <4/60 <4/70> <4/80>

jiji

jjjj

10 0

18

2

1

16 2

10

20

8

13

0

0

1

1

3 8

6

1

6

1

4 4

IM

iiij

ikik

0 0

8

6

1

:,)" `p'

0

t'

0 0 0

t) .,

3

p

kjkj

kkkk

20 0

23

26

1

2

23

6

16

26

1

2

2

2

7

2 2

5

8 8

jkjk

7

kiki

5

Here the tetrads have been given in terms of 1, j, k so that the results of the formulae may be checked by inspection. From the table, for 81 = 2, 82 = as = 6, hence <4/20> = <4/60> _ <1/2X3/6> = kiki I"

I'

<4/20> = <4/60> - <3/20X1/0> - kilo

35

VECTOR ALGEBRA

Vectorially written, `1 is seen to be symmetrical '1

1<3/0> + (3/10> + <3/20>1i +

[<3/3> + <3/13> + <3/23>Jj + a

[c3/6> + t3/16> + <3/M k i[<3/0> + <3/10> + 0/20>1 + j[<3/3> + <3/13> + <3%23>1

+

k[c3/6> + <3/10 + 4/26>1

where the values of 8 from which the terms derive have been written above them. Triadically written, '1 is also symmetrical:

'1 = i<3/0> + j<9/3> + k<3/6> +

41

i<3/10> + jd/13> + kd/16> + i<3/20> + jd/23> + kd/26> = <3/0>i + <3/3>j + d/6>k + <3/10>i + <3/13>3 + <3/16* + <3/20>i + d/23>3 + <3/26>k

Notice that only nine of the 27 unit triads occur.

Example 2.8-2. j3exads: n = 3. Let q

2.2n-q=6-24

<2n/s(1 + 3°)> _ 4/288> <3/a>t3/8> <2/px4/0 - <4/f>c2/p'> Range of a is 0, 1, 2, ,26. 8(I + 3") = 28a. 28a

9ai-26pi [al + 26pil/9

36

VECTOR AND POLYADIC ANALYSIS

Tabulated dyadic and tetradic factors of the centrally symmetric hexads are: s(or al)

0

0/280 (p

2

1

3

4

5

6

7

<6/0> <6/28> <6/56> (6/84) <6/112> (6/140> <6/168) <6/196>

0

0

0

1

1

1

2

0

28

56

3

31

59

6

2 34

W , 04 t '

0 0

1

2

3

4

5

3

6

9

12

15

8 18

7 21

82

0

18 6

15

2 18

11

9

10 12

19

0

9 3

1

ei

10

11

12

13

14

t)

,

p

p'

s(or al)

8

(6/224) (6/252> (6/280) <6/308) <6/336> <6/364> (6/392>

C6/280 ( p,

t) n

p

(P , t 82

x3

(or at)

p,

(P' ,

t) n

2

3 65

4

4

4

12

40

68

2 34

3

37

4 40

5 43

21 7

4 10

13

4

13

22 16

17

18

19

20

21

3

62

3 9

8 24

0

1

28

31

20 24

3

12

1

15

16

37

(6/420) (6/448) (6/476) <6/504) (6/532) (6/560> (6/588>

:6/280 (

9

21

p t

6

7

74

21

5

5

5

6

15

43

71

18

8

7 49

8 52

0

1

2

3

56

59

62

65

14 22

23 25

6

15 5

24

7

8

11

46

s:

5

81

19

2

6 46

VECTOR ALGEBRA

37

s (or a,)

22

23

24

25

26

<6/280

<6/616>

(6/644>

<6/672>

<6/700>

<6/728>

(p,

t )n

p

7

7

8

8

8

t

49

77

24

52

80

68

71

74

77

80

16

25

26

17

8 20

17

14

23

26

(P" t') n { sa

'

For al= 8; a2=20,x3=24 <2n/s(1 + 3")> = <6/224> = <2/2><4/62>

<4/24><2/8> = ikkikk 21

X

<6/224> = <4/62><2/2> = <2/6><4/74>

12

= <6/560> = kikkik

<6/224> = <2/8><4/24> = <4/74><2/6>

= <6/672> = kkikki Written dyadically, according to the table, 0

2

1

61 = ii[<4/0> + <4/28> + 0/56>1 +

i

6

4

ij[<4/3> + <4/31> + <4/59>] + 7

0

8

ik[<4/6> + <4/34> + <4/62>1 + 11

10

9

ji[<4/9> + <4/37> + <4/65>1 +14

13

12

jj[<4/12> + <4/40> + <4/68>] + or 0

9

1$

°1 = [<4/0> + <4/28> + <4/56>]ii + 1

10

19

1<4/3> + <4/31> + <4/59>]ij +2

11

20

[<4/6> + <4/34> + <4/62>]ik +

.

38

VECTOR AND POLYADIC ANALYSIS

where the values of s from which the terms derive have been written above them. Note that the pre- and post coordinates are identical as must be the case because the idemfactor is dyadically symmetrical. If the q-adic $1-1

°Q ° EP-0AP

(2.8-41)

is q-dot multiplied by '"I, the result (0 < q < 2n) is s'" 1

A

(2.8-42)

3P

where 22*1JP is the p'th (2n - q)-adic coordinate of "1 q-adically written. When q < n, or equivalently when 2n - q >- n, there are 3" of these coordinates, one for each value of p from 0 to 3' - 1; each is the sum of a different set of 3"-' unit (2n - q)-ads: P--0

<(2n - q)/[p3" -I- µ(1 + 3")]> (2.8-43)

The 3' unit (2n - q)-ads occurring in these sums are all different so that if q < n, 2w1 ('Q is never zero unless 'Q is zero. When q = n, one has in 2.8-42 a special can of 2.8-34.

When q > n, =*'$P is zero except when p is zero or is the largest integer contained in e(1 + 3")/32"-', s - 0, 1, 2, .. (3" - 1) ; then

i*-13v = <(2n - q)/(s(1 + 3") -

p3lw-'J>

(2.8-44)

which is a single unit (2n - q)-ad. There are 3" of these non-zero coordinates; the set consists of 3"-" each of the 31' unit (2n - q)-ads. If the A,'s of 2.8-42 corresponding to non-zero s*-°%P's happen to be zero,'"1 © QQ will be zero when 'Q may not be zero. In the special case of q = 2n, formula 2.8-42 results in the sum of the coefficients on the diagonal when °Q is written in square array, i.e. in the scalar of 'IQ: n1 2N !nQ 2nQ'S

=

(2.8-45)

This sum may be zero when none of the diagonal At's is zero. Written q-adically

_

4*~'3, P-0 (2.8-40)

= P-0



VECTOR ALGEBRA

39

Example 2.8-3

a) Suppose 'Q = 'Q = Aoiii + A1iij + Aujji = Ao <3/0> + Al <3/1> + An<3/12> then `1

; 'Q = A. <1/0> = Aoi

because when 41 is written triadically triads 1 and l e have zero coefficients as found in Example 2.8-1. b) 41 : 'Q - Aotii + Aliij + Aisiji is the case q = n which reduces to equation 2.8-34.

c) '9 ; 'Q = Ao [<5/0> + <5/82> + <5/164>1 + A1[<5/3> + <5/85> + <5/167>) + A11 [c5/36> + <5/118> + <5/200>1

in which the quantities in brackets are iJ, for p = 0, 1, and 12.. Equation 2.8-42 may be rewritten in the form s.-1 °+01 (9) aQ Y-0

A, Qom, = ;Q

(2.8-47)

if a is written for 2n - q and is restricted to values such that q + a is even. Evidently, if °Q is held constant and a is varied, this formula generates a set of a-adios which are characteristic of °Q. They may be called the a-adios of °Q and will be denoted by QQ. Now, if 0 < x < y and x + y is even, from 2.8-46 is-1

1 ®,r+s1 = E

,

s-o

=

v

(2.8-47a)

3(rs)/2

where the final result follows from the discussion below equation 2.8-44. Therefore the q-adic 1

p 4m has the same a-adic as oQ. It is called the principal a-adie part of 'Q and if subtracted from Q the remainder is. a q-adic with a zero a-adic : C

('Q).=°Q -" {221

1 3io-T"'11

®;o

l a IQ - 33t,-.u* 0`10 0+-i

(2.8-49)

40

VECTOR AND POLYADIC ANALYSIS

which is the deviator of 9Q from its principal a-adic part. The subscript a in 9Qa will be written in roman numerals. The 2q-adic operator in braces in equation 2.8-49 may be called the 2q-adic devia-

tion factor with respect to a-adics; it will be denoted by 19D.. so that equation 2.8-49 may be written O

(9Q)o = 2gD,a © 4Q

(2.8-49a) When a > q, 2Dva = 0; the principal a-adic part of °Q is then the whole

of 'Q.

Example 2.8-4 The A -adics and deviators of a tetradic: (For the matrix of the tetradie 'Q see Fig. 2.8-1, p. 25)

a = 0: 4+a1 = 41; 24D. = 6Do = 61 -- i'1 41

Scalar of 4Q = 04Q = '1 ® 4Q = 'Qg

= Ao+Aio+ A20+Aso+A4o+A6o+Aso+A7o+ Aso

Scalar part of 4Q = .41 4Qg

Deviator of 4Q from its scalar part = 8D,o ® 4Q = (4Q)o. The co-

ordinates are those of 'Q, see Fig. 2.8-1, except that 3. 'Qs is subtracted from each of Ao, A,o, Ago

Aso

,

.

a = 2: 9+a1

= 61;

2°D. = 8D,, = 81 - j 61

: 61

80

Dyadic of 4Q = 61 ®4Q = 4Q = E Ap 63p P-0

_ (A0 + Ate + Ar,6)ii + (As + As, + Aso)ij + (A6 + An + A8)ik + (A, + Arr + A66)ji + (A12 + A4o + Aes)jj + (Air, + Aa + A71)jk + (Ais + 4446 + Av4)ki + (4421 ± An + An)kj + (A24 + A52 + A8o)kk

VECTOR ALGEBRA

41

Note: The values of p for which e3, is not zero are the values of t in Example 2.8-2; the values of e3, are the <2/p>'s of Example 2.8-2. Principal dyadic part of 4Q = 61 : ;Q 1(Ao + A26 + Aa) [<4/0> + <4/28> + <4/58>1 + *(As + A31 + A59) [<4/3> + <4/31> + <4/59>] + } (A o + A s4 + A62) [<4/6> + <4/34> + <4/62>] + etc. Note: 61 is expressed dyadically in Example 2.8-2. Deviator of 4Q from its principal dyadic part is O

60.2 ® 4Q = (4Q)II = 4Q - i61 : 40

obtained from Fig. 2.8-1 by subtracting the principal dyadic part given above.

a=4: Tetradic of

a+a1 = s1; 4Q

=

s1

2cDoa

= 6D.4 = 0

®4Q = 4Q

Example 2.8-5. The A-adics of Idemfactors: 21

a = 0 a = 2

°21 = 21

11 =

41

: 21 = 3 : 21 = 21

i1 =41®41 = 9 21 = 61 ® 41 = 21

41 = 61 ®41 = 41

01=61®61=27 61

61 = ® 61 = 21 101 ® 61 61 =

<4/0> + <4/40> + <4/80>

= HE + jjjj + kkkk a = 6

61 = 121 ® 41 = 41

42

VECTOR AND POLYADIC ANALYSIS Note that the deviators are zero for the first and last of each set

but not for the others. The operators '+ai Q and "D. © give intelligible and useful results when applied to polyadics of rank higher than q;; thus for w > q, a._1

a+a1

©'K E i+ais . f Bs s-o

(2.8-50)

= QV+*K a.-1

`K

QAD a+.3a = .,: +`K

P-Q

(2.8-51)

which may be called respectively the post- and pre- a-adic sums of q-adically written '°K and 24D +a

(9) '°K m- WK - 31-b /a %+1 ® ae +aK ((a.o

= ("K)a "K (g 19D..

l

- "K - e

(2.8-52)

1! nr q "K q) q-1-01 (2.8-53)

- (pK)a are the respective deviators of "K, q-adically written, from its post4,0

and pre - a-adic parts. The w-adic (-K)., when q-adically written, has o,q

zero as a post-a-adic sum; (' K)a , q-adically written, has zero as a prea-adic sum. The subscript a will be written in roman numerals and will be omitted when it is zero. When a is zero equations 2.8-50 and 2.8-51 reduce to a*.-1

s"1

--o WK

v

_2%B

.(1+>:)

n

(2.8-54a) (2.8-54b)

.-o

which are respectively the sums of. the diagonal (w - 2n)-adic post and pre-coordinates of W'K when it is 2n-adically written.

The diagonal sum of the 2q-adic coordinates of a(Q+e1. 2t-adically written, is a t-ad transposer for q-adics if t < q. That is, the backward transposing operator "'Ti-t which satisfies

VECTOR ALGEBRA

43

'cTre a QQ = QQ

(2.8-55)

for all QQ is 2QTr7

=

2=1

=(2+2)1

=

scQ+t)1 O tt1

at-i m E 7(Q+t) g i+$')

(2.9-56)

To show this, we observe that

_

(2.8-57)



because this sum, operating q-adically on QQ will convert into its t-adic backward transpose each of the q-ads of which 'IQ is the sum (cf. Eq. 2.8-7, p. 24). Now

(p _ x3 l' + y)



= <(q - t)/y>,

1T--l

= <(q - t) /y> = ,

(2.8-58)

(p' = y3' + x) (2.8-59)

= <(q -- t)/y><2t/x(1 + 3t)><(q - t)/y> _ <2q/ix(1 + 3`)3Q-` + y(1 + 3q+=)]>

(2.8-60)

(2.8-60a)

If the last expression is summed over y from zero to (311--' - 1), comparison with equation 2.8-4:3 shows the result to be which is the 2q-adic coordinate of 2'+`)1 with respect to the centrally symmetric 2t-ad, <2t/x(1 + 3`)>. If in addition we sum over x from zero to (3' - 1), the result is 2.8-56. The q-adic transposes of 'Trt, which are identical inasmuch as it is a 2q-adic, are the same as the backward (q - t) -ad transposer:

an

na

2QTr; ME 20Tr; = I°Tr

(2.8-61)

In summary

=m ,IQ

._ 2QTri ®

QQ = 2QTr

QQ

= QQ ©

r7a-t 2QTra=,

= QQ

© QQ = aQ © 2aTre' = QQ

(2.8-62)

VECTOR AND POLYADIC ANALYSIS

44

Example 2.8-6. Transposers for use as prefactors with q-dot multiplication 2gTrt

For dyadics: q = 2, t = 1 'Tri = 21 : ®1 = iiii + jiij -!- kiik +

ijji + jjjj + kjjk + ikki + jkkj + kkkk For triadics: q = 3; t = 1,2 21 6Fr1 = : 81

;

°Tr2 =

41

® 101

For tetradics: q = 4; t = 1,2,3 &Trt = 21 : 101;

gIrt

$Tr= = 1 © 121

61 © 141

= The backward t-adic transposer'Trt, which when used as a prefactor

'r-1

in q-dot multiplication with an arbitrary q-adic 2K produces 2K, is the t-th q-dot power of 2'Trt ; that is, 2QTr4 ® to t factors = 2gTri `® (2.8-63) 2gTri = 2gTrt This formula follows from a consideration of the effect of the operators of the general q-ad. If t = q or an integral multiple of q, the effect of the operator on the q-ad is nil: q successive transpositions of the first direct factor to the rear results in the original q-ad; thus 221rt "'°®= g°Tr. = 2gTr. = 24Tr1 90 = 221 (2.8-64) if m is zero or an integer. By definition (cf. Section 2.8-31) forward and backward transposes of the same rank and order are reciprocals of each other with respect to dot multiplication of half their rank: (2.8-65) 2gTrt ©2QTrj = 291 = 24Tri ©"Trt thus, in view of 2.8-64, 2QTrt = 2gTrt = 2gTrg-c = 2gTri (e-')® (2.8-66) to 2gTrt = 2gTri (-g)® = 2gTrg = 22Tri From 2.8-63

Zq o 24Tri

=

2gTr +e

24Tr: © 2gTr; = 2gTr

(2.8-66)

VECTOR ALGEBRA

45

TABLE 2.8-1. MULTIPLICATION TABLE FOR IDEMPACTORS N

1

'I

41

1.1

"1

141

1

1

41

41

81

101

121

141

1:

3

'1

4Tr i

'Tr 1

IT, i

10rr i

l2Tr i

41

41

3'Prl

41:

21

41

'1

°1

101

a1

141

3 21

41

9

21

'Tr 7

2Tr =

1°Tr i

161

121

141

41a) 41@

_.

-

'1

'1

61:

4Tr i

'1Qs

41© $1Q

°1©

81

'1: '10

'Tr 1

e1®

9 10 Pr1

---

$1

'1 41

'1

--

--

It Pr`

41 - 4Tr

3 ' Pr: 61

91

3 41

'1

9'1

41

iiii + iiji

27

+ kkkk

'1

°

3f°1

°Tr i

'rr i

27 9 17 Prt 14 Pr1

'1 41

41

3 10 Pr= Si

41

r'1 ©' 1

if

a < q;

101

- 321 if

121

q

141

a

From 2.8-63 and 2.8-64 a transposes of order n = aq + r, where a is an integer and r < q, is identically the transposer of order n = r of the same direction and rank. There are in all no more than q distinct transposes for q-adics if the idemfactor of rank 2q is counted: viz., q. By equations 2.8-66 this set include9 all the for2"Tr,, n = 1,2,3, ward and all the backward transposers. It also includes the q-adic transposes of the transposers; they are identical with the reciprocals of the respective transposers. A multiplication table for idemfactors is given in Table 2.8-1. 2.8-5 Linear Polyadic Functions of Polyadice If one factor of a polydot product of two polyadics is varied at constant rank and the other factor is held constant, the product is a linear function of the variable factor. The linear vector functions of section 2.5

constitute a special case. As in that case, the constant polyadio factor conjointly with the multiplication sign is an operational representation of the functional relationship: for example, if 'K is the constant polyadic in

'K ®'X = "Y

(2.8-68)

46

VECTOR AND POLYADIC ANALYSIS

where 8 < q, p, then "y = f(PX)

and

"K ® V f(

)

(2.8-69)

The multiplication sign must be considered part of the operator because to alter the sign used with the polyadic of an operator changes the relationship represented: an example has been met in the case of the dyadic idemfactor 1,

1:X=Xe whatever the dyadic X. When the multiplication sign is written at the right, the operator is to be used as a prefactor; when the sign is at the left, as a postfactor. Every linear function that defines a y-adic for all values of a variable p-adic may be represented by either of two unique (y + p)-adics which operate with p-dot multiplication upon the p-adic, one to be used as a prefactor and the other, as a post-factor: Let "Y = F('X) be linear and defined for all 'X, p being fixed. Then because every p-adic may be represented in the form 'X

-

E A,n

where the

are a complete set unit p-ads, and, by hypothesis, the value of F(

)

is known for each unit p-ad, one has

"Y = E A. "Y. in consequence of the definition of linearity (Eq. 2.5-1) The coordinates A. of 'X are, however, given by

A. -=

®'X = 'X ®

Hence

"Y=IE"Y..

='X® E

"Y

VECTOR ALGEBRA

47

Therefore F(

) ° E m"Ym

(2 . 8-70)

® 2-, <j;7i;>"Ym M

These two equivalent operators, which are p-adie transposes of each other, will be called the normal prefaetor and postfactor operators associated with F( ) for p-adic arguments; they will be denoted respectively by

v+PFP

7P "Y = F(PX) -

® and ®"+PFP so that +PF®®'X yX

(2.8-71)

1-1 P

®Y+PFP

There is associated with F( ) for p-adic arguments no prefactor operator distinct from "+'F, ®, valid for all p-adios, and operating with p-dot multiplication. If "+'L ® were such an operator then { v+PL - v+PFPJ

®

= 0

for all m. But, if so,, each of the y-adic precoordinates of '+'L is identical with the corresponding pre-coordinate of "+'F, ; that is impossible unare identical. An analogous demonstration apless "+'FP ® and "+'L plies in the case of the normal postfactor operator. Suppose now that the particular function f( ) of equation 2.8-69 is ®.

the same as F(

) of the last paragraph and s is less than p; that is,

suppose QK

® ®V "+PF®

1 1P v+PFP

v

F(

)

(2.8-72)

From the properties of idemfactors,

2s1®PX= PXif z C p; hence

qK Qs v { QK ® 2`'11 ®

(2.8-73)

where a < z < p. With z so chosen, the polyadic in braces will never be zero unless QK is zero because when 2s1 is s adically written there are no

zero coordinates. In particular let z = p, then in view of the uniqueness of "+PF®®,

r+PFP ae QK ®2'1

(2.8-74)

48

VECTOR AND POLYADIC ANALYSIS

Therefore, when a linear function of a polyadic can be represented by a prefactor operator with a multiplication sign of rank s lower than the rank p of the argument, then (1) the polyadic of the normal prefactor

operator is an s-adic multiple of the 2p-adic idemfactor, and (2) an equivalent prefactor operator exists with a sign of each rank from s to p. An analogous situation occurs for postfactors operators when a postfactor operator with a sign of reduced rank exists. Each operator of an equivalent set has the same value r for the excess of the rank of its polyadic over twice the rank of its multiplication sign because r is the difference between the rank of the product and that of the argument. For example, for the operators of the previous paragraph,

r - (y+p) -'2p=q-2s= (q+2z-2s) -2z=y-p (2.8-75) It will be convenient to call r the net rank of the operators and of the functional relation they represent. The lowest possible value of r is evidently -p because y is non-negative. From equations 2.8-75 it is seen that, if on passing from one operator of an equivalent set to another there is an increment Az in the rank of the multiplication sign, there is necessarily an accompanying increment 2Az in the rank (q + 2z - 28) of the polyadic of the operator. Because z is integral, it follows that the ranks of the polyadics of the operators of any equivalent set differ by multiples of two and are therefore either all even or all odd. It further follows, when we note that all of y, p, q, e, and z are non-negative and

integral and s < q, p, that an operator representing a function of net rank r must have a polyadic with rank at least I r and, if such an operator with a polyadic of rank ; r I occurs in the set representing the function, its multiplication sign is zero when r is positive and -r when r is negative.

It is natural in an equivalent set to call the member which has the multiplication sign and polyadic of lowest rank the "lowest operator of the set." Whether or not the lowest operator actually has a polyadic and multiplication sign of ranks as low as the limits set by the net rank of the function depends on the possibility of satisfying equation 2.8-74 (or its

analog for postfactor operators) with the limiting values of q and s. There are quite evidently many (y + p)-adics that cannot be expressed in the forms "K 2 PI or 2P1 F-'K

when y > p, or in the forms r-vK p^ y 2P1

or 2y1 P- y n-vK

VECTOR ALGEBRA

49

when y < p. In such cases the lowest operator is determined by the lowest values of q and s for which the polyadics of the normal operators can be expressed in the form of equation 2.8-74. If the polyadic of the normal operator can be expressed as in equation 2.8-74, then, Y+vFp

2L--S /1 = °K O$ 29 2P - B 2p1

which when rewritten with z = 2p - a is V+pFp ® '+'1 = 9K (2.8-76)

3p-' 2K

by equation 2.8-47a. Comparison with equation 2.8-51 shows that this is the post s-adic sum of the (2p - s)-adically written normal operator and from equation 2.8-53, a+1Fp if (2p - s)-adicallv written is identical with its post a-adic part. Hence if 41-f.

factoring in the sense of equation 2.8-74 is possible and °K can be found from equation 2.8-76. Analogously if rlp

r+1Fv

=0

then the normal postfactor operator may be factored in the sense

vFp = 2y1 ®aK, and

°K' = 3p. 1-

111

( 2.8-77)

2p1 E "+vFp

Example 2.8-7. Let F( ) signify multiplication by the number 3. Then, y = p, and 2110 r+1Fv

®=3

f-1 V

® y F1 = ®.3 21

The set of equivalent operators has a net rank of zero and is

329

®v321-21E

v...

321.:c'3

®3291

X3 3 s-1'c'...

.321

50

VECTOR AND POLYADIC ANALYSIS

The number 3 corresponds to 'K of equation 2.8-72; here q = 0 and

s _ 0.

Example 2.8-8. Let the polyadic ++PFP of the normal operator (cf. Eq. 2.8-72) be a triadic. The possible situations are tabulated below: Lowest equivalent operator

Case

Y

I

0

IV

2 3

II III

1

P 3 2 1

0

r

-3 -1 +1 +3

a 8

q 3 1

1

1

0 0

3

In cases I and IV there can be no equivalent operators with signs of rank less than the rank of the argument; in these cases the two normal

operators constitute the entire set. In cases II and III it may be that there are equivalent operators with polyadics of rank unity and multiplication signs of rank one unit less than the argument; if these equivalent operators exist they will be the lowest operators of their sets. In case II the actual occurrence of an operator of reduced sign depends on the possibility of expressing 3F2 in the form V .

41

f--12

or 3F2 in the

W. From equations 2.8-47 and 2.8-48 (read with q = 3, a = 1, and 'F2 = °Q), if 'F2 = V 11 = 41 V, then form 41

so that 3V is the "vector of 3F2", and

its principal vector part, is identical with 'F2. The general triadic is 3T = Ao <3/0> + Al <3/1> + .. A26 <3/26>

and its principal vector part is, by equation 2.848, I''1 41 ; 'T = }[(A0 + A1o + Ago) (<3/0> + <3/10> + <3/20>) + (A3 + A1a + A23)(<3/3> + <3/13> + <3/23>) + (A6 + Ale + A26)(.3/6> + <3/16> + <3/26>)] (In verifying this it will be helpful to note that 'I is triadically written in Example 2.8-1). For this to be identical with 'T it is necessary that

Ao=Alo=A20 A3=A13= An As = A16 = A26

VECTOR ALGEBRA

51

and all the remaining coordinates must be zero. Therefore if, but only if,

°F2 = Ao (<3/0> + <3/10> + <3/20>) +

As (<3/3> + <3/13> + <3/23>) + As(<3/6> + <3/16> + c3/26>)

it is factorable in the form V V.

41

where

V = Aoi+A,i+Ask and in case II there will be a prefactor operator of reduced sign. How-

n=

ever, if 'F2 has this form 'F2 will not be similarly factorable because the unit triads numbered 3, 6, 10, 16, 20, and 23 are not dyadically F-12

symmetric. To be factorable in an analogous manner, 'Fs must have the form given above for 'F.- and if this is to be true, the six unit triads just listed must be replaced respectively by those of which they are dyadic forward transposes; viz., 12, 24, 1, 2, 25, 14. Therefore, the conditions that a triadic have a dyadic forward transpose factorable in the form 41 W are that

Ao-A1As A21=A1z=A14

A,4=A2a=Ass and all other A's equal zero. In case III a prefactor operator of reduced sign exists if

'F1 = V 21 = (Ai + Bj + Ck) 21

A (iii + ijj + ikk) + B(jii + jjj + jkk) + C (kii + kjj + kkk) that is, if 'F1 is the general triadic with

Ao =A4=As As=A1s=A,7 Ats - An = A26

52

VECTOR AND POLYADIC ANALYSIS

and the other A's are zero. A postfactor operator of reduced sign exists if

nl IFi = 21 W = A'(iii + jji + kki) + B'(iij + jjj + kkj) + C'(iik + jjk + kkk)

that is, if

'F, = A'(iii + jij + kik) + B'(iji + iii + kjk) + C'(iki + jkj + kkk) which is the general triadic with A0

A10 = A20

An

As=Ale=A" and the other A's are zero. As in case 11, if a prefactor operator of reduced sign occurs, there is no

postfactor operator of reduced sign in the same equivalent set and vice versa.

2.8-51 Use of Linear Operators with Arguments of Various Ranks When the Algebraic linear equation y - 3x is multiplied by z to give Y = yz = 3(.xz), one has changed the argument of the functional operation of tripling a quantity but the functional relation is itself unchanged. In a like manner, when equation 2.8-68 is u-dot post-multiplied by a polyadic 'W, with u chosen not to exceed either (p - s) or to, the result °K ® {'X ® wWj = "Y Q) 'W

(2.8-78)

may reasonably be said to express the value of F(rX © WW) if °K ® is a prefactor operator associated with F( ) for p-adic agruments. In this

equation without altering the result, °K ® may be replaced by any of

VECTOR A LOBBRA

53

the set of prefactor operators given by equation 2.8-73 for s < z < p + w - 2u; that set is therefore equivalent to °K ® for arguments of all ranks not less than s. This interpretation of F( ) for other than p-adic arguments entails recognition of the fact that F( ) is in general multiple 1

a

valued. For, consider `)J 1K ® "1} which is the normal postfactor opera.tor associated with F( ) for z-adic arguments for the branch of F( ) to which °K ® belongs: Ir

I

°K ® = ® { °K ®2111

(2.8-79)

when the argument is any arbitrarily chosen unit z-adic but if is replaced by the unit (z + 1)-adic a, where a is one of i, j, or k, the left side is simply post-multiplied by a whereas the value of the right hand side depends on what unit z-adic is constituted by the terminal z

unit vectors of a. Thus, in general, ®{ °K ®

2=1

i.

J and its set of

equivalent post-factor operators ( 2'1 ®{ °K ® 2.1 { must represent a branch of F( ) other than that represented by °K ®; the two branches fuse for z-adic arguments. There may be many branches of F( ) because a post-factor branch is derivable for each allowable value of z in the way just given and from each member of an equivalent post-factor set a new prefactor branch may be constructed. Figure 2.8-3 shows one side of the family tree of a linear function F( ) of net rank x - 4 "basically for dyadic arguments." We suppose

n2

that neither of the ultimate parent operators XF2 : and : =F2 is factorable; that is, there is no value of s < 2 such that the prefactor operator can be

written in the form °K ®41 : or the ",,.factor operator in the form : 41 ® 2L. These two operators are the normal operators associated with

) for dyadic arguments. In the diagram each branch of F( ) is indicated by a double line labeled with its parent operator. Single tie F(

lines join pairs of branches at the rank of argument for which they fuse. Thus, for tetradic arguments, :'X = :FZ : e® ® +X xF2 1

I+

'X0`{ Ff :81)

not

2

a

6

4

Sec. Postfactor Branch I IC I81 Cl

srU 111 ('f::'1110

:IA1a

IA2 ('t i (rt::'111 O

See. Prefactor Branch IA

Sao. Postfactor Branch II

UIi O(ars:1'1 1

IIIA

(w

f 1=

Fiouaa 2.8-3. Family tree of a linear polyadic function basically for dyadics. 64

VECTOR ALOBBRA

55

TAiI.E 2.8-2. PARENT OPIDRATOR5 rOR BBAwass or FVNCvON F( ) DIAORAMMRD

IN FIa. 2.8-8 Patfactor Branch (Minor Image of Fit 3.3-3)

Prefector Branch (Shown in Fla. 2.3-3)

r T2

Primary

'F,:

Primary

3

:'F,

=

{'1:'F,)

I

3

1

13

I,

nI

14

IA'

IA 6

41

_____ _

©([61; [sh: I])©101)

1A1

iE

IB

[101 z ['F,:'1 ] 1®

II

®1'F,:'1)

IA 1'

(101 ® { ['1: F,] ;'1))®

3r

'F,1?161)

IB,

1'-T+

1+

31

IIA

1

n=

IIA'

11'1 ® ['F3:'1])(D I

III

II'

3( ., 1 i I

III

11,1:'F:)®

O ('F,:101)

r'-14

and ® {=F3 :'1 } becomes the parent of Secondary Post-factor Branch H. For pentadic arguments, using this new branch,

r _,+

I

+

iX p 1sF3 :'1} _ 'x ® Vol ®{aF2 : 3111 = Vol Q+ 61

T

1

1

14

{`F2 :I'll] 1 (1) $x

and [101 ® {=F2 : '111 ® becomes the parent of Secondary Pro-factor

56

VECTOR AND POLYADIC ANALYSIS

Branch IIA. The diagram is drawn on the supposition that none of the parents of the secondary branches is factorable; this is not untypical as Example 2.8-8 shows. When all the parent operators are thus unfactorable, F(2X) is single valued and the degree of multiplicity of values of F( ) for arguments of higher rank is, in general, the same as the number

of vertical tie lines for arguments of that rank; for example, F(°X) is eight-valued, four deriving from the diagrammed Primary Prefactor Branch and four from the Primary Postfactor Branch.

Example 2.8-9. In Mg. 2.8-3 let'F2: = Ajij:. The parent operators that act on tetradica are then: Ajij

a) "Fs

© A(ijiijii + ljijjij + ijikjik + ijjijji + ijjjjjj

©(`Fs : 4)

+ ijjkjjk + itjkijki + ijkjjkj + ijkkjkk) which form an equivalent pair for tetradics.

A(ijiji + ijjjj + ijkjk)

b) : f`F, : "II 41 1

13

(

I

= A (ijiiIji + jjijiji + kjikiji + ijjujj

jjJj j + kjjkijj + ijkiijk + jjkjijk + kjkkijk) O which form an equivalent pair for tetradics. >-ls

c) : =Fs = :Aijj

41--1 Is (61 : `F:) Q = A{iijiiij + ijjijij + ikjikij + jijjiij + jjjjjij 1

+ jkjjkij + kijkiij + kjjkjij + kkjkkij) ( which form an equivalent pair for tetradics.

VECTOR ALQRBRA

57

81

d) {'1

A(iji + jjjl$ + kjkij)

: `F2}

.

h

© ({°1 %'F:{ : °1] = © A(iijiiji + iijjijj -F iijki)k + jljijji

+ jijjjjj + jijkjjk + kijikji + kijjkjj + kijkkjk) which form an equivalent pair for tetradics.

If the tetradic argument is (ijij + jiji) the four resultant values are:

a) Ajij b) Ajji

c) Aijj d) Ajji

Here branch I fuses with the corresponding branch derived from

r7a `Fi .

If the argument is the triadic Bijj, the only equivalent pairs are 1-----I3

`F2: and

`F=:'1{

and

r ..b "F1

s

MI! and {'1 :1P,

and the two results are AB j j and zero. 2.8-6 Polyaross products (Three-Dimensional Systems) Polycross multiplication of poigads is defined by rules analogous to those of polydot multiplication:

abxcd = (axe)(bxd) = cdXab (2.86-1) abcX49 - (axe)(bxf)(cxg) -efgxabc (2.86-2) and for s-ads,

abod... x'efgh... _ (ax e)(bxf)(cxg)(dxh) ... (-1)' efgh ... X' abed 9 ..

(2.SG-3)

VECTOR AND POLYADIC ANALYSIS

58

It is distributive, commutative only if the multiplication sign is of even

rank, and in general is not associative. It is clear from the definition that a polycross product of two polyads of the same rank as the multiplication sign is zero if the mth direct factor of one is parallel to the mth direct factor of the other. Expressions such as ab x cdef are to be read as (ab X cd)ef; each factor must be of at least the rank of the sign. The q-cross products of the polyadics "K and 'Q are found by first writing the polyadics q-adically as in equations 2.8-12,

"K =

n-QA," _ m

in

'Q =

s

p-QB,,,

'"a. _ E e p-9b,

and then forming the sums

"K X° PQ = E E : QA," X° p'Qb. 4

PQ X "K = E FIn Qa X WI

Q -'B..

(2.86-4)

.

The rank of the products p + n - q. The scalar of the double cross square of a dyadic is twice the sum of the products of its characteristic numbers taken two at a time: 1

: (K x K) = X1x2 + X1X3 + a2X

(2.86-5)

This is the so-called second scalar invariant of the dyadic K. By multiplying out, it is easily verified that (K X K) : K = 6 det K

(2.86-6)

and that

=IdetK

(2.86-7)

A formula for the reciprocal of a dyadic follows from equation 2.86-7:

J(KXK)/det K f

(2.86-8)

K^' _ (K xx K) /det K 2.9

Versors: Rotation Operators (real vectors)

Suppose i, j, k, and a, b, c are two distinct sets of mutually perpendicular unit vectors, both chosen to be right-handed so that by rotation one

VECTOR ALGEBRA

59

set of axes may be swung into coincidence with the other. In these circumstances a, b, c are linearly related to i, j, k:

a b

aa aa, aak

c

ae{

ab{

abj abk

acj ack

i j

(2.9-1)

k

where the a's are the set of direction cosines and are related by the following equations n

aan

abn

9

x

ani =

E am aba x

Eani anj x

1

x

anj =

n

aan aen

ank == 1 n

E ax: ask = x

(n - a,b,c)

4

Z

n

(n = i,j,k)

a!,.

n

x

n

(2.9-2)

abx a,n = 0 (n = ij,k) anj ank = 0

(n = a,b,c)

(2.9-3)

The determinant of the a-matrix is -l-1. Consider the two different vectors of equal magnitude

V = vai + W = vya +

v.k

(2.9-4) (2.9-5)

v.c

= i(v--aai + vyabi + v.aa) + j(vzabj + vvabj + v,aoj) + k(v.am + vabk + vsaek)

(2.9-5a)

Geometrically speaking, W may be obtained from V by the same rotation as that required to swing i, j, k into a, b, c. A dyadic for use as a prefactor to perform the conversion of Y into W is, by inspection,

R;; t = ai + bj + ck

(2.9-6)

If used as a post-factor, it performs a rotation through an equal angle in the reverse sense :

V

(2.9-7)

Its transpose is also its reciprocal: 1

R:R=1s=3 Dyadics of this type are called versors.

(2.9-8) (2.9-9)

60

VECTOR AND POLYADIC ANALYSIS

The vector of R is

(R)v - a x i+b x j+c x k (a,j-aai,)i+(aaa.i)j+(ao;-a.j)k

(2.9-10)

(a0,-a ,)a+(a.e-'a,c)b+(aei-af)c The first of the two equal expressions is obtained when equation 2.9-1 is used to eliminate a, b, c; the second, if 1, j, k, are eliminated. If R operates on the first, the second is given; if R operates on the second, the first is given. Here, then, is a vector unchanged by R; physically, it must parallel the axis of rotation; vectorially speaking, it must be a multiple of a characteristic vector of R associated with the characteristic number unity. If we let r = B(R) . be this characteristic unit vector and

choose any two unit vectors p and a so that r, p, a is a right-hand n'utually perpendicular set, we can, by use of 2.9-10, express R as given

by 2.9-6 in terms of r, p, a. The result is 2.9-11 if 8 is taken positive when the rotation would advance a right-handed screw in the r-direotlorr; alternatively it may be expressed in terms of r and 8 alone:

R,,e - Ri;;' - rr + (pp + as) cos 0 + (op - pa) sin 0

=rr+(! -rr)cos0+1 x rsin 8

(2.9-11) (2.9-11a)

Equation 2.9-11a leads to

2r sin e

(2.9-12)

hence r is opposite in direction to (R,,); that is, B must be taken negative if rotation is positive in the sense specified. The scalar of R is(R,,#)

a = 11 + 2 cos a8 = a.i + c , + aw

yp

By using 2.9-12 and 13:

8 - 2 tan -'j I (R,,1)V f Al + (Rr.#)s){ In standard nine term form: Rr,r

(2.8-13a)

Ri%' - ar.di + auij + aadk +

ct.ji+aajJ+cr;jk+

(2.9-14a)

a.rki + aukj + akk a.,aa + aajab + a,iac + a.;bc + ab;bb + a, jbc + a.koa + abkcb + adcc

(2.9-14b)

VECTOR ALOEBRA

61

whence it appears that R shares with (R)r the peculiarity of having the same coordinates with respect to i, j, k and a, b, c. It should be observed that the matrix of the eoefficients in 2.9-14 is the transpose of the matrix in 2.9-1. From 2.9-14a and b, the determinant of the coefficients is seen to be ±1. The necessary and sufficient condition for a dyadic to be a versor is that its determinant be + 1 and ite transpose be its reciprocal. The following special formulae may be noted: R,,R,

1

2rr - I

for n even

(2.9-15)

for n odd

(2.9-16) (2.9-17)

Rr,(+")

If a and b are any two unit vectors R,.#

(2.9-17a)

where

p = a x b/sin(/2)

(2.9-17b)

This shows that any versor may be factored into two biquadrantal versors. The validity of formula 2.9-17a is easily established by carrying out the operations indicated by

taking i equal first to a unit vector perpendicular to a and b, and second, equal to any unit vector in their plane.

By use of 2.9-17a the dot product of any two versors, say Re,, and R,,O, may be found. If r and p are the same, formula 2.9-17 applies; if

they are different, let e - r x p/sin (cops' r-p), then Rr,, = if a is a unit vector chosen so that

a

xc=rsin(6/2)

and if b is a unit vector so determined that

cos (0/2)

c xb=psin(4/2)

VECTOR AND POLYADIC ANALYSIS

62

one has also RP,. =

Then Rb,r'Ra.r

(2.9-18)

because R,,Sr is the idemfactor by (2.9-15). Any complete dyadic may be expressed in the form

K=Aai+Bbj+Cck

(2.9-19)

by suitable choice of a, b, c and i, j, k (cf. §2.7, p. 17). It is possible so to choose i, j, k, a mutually perpendicular right-handed set, that a, b, c are a determinate mutually perpendicular right-handed set not necessarily the same as i, j, k and further, that A, B, C are either all positive or all negative. Evidently i, j, k and a, h, c may all be supposed unit

vectors. We may, by inspection, always factor K into a versor and either of two symmetric dyadics

K = (ai -}- bj -{- ck) (Ail + Bjj -1- Ckk)

(Aaa + Bbb + Ccc) (ai {- bj + ck)

(2.9-2t1)

The versor is the same in either method of factoring, but the symmetric factors differ from each other in the same way that the vectors V and W of equations 2.9-4 and 5 differ from each other; one has the same

orientation with respect to i, j, k that the other does with respect to a, b, c. The operation of rotating a dyadic, that is of deriving from a dyadic K known in terms of i, j, k a different dyadic L which is functionally related to a, b, c in the way that K is to i, j, k may be expressed by L=

`R,',e:K

(2.9-2I)

The first form is easily derived by considering R in the form of 2.9-6 and K in the nine term expression in i, j, k. The second is true if

4R,,o = aaii + abij + acik +

baji + bbjj + bcjk + caki + cbkj + cckk

(2.9-22)

The tetradic versor 4R,,o has many properties reminiscent of those of

VECTOR ALGEBRA

63

R,,#. Its scalar, obtained by inserting the double-dot in the tetrads of 2.9-22, is 2

4

2

2

( R,,e)D = aai + ab, + aek + 2(a,cab; + aaiaok + ab;aek)

(2.9-23)

_ (R-,e)s = (1 + 2 cos 8)2 The analog of the vector of R,,e is the dyadic formed when a double-cross is inserted centrally in the tetrads on the right of equation 2.9-22: (4Rr,e) y

= (Rr,e) v(R,,e) v = 4 rr sin2 0

(2.9-24)

The unit dyad rr is not affected when operated on by any tetradic versor for the axis r; this expected result is easily seen by taking K = rr in 2.9-21 and using 2.9-11 as the expression of Rr,e. Thus for every tetradic versor about r, rr is a characteristic unit dyad associated with the characteristic number unity. The column dyadics of 'R,,# in the r, p, a set of unit vectors (cf. Eq. 2.9-11) are, 4R,,e

rr = rr

rp Ar,e = rp cos 0 + rs sin 0 4R,,e : rs = rs cos 0 - rp sin 0 4R,,e : rp = R,,6

4R,,e : pr = pr cos 8 + sr sin 8 4R,,e : Pp = Pp cos' 8 + (ps + ep) cos 0 sin 8 + as sin2 8

4R,,e : ps = Ps cos2 8 - (PP - as) cos 0 sin 8 - sp sin2 0 4R,,e Sr = sr cos 0 - pr sin 0 4R,,e sp = sp cos2 8 + (as - pp) sin 0 cos 8 - ps sin' 8 4R,,e

as = as cos2 8

These results enable us to write the tetradic versor in the form analogous to 2.9-11. Each of the principal dyadics is symmetric. The "double-dot magnitude" of each row and column dyadic is unity. The determinant of the versor is unity. The tetradic versor does not affect a dyadic versor about its own axis; whatever may be the angles of the two : Rr,e = R,.e = Rr,e+d

R,.m

Re

Rr,-e = Rr,

(2.9-25)

VECTOR AND POLYADIC ANALYSIS

64

This is to be expected because R,,# is a function only of rr and scalars and should not be altered if rr is not. The array of the coordinates of 'R,,, in i, j, k is given in Fig. 2.9-1 in terms of the a's of equation 2.9-1. It is the transpose of the a-matrix in terms of ii, for the linear transformation expressing aa, ab, ac,

ij, ik,

.

as

ii

ab

ij

ac

1k

ba

bb

ji

= IlaII

ji

be ea cb

jk

cc

kk

(2.9-26)

Id

kj

which is analogous to Equation 2.9-1.

Versors for polyadics of ranks higher than two are readily written down by analogy and have properties analogous to those of R,,, and `R,,, ; the rank of the versor is twice that of the n-adic on which it is to operate. ii

ij

ik

ji

ji

jk

ki

kj

kk

1i

asi

a«abb

a.taet

abiaoi

abi

abtaei

aseasi

aetabi

aet

ij

asiasi

astabi

asta.i

zbiasf

abiabi

abuts

aetasi

asiabi

ik

asiaab

a,tabb

as/aek

abiaak

abiabk

aeiaab

ae a

assay aetaeb

ao/aci aaiae/

ab/a.i

a; abi

aHaak abfaei

asiasi a.faai a.laab

avow.

abbaei abbaei

asbasi

aababi

aeWt.i

adasi

a.babi

**Wei

abbaek

aeba.b

aebabb

aJ

ji ij

jk ki kj

kk

aaiaet

a; f ayaak aaba.i aabay a4k

aaja1j a.iabb aakabt

a.babi aababb

ab/aa/

abiaak abbaai aabaci abkaai a.kaek l abbaak aaiaeb aakaei

abiabb abbabiabbabi ask

abiaei abuse

aaiabi asiaib

aelasi aei

Figure 2.9-1. Coordinates of 'R,,, in i, j, k.

2.10 Invariants-of- Vectors and Polyadics

A scalar invariant of a tensor is a scalar number, determinate when the tensor is given and independent of the way in which the tensor is specified; in particular, for the class of tensors treated in this book, its value is unaltered when the axes with respect to which the polyadics are defined are changed by linear transformations of the type given

65

VECTOR ALGEBRA

in Section 1.2. It is a property of the tensor per as and every adequate description of the tensor must directly or indirectly furnish all of its invariants. For every polyadic, the magnitude I 'K I is a scalar invariant: the positive square root of the sum of the squares of the absolute values of the coordinates--

I°KI _ + K

(2.10-1)

v

No polyadic other than zero has a zero magnitude. In the case of a vector V no scalar invariant distinct from 1 V I exists. A dyadic K has, in general, as many independent scalar invariants as it has dimensions. The characteristic numbers X1, X, , X, , X4 , . X are scalar invariants. It is, however, more usual to take certain functions of the X's as the primary invariants, largely because these functions are more simply expressable in terms of the coordinates of the dyadic than are the X's themselves. When a dyadic is fully defined its coordinates with respect to any given set of mutually perpendicular unit vectors are easily found (cf. 12.7). In the following list the standard invariants are given in the order in which they are commonly named, viz: K1, the first invariant; KB , the second invariant; etc. The first and the last are often

given special names: the "scalar" and the "determinant" of the dyadic, respectively.

I. K1- K8 = K : 1 = X1 + X9 + ... + aN

(2.10-2)

This is the sum of the diagonal coordinates of the dyadic in the canonical form (Eq. 2.7-7 or 2.7-8) II. K, = X1X, + X1X3 +

X,X1 + X,X +

+ 4_14 N

(2.10-2a)

This is the sum of the principal two-rowed minors of the determinant formed by the coordinates of the dyadic in canonical form (Eq. 2.7-7 or 2.7-8) Ill. Ka = X1X2X. + X1)12)14 +

.

(2.10-2b)

= sum of all the different products formed by taking the X's three at a time. This is the sum of the principal three-rowed minors of the determinant of the dyadic.

N. KN m det K = X1X,X2 ... XN

(2.10-2n)

66

VECTOR AND POLYADIC ANALYSIS

This is the determinant of the coordinates and is usually called the determinant of the dyadic. The characteristic numbers and the primary invariants of the adjoint of K are the conjugates of those of K. For a Hermitean dyadic these invariiants are all real numbers because the characteristic numbers X of a Hermitean dyadic are all real, (cf. Eq. 2.5-14). The characteristic numbers of the mth single dot power of K are the mth powers of those of K.

The characteristic numbers X of a dyadic K satisfy the characteristic

equation AN - K8XN-1 + K2XN-a ... + (-1) NKN = 0 (2.10-3) The left hand side of this equation is simply the formal expansion of the determinant of

Z=K-X1 which is a "singular" dyadic (i.e annihilates vectors of at least one direction (see p. 12) ). To demonstrate the singularity of Z, observe that if v is the (as yet unknown) normalized characteristic vector with which X is associated, one has by definition (Eq. 2.5-13) v

(2.10-4)

The determinant of a singular dyadic is zero: Using Z as an example, let

the basic set of mutually perpendicular unit vectors (v, p, q, chosen to include v; then Z = a..vv + a.pvp + a,avq + ... +

ap pv + apppp + avepij +

+

) be

(2.10-5)

a,qv + agpgp + aggq;I + ... + and we see that

is necessary in order that equation 2.10-4 be true. Thus det Z has a complete column of zeros and therefore vanishes. Usually K is given in the form typified by equation 2.7-8:

Av,ji + A,Jjj + A,,,jk +

A.ski + A, kj +

Akk

(2.7-8)

VECTOR ALGEBRA

67

In this i, j, k basis the coordinates of Z are, except on the diagonal, the same as those of K; it is seen from equation 2.10-4 that the diagonal coordinates are (A.. - X), (A,,,, - X), etc. Thus, when K is given, det Z is known for the same frame of reference in terms of X. After finding X1 , X2 , the corresponding characteristic vectors may be determined as follows in terms of the basis in which v1 , vs , K is known. Again let us use equation 2.7-8 as the example. Let

vi = x1i + ylj + z1k, then K

v1 = (A xi + Axvyi -I- Ax.z1) i +

(A,.xi + Al,,,yi + A,.zi) j +

(2.10-6)

(Axi + Avyi + A..z1) k In order that this shall be X vl as required, one must have separately,

(A..-X1)x1+Any, +A=.z1=0 (An - Xj)yj + A,.zi = 0

(2.10-7)

A.=x1 + A.,,y1 + (A.. - X1)zl = 0

That these three equations are not all independent is known from the fact that det Z = 0. Any two may, however, be solved for yl/xl and zi/xi . In result one finds the rule: the coordinates of v1 are in the ratio of the

cofactors of the corresponding terms in any row of the matrix of the coordinates of Z for X = X1

.

x1:y1:z1 = Co(Au X1):Co(A,r=)

(2.10-8)

= Co(A1.,):Co(A.):Co(A.. - X1) The co-factor of a term Am in a matrix is the determinant of the terms of the matrix remaining after striking out those in the row and column containing A.,, ; the sign of the determinant is taken as (-1)` where m is the number of the row and n of the column in which A.. occurs. The characteristic equation (2.10-3) of a dyadic is satisfied if the dyadic itself replaces X provided that the powers indicate single dot powers of K (e.g., if K' = K K K, etc.) and the last term on the left is multiplied by 1. The result is known as the Hamilton-Cayley equation. By its use the higher single dot powers of a dyadic may be expressed as linear combinations of the first (N - 1) powers. Consequently

68

VECTOR AND POLYADIC ANALYSIS

a single-dot polynomial in a dyadic K is at most of (N - 1) degree in K.

A polyadic of even rank t'K in N dimensions has, in general, Na characteristic numbers A that satisfy

dot (t'K -- At'1 j = 0

(2.10-9)

which when multiplied out will be an equation analogous to 2.10-3 but of the N' degree in A. For these values of A the "characteristic 2q-adic"

16z - "K - A41 (2.10-10) is singular with respect to q-dot multiplication. From the N' values of A, one may construct N' scalar invariants analogous to those given above for dyadics. The analogous Hamilton-Cayley equation for a 2q-adic is '°K1" - -

@Vg"K'"-'

+

t°Ktt°K'r'-t

... (-1)rr

a0

(2.10-11)

wherein the powers of t'K are q-dot powers. A dyadic is, of course, a 2q-adic for q = 1.

III. Vector and Polyadic Calculus (For real quantities unless otherwise specifically noted) 3.1

Differentiation and Integration with respect to Scalars.

Differentiation and integration of vectors and polyadics with respect to scalars follow the rules of ordinary scalar calculus except that care must be exercised to preserve the order of the factors of a product. For example,

(cxb)=cx L+ d(ab) d(abc)

xb

t1

= a db + (do) b adb + b da _ (do) be + a(db)c +.ab do

r-11

I-f1 bcda+acdb+abdc

(3.1-1)

(3.1-1a)

(3.1-1b)

Notice that we may avoid parentheses by the transpose notation. The solutions of linear differential equations in which the independent variable is a scalar and the dependent variable is a vector, or a dyad, or other tensor, are obtained by the usual procedures of scalar calculus. Constants of integration are, of course, tensors of the same order as those to which they are added. The differential dR of a vector R is a vector and, in general, has a direction other than that of R; similarly the differential dW of a dyadic W is a dyadic and, in its nine-dimensional space, generally has a direction different from that of W; and so on. 3.2 Differentiation with Respect to Vectors and Polyadics when a scalar S is a function of a vector R, its vector derivative with

respect to R is usually denoted by VS and is defined by the equation

dR VS = dS

(3.2-1)

Often the independent variable is clear from the context and need not be indicated in the derivative symbol. If identification of the independent variable is neoessary, the usual derivative symbols of calculus are used.

Thus VS is sometimes written as dS

dR 69

70

VECTOR AND POLYADIC ANALYSIS

or, if S depends also on other variables, the usual partial derivative notation is used. The usual derivative symbols will be used when the independent variable is a polyadic. The symbol V is read "del" in the more usual American usage; it is sometimes called "nabla". When S is a function of a dyadic W, its dyadic derivative with respect to W will be denoted by dW and defined by dW:dW = dS

(3.2-2)

From the respective equations of definition VS is seen to be a vector and dS/dW to be a dyadic. If, for example, S is the temperature in a solid conducting heat, dS is the increment in temperature if the point of observation is changed to one removed from the first by the differential displacement dR. Suppose the absolute magnitude dp of dR is held constant so that dR = r, dp where r, is a unit vector of adjustable direction. It then follows from 3.2-1 that, if 8 is the angle between VS and r,, r,- VS = aS = 1 VS 1 cos B

(3.2-3)

P

is the scalar rate of increase of temperature with distance measured in the direction of r, . The directional derivative given by 3.2-3 usually varies in value with the direction of r, ; its greatest value (if there is a greatest value) must occur when r, is chosen to have the same direction as the vector V S because the cosine of the angle a between r, and V S will then be unity, its greatest value. The directional derivative then becomes I VS I . The vector VS, when it exists, thus has at each point the magnitude and direction of the greatest rate of increase of S with change in position in the space for which R is the position vector. It has therefore quite logically come to be called the gradient of S with respect to R and is sometimes written "grad S". For analogous reasons d W is called the gradient of S with respect to the polyadic °W. The same notation and terminology are used for the vector and polyadic derivatives of vectors and dyadics. Thus, for example,

dG

(3.2-4a)

dW:dW = dG

(3.2-4b)

VECTOR AND POLYADIC CALCULUS

dW:A =dL

71 (3.2-4c)

and W are called gradients with respect to the wherein VG, indicated independent variables. If those equations of definition are to be consistent with the meanings previously assigned to dot and dcuMedot products, VG must be a dyadic; dW , a triadic; and dW , a tetradic: dW,

the first derivative symbol has the same rank as does the independent variable. Observe that the order of the factors on the left of these ecquations may not be changed without altering the meaning. Formally, the operator V behaves like a vector in that when it multiplies a scalar S to form VS the result is a vector and when it multiplies a vector G the result is a dyadic. From equation 3.2-3, it appears that the "coordinates" of V with respect to any agreed set of Cartesian axes i, j, k may be obtained by the usual formulae for resolving a vector into its components (cf. § 2.2 below Eq. 2.2-9) :

i v = a(ax ) j , v = a(

)

(3 . 2-5)

cly

k.v a Z

whence, V is formally the vector

v(

)

+kaax

(3 . 2-6)

Further, it has in common with other vectors the property that its value is independent of the origin and orientation of the coordinate axes which

may have been adopted in a particular problem. This is true also, in general, of the quantities formed by operating with V. For example, if in equation 3.2-1 dS is, as supposed above, the difference in temperature

between two points displaced one from the other by dR, the physical quantity measured by VS surely is independent of the location of the origin of R and of the orientation of i, j, k. Quite analogously, the operator dW behaves like a dyadic. Its coor-

VECTOR AND POLYADIC ANALYSIS

72

dinates with respect to the nine dyads of the system i, j, k are d(

= a( ) ' dW 84,. )

ij:d(dW) = a(8A,,) )

ik.d(

dW

.

(3.2-7)

a(

M.

if we suppose W is in the form 2.7-8 so that

d W = ii dAs, + ij dA,,, + ik dA,. +

ii dA,,,+ii dA,,,+jkdA,,.+ Id dA. + kJ dA. + kk dA.

(3.2-8)

Thus d(

)

dW

=ii a(aAr,)+i) a(aA,r)+ika(M.) +

ji a(aAr,) + jj a(aAr,) + jk a(aAr.) + kia( ) +kja( )+kka( ) If the operation d W : d(

dW

)

(3.2-9)

is formally carried out using 3.2-8 and

3.2-9, one obtains aA M.

dAu

+

sA ,r

dA +

aA ,,

dA, +

sA r,

dA,,, + - ..

which, if S is inserted in all the parentheses, is the usual expression of scalar calculus for the exact differential of S when (3.2-10)

In essence, equation 3.2-10 states just what is meant when we say S is a

function of the tensor W so that dW : dW has a meaning consistent with scalar calculus as it should. The first derivative with respect to a polyadic may be regarded as a

post-factor linear operator that converts the different ;al of the inde-

VECTOR AND POLYADIC CALCULUS

73

pendent variable into that of the dependent variable. From this point of view (cf. j 2.8-5), equations 3.2-4 are seen to be written with the operators in "normal form"; that is, with the rank of the multiplication sign the same as that of the argument. As with other such operators it is sometimes possible to find an equivalent operator with a multiplication sign of rank less than the argument. Thus, if 3.2-4c can be factored into `1

a

d

of equation

'Z the equation may be written

dL - dW :'1 'Z - dW 'Z

(3.2-11)

because then

The second derivative of a function of a polyadic °W with respect to the polyadic is the gradient of the gradient of the function. It is a postfactor linear operator that converts d°W) into the differential of the first derivative : d

[d W] = a4w

(deW)(d}W)

(3.2-12)

When °W is a vector, one usually writes the second derivative in terms of V, thus V V'Y would then replace the expression at the right of the multiplication sign. The second derivative symbol itself behaves formally like a centrally symmetric polyadic of rank 2q if there is but a single independent variable of rank q. The rank of a partial derivative symbol is the sum of the ranks of the variables with respect to which differentiation is indicated. Differentiation of composite functions is closely analogous to scalar calculus although, because certain of the derivatives may be polyadice, attention must be paid to the order of the factors. For example, let W be a function of V and let V itself be a function of R and S, and suppose all the functions are differentiable. Then,-

dW = dV dW

(3.2-13)

(3.2-14)

dW = [dR (8V Is

+

dS ()]

dV

(3.2-15)

VECTOR AND POLYADIC ANALYSIS

74 and

(8.2-16) (aR }8 (aR)s dV The rules for differentiation are essentially those of scalar calculus.

They are illustrated in Section 3.5 below.

3.3 The Divergence and Gauss's Theorem Physically, the divergence of a vector or of a polyadic quantity is the rate of escape per unit volume, from the vicinity of a point, of the quantity of which the vector or polyadic represents the flux density. Let us suppose G is the local mass velocity of a fluid and compute the efflux of fluid from a sphere of radius ip with 0 as a center. Let r

be a unit vector of variable direction extending from 0. Thus rip is the. position vector of a point on the surface of the sphere and r is a unit outward normal to the surface at this point. If G is the mass velocity at 0, then at rip the mass velocity is, G + rip

VG

and, per unit surface area of the sphere at rip, the mass rate of efflux is:

r- (G+rip VG) The total mass rate of efflux from the whole spherical surface is therefore:

ip r VG-r) dS

(3.3-1)

if dS is the scalar element of spherical surface at rip and the integral is taken over the whole surface of the sphere. In this integral G and VG are not functions of S (or of r); the values of these quantities at 0 do not depend on the location of a point on the sphere. It follows that:

For every element dS there is a diametrically opposite element of equal area for which r G is reversed in sign because the sign of r is reversed : thus the integral over one hemisphere just cancels the other. The remainder of integral (3.3-1) is: 3

(ri

r:

r

dS

VECTOR AND POLYADIC CALCULUS

75

where r, , and r, are unit vectors so chosen as to form together with r a

mutually perpendicular set. From equation 2.7-13, the quantity in parentheses is the scalar V G of the dyad VG and the value of integral 3.3-1 is therefore:

(V.G)bv

1

3

e

(3.3-2)

where by is the volume of the sphere of radius Sp. The divergence of a vector at a point is therefore the scalar of its gradient at that point:

dive = V.G

(3.3--3)

The notation V G is that usually used whatever the rank of G, A vector or polyadic for which the divergence is sero is said to be "solenoidal." From the present point of view, Gauss's Theorem, s

dS = cc v X dv

jr

(3.3-4)

when X may be either a vector or a polyadic, is almost obvious. On the left the integral is taken over the complete surface which bounds the

volume over which the right-hand integral is taken; on the left X is evaluated at dS, on the right V X is evaluated at dv. The unit vector n is the outward normal of dS. Expressed in scalar calculus:

aG. aGr aG. v .G a ax + ay + az

(3.3-5)

where x, y, z are any set of Cartesian coordinates and G., G,, G. are the magnitudes of the components of G along the coordinate axes. Equation 3.3-5 is readily derived by forming the scalar product of the right side of equation 3.2-6 and

C - iG. + 3Gv + W.

3.4 The Circulation, the Curt, Stokes's Theorem. The circulation of a vector G around a closed path is, by definition the value of the integral:

i dC is an element of the path and G. is the value of G at that ele-

76

VECTOR AND POLYADIC ANALYSIS

Figure 3.1. The unit vector n projects upward out of the paper.

ment. The circulation of G per unit enclosed area around the boundary of an infinitesimal plane element of area, taken perpendicular to the local axis of rotation at a given point, is the magnitude of a vector called the curl of G. The direction of curl G is the positive direction of the local axis of rotation. Curl G is thus a measure of the local swirl of the flow represented by G. If G is the linear velocity at a point, curl G is twice the local angular velocity. Suppose G is the local mass velocity of a fluid. In a plane perpendicular to an arbitrarily selected, but fixed, unit vector n, construct a circle of radius dp with its center at the point of interest. In terms of the values of G and VG at the center, the circulation around this circle is:

C=

(3.4-2)

where r is a unit vector of variable direction in the plane of the circle. Let the direction of integration be such that r swings around the circle as would the head of a right-handed screw when the screw advances in the direction of n. In the first place:

f Sp

0

(3.4-3)

VECTOR AND POLYADIC CALCULUS

77

because G is not a function of r and in integrating around the circle each

dr is cancelled by the oppositely directed dr' of equal magnitude diametrically opposed to it. Further, the remainder of the integral 3.4-2 may be written:

r,

C

(3.4-4)

if r, is any unit radius vector of the circle other than r, because the value of 3.4-2 is the same whatever the radius at which integration begins and ends. If r, is taken perpendicular to r and on the counterclockwise side of it, see Fig. 3.1, one has:

dpdr = r,ds apdr,

(3.4-5)

-rds

where ds is an element of the circumference. Equation 3.4-4 then becomes : X

C=2

(3.4-6)

X

_

ap r V G r, d8

_

(V x G) x

_

X

bp[r (V G + V G) r, - r, (V G + V G) r] ds

(3.4-6a)

ds

(3.4-6b)

(V x G) r x r, apt s

(3.4-6c)

2

_ (V x

(3.4-6d)

sA

In the above transformation 3.4-6b follows from 3.4-6a by equation 2.7-14; (c) from (b) by the vector identity:

axbc=abxc

(3.4-7)

and 3.4-6d from 3.4-6c because r, r, and n are so positioned by construction as to form a standard mutually perpendicular set of unit vectors and the integral a_ p d8 = aA 2

is the area of the elemental circle. From this result, equation 3.4-6d, the vector V x G which formally is the vector of the dyad VG is such that, its projection on the normal n

78

VRCTOR AND POLYADIC ANALYSIS

of the plane element gives the circulation around the element per unit enclosed area. If the element is so oriented that n has the direction of V x G the circulation around the element is the magnitude of V x G and is the maximum circulation for various orientations of the surface element at the given point. Physically, it is evident that this situation occurs when n has the direction of the local axis of rotation. Thus, it has been found that

curl G= V X G- dR x G

(3.4-8)

is consistent with the interpretation of V as a vector. If X is either a vector of a dyadic, Stokes' theorem states that:

i

i

x XdA

(3.4-9)

where the integral on the right is taken over the whole of the surface A which is bounded by the curve C around which the integral on the left is taken. The variable vector n on the right is the outward normal of the surface element dA. Neither the surface A nor the curve C need be plane. The proof of this theorem, which is almost obvious from the above discussion, consists essentially of building up the finite surface A by piecing together the results of equation 3.4-6a for a series of elements 5A. A vector or dyadic of which the curl is zero is said to be "irrotational". 3.5 Operations with V and dW

The operators V, V , and V x are all distributive; e.g., the divergence of a sum of vectors is the sum of their divergences. In a general way operations with them on composite functions resemble those with d/dx. This is also true of dW and the operators derived from it.

Gradients are usually easy to find from the equations of definition, Thus, if R is the independent variable to which V refers, one sees at once from 3.2.4a that VR = 1; and if i denotes, as usual, the vector unit in the direction of a particular x-axis, Vi = 0 because i is a constant vector. The close analogy to scalar calculus appears in the following

evaluation of V(R R) and V R E

R.RIR

:

d(RR) =2dRR=21RIdIRI

(3.5-1)

VECTOR AND FOLYADIC CALCULUS

79

whence on comparison with equation 3.2-1

V(R R) - 2R

(3.5-2)

VJRJ -R/JRJ=r,

(3.5-3)

where rl is a unit vector in the direction of R. The unit vector rl, unlike i, is variable in that its direction varies with R; to find Vr1 we may illustrate the rule for operating with V on the product of a variable scalar and a variable vector: VR

V(JRJri)

(VIRJ)rl*+IRIVr, (3.5-4)

r1rl -I- J R J V ri

\7 r1 =

1

11 - rlrA

Incidentally, comparison with equation 3.5-3 shows that the expression just found is also the second vector derivative V V J R J . If W is the

independently variable dyadic with respect to which dW is taken, a parallel set of results is obtained: dW

dW dW

(W:W) - 2W dW { W d

W$

WW

(3.5-6)

W/JWJ = wt, a unit dyad. -w,w:1

F-W

d2 l W I

dws

(dW) (dW)

dW

(3.5-7)

(3.5-8) (3.5-9)

If U and V are functions of R, (3.5-10)

N

^+

= V. VU + U. VV

(3.5-10a)

80

VECTOR AND POLYADIC ANALYSIS

because we have

d(U-V) = (dU)-V + U-dV

(3.5-11)

_ (dR- VU) - V + U-(dR- VV)

(3.5-12)

= dR-(VU)-V +

(3.5-13)

where in the last term of 3.5-12 is converted into the last term of 3.5-13 by observing that the dot product of the two vectors U and (dR - V V) may be written in either order. Comparison of equation 3.5-13 with the definition of a gradient gives 3.5-10. Observe that in equation 3.5-10a, the use of the transpose of V U and V V avoids the use of parentheses to indicate that V is intended to operate only on one of the vectors U and V. Further

N

V(UV) = (VU)V + UVV

(3.5-14)

because

d(UV) = (dU) V + UdV

= dR-(VU)V+ U(dR)-VV = dR-( VU)V + dR-(UV)V

(3 . 5- 15)

(3.5-16)

where the transformation of the last term is effected by treating V as a formal vector and using the following vector identities

a(b-cf) se ab-cf w a(b-c)f sm (b-c)af m b-(oa)f

= b-(ac)f an b-acf

with a = U, dR = b, V - c, V = f. Use of the form b-caf would imply that V was differentiating U(. a) which is not true in 3.5-15; use of b- cf - dR-UVV indicates that the differential operator part of V is acting on V alone but the vector part of V is entering the triad

ti

....

U V V as a prefactor of UV.' * This notation is due to Chapman and Cowling. Its internal consistency may be seen by formally writing out in nine term form the dyad UV, transposing it, and

forming the triadic UV V. Dot multiplying this triadic with dR as prefactor will give the dyadic UdV.

VECTOR AND POLYADIC CALCULUS

81

The operations of the preceding paragraph with V as well as opera-

ations with V- and V x are sometimes hastened by imagining V factorable into a scalar differentiator a and a hypothetical vector a so that, formally, V = aa. Then V(UV) = aa(UV) = a[(aU)V + UaVJ

ti

_ (VU)V + UaaV

ti

(3.5-14a)

_ (VU)V -l- UVV V. (UV) =

a.[(au)V + UaV)

(VU)V+

(3.5-17)

U. VV

V x (UV) = as x (UV) = a x [(aU) V + UaV]

= (a x aU)V - (U x a)aV

(3.5-18)

=(VxU)V-UxVV That the latter two results are correct may be shown by use of the equations of definition. It will be observed that one first operates with a exactly as if it were "d" and then with a as if it were an ordinary vector.

By following the rules of operations with vectors, any products containing a as a factor are then reordered so bring a and a together. This somewhat heuristic method may be extended to more complex cases. For example,

V(WV) = aa.(WV) _ (V-W)V + aWaV

(3.5-19)

The first term is perfectly understandable. To reduce the second term note that 0.0/ j a f' is unity, so that

aWaV = 6W.aVO.0/ 16 (3.5-20)

= a.w. vv-all 6 12

ti

[W.vVj.0O/1612

(3.5-21)

Now V, as has been pointed out on page 71 has coordinates independent of the orientation of the coordinate axes in use; therefore this is true of a and also of the unit dyadic 66/1 6 3'. The latter has coordinates

82

VECTOR AND POLYADIC ANALYSIS

independent of every rotation and must therefore be the idemfaetor I which is the only dyadic with such properties. Hence, from equation 2.8-45 and the definition of a double dot product,

ti

[W- VV]" = W:vv

(3.5-22)

and

W: VV

(3.5-23)

which may be proved correct by formal expansion of both sides.

3.6 Mixed Derivative Operators Inasmuch as V W is a vector, one may of course find its gradients W

V

V V W] and so on. Evidently the possible operators of order higher than the first are very numerous. Among them V V. V (

),

V [ V x ( ) ], V x V( ) and VV-( ) maybe particularly noted. The operator V- V( ), usually written V'( ), is the scalar Laplacian operator as may be easily seen by multiplying out when V is expressed by equation 3.2-6:

- C-+

V'( Th e operator d

:

dW dW 1a' aAs,

. j

(3.6-1)

x

[. dW]4'6 the nine term scalar expression

+

a'

82

+ aA.' +

(3 .6-2)

The operator V V ( ) is expressible in term s of two others: VV -( ) - V X [V x ( )] + V'( ) (3.6-3) as would be expected from the vector identity

b x (a x c) + if V is read for a and for b. Certain combinations vanish:

V-[ V x ( )] = 0 V X V( ) = 0

(3.6-4) (3.6-5)

VECTOR AND POLYADIC CALCULUS

83

Therefore, the curl of a vector or of a dyadic is nondivergent (i.e., is "solenoidal"); it is also true that an solenoidal vector or dyadic may be regarded as the curl of some vector or dyadic. The second equation

shows that a vector gradient has no curl (i.e., is "irrotational"); any irrotational vector or dyadic is therefore the vector gradient of some scalar or vector, respectively. 3.7

Maclaurin's and Taylor's Series

The analog in polyadic functions of the scalar Taylor power series is, if "Y is a function of 'X,

"Y = "Ko + [=X - -A) ® v+-K, + ...

+ {'X - ,A]x Nx

r+N:KN

+-

(3.7-1)

wherein 'A is a constant x-adic in the range of =X over which "Y is de-

fined and the several 'K's, which are constant polyadics of the ranks indicated, are appropriate multiples of the values at 'X = 'A of the successive derivatives of "Y with respect to =X. The conditions for the validity of this expansion are analogous to those found in scalar analysis; in particular, the successive derivatives of WY must exist at sX = =A. Differentiation of the series with respect to 'X gives

d"Y=

'

d X

zf ®'"'Kl+ {=X-zA}®{''Tr;+ (3.7-2)

which, as sX -- --A, reduces to the first term so that the coefficient of the linear term in 3.7-1 is, as in the scalar case, simply the value of the first derivative at =X = 'A. A second differentiation gives C

ds"Y 1 d ¢X d rX X--A - {`Tr: +' 1 } 2x

y+::Ks

(3.7-3)

and, again as in the analogous scalar case, we find that the result

d'''Y 2 d-Xd'X -X='A

,,+2 K2 = 1 C

(31-4)

satisfies 3.7-3. To make this clear, write V. for the dyadic derivative operator so that the second derivative is ] V: V. "Y1 and note that the transposer `Tr: 2x behaves like an idemfactor when it acts on the

VECTOR AND POLYADIC ANALYSIS

84

x-adically symmetric 2x-ad V:0:. In general, the result for the N-th term is

d'rX - 2A]_

dN "Y

(d xX)N

and dNIXX

-

X==A

=

:A]N

(d-X)N

r+NsKN

(3.7-5)

[(d zX)N

O

= 2X1 ®(`"Tr; +''1 } 27C ("Ti;

ax ... , - x)g O.

(:":Tra

0-1

+E

!

uc.+u1O 2a(lr u1(r-1)s (3.7-6) 1

.

of which the terminal factor behaves as N "0"1 when used as a prefactor dN in NX-dot multiplication of (d -X)N' the next to the last factor behaves

as (N - 1) '"21 with (N - 1)%-dot multiplication, and so on. Hence "+NZ KN

=

_L[ d[`Y] N! (d -X)NIX

(3.7-7)

Therefore, if the direct power series 3.7-1 does converge to `Y, the values

of its coefficients are formally given by the customary formula for the coefficients of Taylor's series. As in scalar analysis, the corresponding series with 'A = 0 is a Maclaurin's series. Questions of convergence for series like 3.7-1 are handled much as are those for series of complex variable terms in scalar analysis. Every term of 3.7-1 is a y-adic and may be expressed as a sum of unit y-ads. The coordinate of "Y along a particular unit y-ad, say , is the sum of the coordinates along the same unit y-ad of the several terms; all of these scalar series must converge simultaneously if 3.7-1 is to converge. If we consider any given term of the series it is clear that the coordinate of each y-ad of which it is the sum is not greater than the absolute magnitude of the term. It follows therefore, that if the polyadic power series 3.7-1 is "absolutely convergent" in any range of aX (i.e. is convergent when each term is replaced by its magnitude), then the series is convergent as it stands in that range. (cf. Thomas, "Calculus and Analytic Geometry, p. 637 ff.).

Appendix A: Summary of Notation and List of Formulas

General notation In the text of this book and in the list of formulas below light face italic type is used for scalar numbers, bold face italic type for vectors,

bold face roman type for unit vectors, and sans serif type for polyadics (tensors) in general. An exception arises in the cases of the "scalar", the "vector", and the "scalar invariants" of a polyadic: these quantities are denoted by the sans serif symbol of the polyadic with an appropriate subscript. A superior number or letter written before the symbol serves to indicate its rank; thus 2"K is a polyadic of rank 2n.

The index of rank is omitted in the case of dyadics (rank = 2); the index unity is not needed because a polyadic of rank unity is a vector and appears in a different type face. When, without specific definition,

the same letter appears in the same equation or context both as a vector (e.g. B) and as a scalar (e.g. B), the scalar is the magnitude of the vector. The multiplication signs used to indicate the three types of multiplication met are: Direct multiplication (page 15) : No sign; thus abc. Dot or scalar multiplication (pp. 7, 21) : , :, ; , and an encircled number or letter for a number of dots in excess of three, e.g., ®, 1'

Cross or vector multiplication (pp. 9, 57) : x, x, and, for more x 's than two or three, X' where the exponent indicates the number of x 'a. Powers of vectors and polyadics are indicated by an exponent as in ordinary algebra but, because it is here necessary to show the type of multiplication, the exponent is followed by the sign to be used; thus,

ab=aaaaa;

'K'O = "K (9 "K

K

An exception is made in the case of 7' which by long standing usage has come to mean V V. V.

In the lists below, the symbol or formula is followed by the number in Italics of the page on which its explanation or derivation is given. When .% pertinent formula appears in the later lists in this Appendix, its number is given in bold face. 85

APPENDIX A

86

Special symbols :-QA.,

24.

"rOB.. , 24.

'"Tr: , 28, 42, 2"Tr:

,

10, 11.

98, 42, 10, 11.

1, 12, 19,81, 1. g, 7.

=°1, 81, 2.

aw Q0p,28.

6,81.

R7;, 59,.17.

V, 69,

R,.#, 60, 18.

V x, 77, 84, 90.

4R,,/, 62,21.

75,73, 82.

(R, 7.

Markings and subscripts The particular letters marked below have in themselves no especial significance; they serve here to exemplify the meanings of the marks and subscripts. 1F, a = complex conjugate of a, a.

[3]

symmetric part of K.

R = transpose of K.

[13, 43]

[12, 41, 42]

transpose of the matrix 11 A 11.

[41

adjoint of K = transpose of the complex conjugate of K.

[13]

x

[13, 44]

K = antisymmetrio part of K. O

O

K, ('Q)0 = deviator of K, QQ from its scalar part.

[20, 89, 9b]

0

(QQ). - deviator of 'Q from its principal a-adic part.

[89, 91

n- xX

'K, "K = n-adically symmetric and n-adically antisymmetric parts of 'K.

[29, 40]

87

APPENDIX A 9,0

0,Q

"K),, ('°K), = deviators of '°K, q-adically written, from its pre- and post-a-adic parts. [421 Of-I [27, 38] aQ = backward n-adic transpose of IQ.

FT IQ = forward n-adic transpose of Q. - m.-th unit a-ad.

[27, 391 [08]

K,s = scalar of K.

[20, 65, 471

Kr = vector of K.

VIl

, KN = first, second, ., Nth scalar invariante of K. det K = determinant of the matrix of the coordinates

K, , K2,

[851

of K.

K = absolute magnitude of a, K.

a

Idemfactors and other operators aq--1

Let to * 1. 21

2.

p-0

A, and let K be an arbitrary real dyadic ;-

1 = ii+33 +kk

[10, 19, 81]

2*-1

2"1

= E8-0tn/8>

2"1

= E l.JQJ,

[311

19-1

9-0

[33]

9-0

4.:+1 ®r+:1 = 3(r5.

2n,

1*1

(0 < x < y, x + y even)

[891

(m ? n)

[811

(q + a even)

[39]

mQ ®I"1 = Q ®'"GQ = f.-1

6. "414 0Q

6a. asI 7.

= EP-0A, Q+*3P ` 'Q :*Q=="a,MLQ

3teI s r+ I ®QQ

Qrr.

[881

(q + a even)

= principal a-adie part of IQ.

[391

88

APPENDIX A

8. 22Ds. _ 21 - 3191,/2

+s1 9

9-1-61 OA

0

9. (qQ). = 22D ®4Q

(q + a even)

[401

(q +a even)

[401

= deviator of 4Q from its principal a-adic part. 0

0

9a. K = (2K)o = 4D,o :K

[401

0

9n.K=K-}Ks1

1201

e11

10. 2gTr; ®QQ = gQ = QQ Q 24Tri

[291

1F-1

10a.`Tri:Q=Q=Q 11. 2QTr-,

©,,Q

= QQ = QQ ®2QTre 2(9+n1

12. 2QTri =

I

9M

F79

13. 2gTre ^ 2gTrg

= 2QTrQ ee

=

2(4+e)1

[29,441

O

2(1

[431

= 2VTr;

[431

14. 4Trt = 24Tri e® 15.

2qTr1 «,4®= 2QTrQ

16. 2gTri ©

24Tr-j

[441

= :4Tr, = 24Tri Q®_ =41

= 241 =

2QTre

(m integral)

®2°Tre'

[441

17. R{;: = ai + bj + ck 18. Rr,e

[44]

[591

R;,°k = rr + (1 - rr) cos B + 1 x r sine

[601

where

r=- 2sin0 (axi+bx j+cxk) 1

=1

19.

19a. R,,,,, = 2rr - 1

(n even)

[611

(n odd)

[61]

R,,,

20.

where

P = a x b/sin(#/2) = 2 cos 1

[611

APPENDIX A

89

21. `R,,, = aaii + abij + acik + baji + bbjj + bcjk +

[62]

caki + cbkj + cckk

22.

[62]

`R,.#

Algebraic Expressions Let a, b, c, --- be real vectors with magnitudes a, b, c, - and. coordinates with respect to the right-handed rec[a. , ay , a.], IN , b, , b.], [C, , c, , c.], tangular frame of reference OXYZ along the axes of which the unit vectors are

i, j, k; and

t"-1

$*-1

E L, with the unit q-ads and unit Let fQ - E Q, and -L s-c p-c n-ads
[Ia+bab1']

[7] [7]

where a = angle between a and b.

25. a b = a b. + ab, + ab, 26. a-5 = I a I' = a = 5-a

[8]

(a real or complex)

27. °Q Q °L = °L Q °Q = J[ 1°Q + °L 12 - 19Q

12

-- 1'L

[8]

}']

(°Q, °L real)

[21]

f.-1 1'

28. °Q Q °Q = 19Q

_

V-0

I Qa

12

[24, 27]

29. a x b = --- b x a

[9]

30. a x (b x c) =

be) =

31. (a x b) x c = c (ab - ba) = 32. a x

x [9] X

b x c= x

[9]

b x

d) = cdXab

[67]

APPENDIX A

90

34. °Q X° aL T-1

35.ab 36. 37.

6 in

(-- 1)°`t X`qQ

rl - ba

167).

is8)

Fill

mm cab

abc

bca

r'i= abc

S

3w

`Q

N!i

nx

39.

r-j

4d®"L

r-1 40,

40. if dQ on

qQ s 9Q

here QQ

(QQ +'Q)

41. it.a 42. K,a

1111

Vol

43. K,, J(K+

1151 1151

41.K=(K_)

45. Ks 'a Ke

r,+K+}Ke1

46. K

47. If K

'COU +

+K%

K&Ji+K33+K,31c+ then

ICX+K,19i+Kkk Ko, + Si) + Kew

Kli+"+K + Kbi+ K*3 + K.1k

Ks

Ke + K` + 94

APPENDIX A

91

and

Kv = (Ks -- Ki)i + (Ks -- Ks)j + (K1 - K3)1 ab:K = K:ab 48. 49.

[$1]

[*0,o1]

[a]

Gradients Gradients are derivatives with respect to vectors or polyadica of scalar, vector, or polyadic functions. The primitive functions of which the gradients are given are, of course, the integrals of the corresponding gradients with respect to the independent variable: in reverse, the list below is a table of integrals. The nsga-

tivee of the primitive functions, if scalars, are the "scalar potentials" of the derivatives listed. In formulae 50-72, unless otherwise stated:

12, R - respectively an independently variable polyadic and vector sX, 'Y, 'K, W - variable polyadics of the ranks shown

a, b, U, V - variable vectors M, N - variable scalars A - constant scalar B - constant vector

50. By the definition of a derivative, if `Y = f(°Z),

4-1 (a) d(°Z) a d(Z) d( Z) © d(Z) = d(Y)

or if 'Y = f(W), (b) dW:d(Y) =sd dW

dW

:dW - d(`Y)

or if 'Y = f(R), (e) dR.dCY) =ld(`

dR

dR

-dR = d(rY)

51. If K = f (°Z, ''X, ... ),

d('Y) = d(4z) & Ca(z)lx + d(1X) ®Ia(X)I + ...

[89 ff.]

APPENDIX A

92

52. If `Y = f (QZ, 'X) and QZ and "X are each functions of °U and 'V which are here independent variables,

d(QZ) = d("U) 0

La("U)Jv

+ d('V) 0 La('v Ju

d('V) 0 [8(V)]u

d('X) = d(*U) ® Lad d(`Y) = d("U)

® C8(UU)'v La(Z)]x +(a(,X)]v ® [a(Y)1 1 a("U)

8('X) z

+ d(v) ©

a- Y)

iFa(-V)1V a(Qz)

a{px)

a(oz) xx {{ a(`Y)

Formulas 51 and 52 illustrate two of the many formulas analogous to those of scalar calculus; unlike the scalar formulas, the order of factors is here important.

53. d(abc) = (da)bc + a(db)c + ab do 1--11

1-11

= bcda+acdb+abdc =

1-11

be +

[691 n1

54. VU=dR=grad U

[69, 701

it V" and "grad" are often used as the vector derivative symbol when there is only one independent vector variable. 55.

d(A'Y) = A d`Y dQZ t

56. dQZ ('Y + 'K) =

57.

d(MN) daZ

dZ +

e

dd K QZ

= M dN + N dM dQZ dQZ

APPENDIX A

93

e

58. ddZ(M`Y)

(dZ) Y+MZ

59.

[79]

ni

=

Mi

x (V x U)+Ux (V x V), (U,Vreal)

V

60. V(U x V) _ (VU) x V - (VV) x U,

(U, Vreal)

dZ(`Y®x) =dz®x+ddZ®tY

61.

62. V(BU) = 63. V(UB) = (VU)B

f7i

64. V (UV) = (V U) V + (V V) U1

[80]

ni

= (VU)V+UDV 65. 66.

ad-"Z

=

2°1

[78,791

66a. VR = 1

[78]

2R

67. 68.

d

ZZ

U(m)] = PM)

[78]

dZ

69. d I Z, _ °Z/I °Z 1 ,

(°Z real)

70. VR" aa V I R I" = nR"-2R,

(R real)

71 .

(q

[79]

021-

d Z = °1

72 . V (

)

d(

dR

)

=

a(

ax

a(

)

+

ay

)

+ k a(az

even)

)

if R=ix+jy+kz

[71]

94

APPENDIX A

Divergence

[741

In formulas 73-94 R is an independently variable vector (in physical applications it is usually the "position vector" in real space) and X - either a real dyadic or a real vector variable

M - a variable scalar W - a variable dyadic U, V- real variable vectors A - a constant scalar 8- a constant vector

73. V X 74.

div X

X2) =

75.

M

76.

-A VX

77.

V

V

78. V. (BU) = B_ 17U 79. 80.

VU- VM +

81.

MU- VV

82.

83.

[751

W:VV

=

Curl

[76

The symbols in formulae 84--90 are explained above formula 73.

84. V

[81]

xXmdRXX=curl

X

85. V X (XI + X2) = V X X1 + V X X: 86. V x (MX) MV x X + (VM) x X 87. V X (UV) (V X U) V -- U X VV

APPBNDIZ A

8S. V X (U X V) =

a9.VxR=0 9O.vx(

95

U. VV +

+k,x(

+3

)

Mixed Derivative Operators

[881

The symbols in formulae 91-14 are explained above form'nls 73.

an V'(

)-8'(

)+8'( )+e'

Ole

112. V X VM=0

92x.VxVU-0 93. V. V X X-0 94.

VX

X X) + V'X

Appendix B: Ternary Numeration In the everyday decimal system of numeration the chain of digits 17710 means 1(10210) + 710(101) + 71o(10io) ; in the ternary system 1221s

means 1(3io) + 2(3io) + 2(3io) + 1(310) where subscripts have been used to indicate the system of numeration in which the numerals are to be understood. The digits 3, 4, 5, 6, 7, 8, 9 do not appear in the ternary system; they are expressed by 103 , 1l3 , 123 , 2C6, 213, 22s , 100s. The rules of arithmetic are the same in all systems. To convert 1771, to the ternary system, we first observe that 101o = 101, and 710 = 213 ; then 17710 = 1(101o)(l(ho) r 710(''-f`1o) -r 7so 1) = 1(1Ols)(1013) + 21s(1Ols) + 21a = (122,) (1013) + 213 = 201203

A routine method of effecting the conversion is illustrated at the right. The first divisor is the highest power of three contained in the number.

-

81)177 (2 162

27) 15 (0

0 9) 15 (1 9

Table B-1 illustrates the two systems.

96

3)

6 (2 6

1)

0 (0

APPENDIX B

97

TABLE B-1 Symbol and Ternary Number of Unit Polyads of Rank Given Number or

Decim l

i - Vectors

2 - Dyads

3 - Triads

4 - Tetrads

Numeration) Symbol

Ternoary

0

f

0

1

3

1

2

k

2

N.

Symbol

Ternary

ii

00

ij

No.

01

Symbol

if@

iii

Ternary No.

Symbol

000

ifii

001

iiij

ik

02

ilk

002

ifik

3

ji

10

i ji

010

fiji

4

33

1!

5 6

jk

12

ki kj

20

ijk iki

21

ikj

021

kk

22

ikk

022

iikj iikk

iii

100

jjii

7 8

9

ijj

to

iii

11

jik

12

jjl

011

iiij

012 020

filti

101

102 110

13

iii

14

33k

112

15

jki

120

16

Ski

121

17 18

jkk kii

19

kij

20

kik

111

iijk

ijij

ifik fiji

fiji

122

ijjk ijki iiki ljkk

200

lkii

201 202

ikik

ikij

Ternary

No.

0000 0001 0002 0010 0011

0012 0020 0021 0022 0100 0101

0102 0110 0111

0112 0120 0121 0122 0200 0201 0202

?nanP

h se(C. ', John Wi1.7 & Book Inc.,1110.

(a.b) (a ( b)

ab

(ab:dc)

' When eompi.z vectors are admKMd, nrnasban writes (sib) to mean lob in the usage of this book.

Cf. ss Qrt

eMahe

Murnaghan (9)**

Lorentst

Chapman & Cowling (7)

Expression used with same meaning byBird, Stewart, & Lightfoot*

Usage of this book and of Gibbs (4), Malgenau and Murphy (8), Page (3), Phillips (1), Weatherburn (6), Wills (5), and Wilson (2).

Ka

[K-al

(Nu ber. In verenth..es refer to the Let of references on pose 101.)

Appendix C: Multiplication Notation of Various Authors

(K:ba)

Vab

(a x b)

a

[a x bl

axb

Appendix D: Miscellaneous Exercises Note: Except when otherwise specifically stated, all the vectors and polyadics involved in the following problems are real and 3-dimensional. Match stick models

are often helpful at first as an aid in visualizing vector relations.

1. Let A, B, C, and D be the position vectors of the four distinct noncoplanar points A, B, C, and D. What geometric figure is formed if the points are joined in the order ABCDA by drawing straight lines,

a) IfB -- A= C- D?

b)If B--A=D-C? c)If3(B--A)=C-D? 2. The vectors A and B have the following sets of co-ordinates relative to a certain set of Cartesian coordinates:

B: (4, 3, -2)

A: (1, 2, 3) a) Show that:

JAI= =51 =35

JBJ =

JA+BI =

51

JA-B

35

B) _ -15

(A -

The cosine of the angle between A and B is 4/

(14)(29). b) Verify that the vectors listed below have the stated sets of coordinates:

(A - B) :(3, 1, -5)

(A + B) :(5, 5, 1)

A X B:(-13, 14, -5) 3. A rigid Cartesian coordinate frame OXYZ is converted into a new Cartesian frame by (1) rotating OX toward OY through 30° about OZ, (2) then swinging OZ into the original position of OY, and (3) then translating the origin three units in the new direction of OX. a) If x, y, z are the coordinates of a vector relative to OXYZ in its original position and x', y', z' are the coordinates of the same vector 99

A['PENDIX D

100

verify that the matrix 41 T 11 of

relative to the new position of the linear transformation on pags 3 is 11 T11

n,

1,

mi

4

m, n, nip

i

cos 30°

0 - sin 30°

-sin 30°

0

-coq 30°

0

i

0

t

ni

b) How by matrix multiplication should one manipulate with 11 T 11

and

x y z

w'

to obtain

71!'

z'

c) Write down the matrix 11 7" 11 of the inverse transformation which

converts the set of coordinates (x', y', z') back into (x, y, z). d) Form the matrix product 11 T' 11

Il

T.

11.

e) Find the values of the determinants of

11 T 11,11 7" 11, and of

11T'1111T11

f) Suppose the original coordinate frame OXYZ is that with respect to which the coordinates of vectors A and B of Problem 2 are given. Find the coordinates of these vectors with respect to the new frame of reference and find the new values for any of the results of Problem 2 that are altered by the changes in coordinates. 4. Let i, j, k and c', j', k' be two distinct sets of mutually perpendicular unit vectors and let K be a linear vector operator such that

a sum of three dyads, each formed by choosing one K unit vector from each of the sets (i, j, k) and (1', j', k'). b) If A is a vector with the coordinates (As , A,, , As) with respect to the system of axes OXYZ along which i, j, k are the respective unit vectors, show that A

has this same set of coordinates but with respect to the system of axes O'X'Y'Z' to which i', j', and k' belong.

c) If OXYZ and O'X' Y'Z' are respectively the original and the

APPENDIX D

101

modified frames of reference of Problem 3 find the set of coefficients in the vector equations

i' = ali + a2j + ask j' = bli + bd + bsk k' = cliff + c 2i + cak

and express the matrix of the set of coefficients in terms the matrix jj T 11 of Problem 3.

d) Express K in the form of Eq. 2.7-8 using i, j, k as the basis and also using V, j', k' as the basis. e) Find the "scalar" of K and the coordinates with respect to each frame of reference of the "vector" of K.

5. For the linear vector function K of Problem 4, a) Write out in the form of Eq. 2.7-8, using i, j, k as the basis, the dyadics X 0

K, K, K, and K

b) Show that (K)v is a multiple of a "characteristic vector" of K that is associated with the "characteristic number" unity.

c) Find the values of the determinant, the magnitude, and the "second invariant" of K.

d) Find the characteristic numbers of K. 6. Prove that for all real values of A. B, and C, X

Ax(3XC)=2 -CB (Why is this true for all real values of the vectors if it is, as it is, true for all possible combinations of i, j, k?) 7. A particle of mass m is attracted by the origin with a force equal to b2m times its distance therefrom. Initially its vector velocity is Yo in a direction that does not pass through the origin and the initial position vector of the particle is Ro . Find the vector equation of its velocity and of its position vector at time t. Make a sketch of the path.

8. If R is the position vector, show calculating directly from their definitions (cf., §§3.3 and 3.4) that for three-dimensional vectors the divergence of R is 3 and the curl of R is zero.

APPENDIX D

102

9. Let R be the position vector xi + yj + zk and let R be its magnitude. Find the gradient of each of the following expressions by forming

the differential of the expression and factoring the differential in the form dR ( ). See Eq. 3.2-1.

a) R' b) In R

d) R'+hhR' e) R'R

c) zyz

f) sin 7R2

g) Ri+R'k h) R/R'

10. For each of the vectors in Problem 9 find the divergence and the curl.

11. From the definition of the adjoint of a general linear vector function K, not necessarily real, (cf., Eq. 2.5-12), prove that the dot product of K by its adjoint in either order is always Hermitean.

12. Prove that the characteristic numbers of an antiself-adjoint linear vector function are purely imaginary. 13. Let S be a symmetric dyadic with the characteristic numbers Ar , A2, A, , and the principal scalar invariants (S)a, (S)t, and (S )s (cf. §2.10).

Show that a)

1$ I'

b) (S), - (1/2){(S)e c) (S), is always negative. d) (S), - (S), -- (1/3)(S)s(S)s ± (74t)(S)s e) The characteristic numbers of S are less than those of S by onethird of (S), .

References on Vector Analysis The following books are listed roughly in order of increasing difficulty of the vector treatment therein. 1. Phillips, H. B.: "Vector Analysis", John Wiley & Sons, 1933. 2. Wilson, E. B.: "Vector Analysis", Yale University Press, 1901. Reprinted by Dover Publications, Inc., 1961. 3. Page, L.: "'Introduction to Mathematical Physics", D. van Nostrand, 1935.

4. Gibbs, J. W.: "Elements of Vector Analysis", in the Scientific Papers of J. Williard Gibbs, Longmans, Green & Co., Vol. II, pp. 17-90. 5. Wills, A. P.: "Vector Analysis", Prentice-Hall, Inc., 1931. 6. Weatherburn, C. E.: "Advanced Vector Analysis", G. Bell and Sons, Ltd., 1944.

7. Chapman, S. and Cowling, T. G.: "The Mathematical Theory of Non-Uniform Gases", Cambridge University Press (Chapter on Vectors & Tensors), 1939.

8. Margenau, H. and Murphy, G. M.: "The Mathematics of Physics and Chemistry", D. van Nostrand, 1956. 9. Murnaghan, F. D.: "Introduction to Applied Mathematics", John Wiley & Sons, 1948.

The uninitiated are particularly cautioned that the notation in vector analysis is not well standardized. The notation and terminology of this book is essentially that of Gibbs, it is also used by Page, Phillips, Wilson, and Wills, and in part by Chapman & Cowling. Phillips is probably the best book for the beginner; Wilson's version of Gibbs' lectures is the next best. Murnaghaa is excellent and includes accounts of several other

useful parts of mathematics; it is, however, a more "mathematical" book than is Phillips' or Wilson's and is definitely not recommended for

the beginner. Margenau and Murphy is of about the same degree of difficulty as Murnaghan but is likely to be more readable for chemical engineers. Several of the better texts on Calculus give very good treatments of elementary vector analysis: one of the best is Thomas' "Calculus & Analytic Geometry" (Addison-Wesley, 1952).

103