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1
20
F %( ( C = (cij ) n- #%7+ #198 7%, 73+ #18#(%1, #+%+ %+ ( #:+) D1 (#+# (+:, , 73): H) H ( , %#( # (, # 671 7# % V . (& /( ( C 2 +(6 %+2.) ) (2 V ! V . H) # %( 7 V 3 #:(, 7(# ##( (E D1+ #%7+%9( 21 # %+ #(+ (+ ( 2 %(. ) %( v :( % (%8 %, 1) %( CvE D1+ %+29( (+ Cv % #: ##( ( 37 (+ v % ( 2 ##(. H) ( 1 C 2 # 67%(6 %1 # #: (H ) (7 (). L# % #( ##( ( (# 7# (a : : : an ) = A) %( v (+ x = = (x : : : xn ) ( + , /( # #( ), ( % % ##( (# -( 7# (b : : : bn ) = B ) ( (+ y = Cx. @7 1
1
1
v = B (Cx) v = Ax = (AC )(Cx) ;1
%, 3(
B = AC
A = BC: (2.4) (HR) (7 7#). ( C # 671(# (, 3(+ ##(% %+7(6 %+ 7# B = (b : : : bn ) 37 #(+ 7# A = (a : : : an ): ;1
1
1
B = (b : : : bn ) = (a : : : an )C = AC bi = 1
1
X
cji aj :
(21 ( % (2.4) 7# 7# 1 D1.) & /( %+ (+ y %( v %+29(# 37 #(+ D1 y = Cx 7 (H ), D1 y = C x, ( 3( # 7 x = Cy: ;1
v = By = ACy v = Ax = ACC x y = C x: ;1
21
;1
C 7, %( 3(-16 % (, 3( <##( ( 71(# # 869 ( + C =, /( (1( 1(3: (# % %1 7 ( ## (H ) 7 7# ## (HR)? ) ( 1 C 2 # 67%(6 (2 %1 # #: ( ) ( # 7% % (: 7 (). ( v, %- % 7# A (+ x, :( % %( Cv, 98 % ( 2 7# (+ Cx: Cv = Ay = ACx: H(, 3( # C |P %+2 ( , ( ##( %(% B = AC , bi = cji aj #% %(# 7# % /( 7# %( Cv ( ( 2 (+ x, v ( % 7# A. L# C | %+2 ( , ( B | 7#, - 21 Cv %(# %(% bi # /DD ( xi . (R) ( # (2 % (: 7 7#). ( C # 671(# (, 3(+ ##(% %+7(6, % 19 ##(1 %(% B :( 7# A (2 C. $ /( #(1 (a : : : an ) 12 # % C ((6 # % /( 2): B = (b : : : bn ) = (Ca : : : Can ) = (a : : : an)C = AC X bi = cji aj : 1
1
1
1
P
( v = Ax = xi ai :( % %( X Cv = xiCai = (Ca : : : Can)x = ACx: %, % ( ) (R) 13(# ( 2 (2 C : V ! V . /( (- ) <13-=, 3 H).
#(( +((6# 7 (6, % #13 D1+
139(#. (6 # (6, 3( +%9( 7% ( ##(+, ( 7% # #(#(% V , 3( (, ( 1 #+%9(#
##(% % 17 (, /( 79(# (+, (, /( 79(# 7#+. S1+ 2 (1 %##(%(6 7 1 1(. 1
22
. 73 21 7% :-( 5(% (% #13 | %(% 7 V ) : ( 3%(#, 3, + # 671+: % <(%+:= (. 8 #13 36 2( ( (3: ( #( M % %%
#(#(% ( 7 : (:. F(%(#(%, 7% M , ( (3 :(, %8 %, % 1 (3, 7+%9( <(33+
7%=, 7 (, ( # (3 #(9(# #(, %( <( 7%=. ( 3, (+ 7% (2 %9(# (33+ 7%, (6 /( 7% % M , % #(%(#(%198 Rn , 9( 73 ##(%+ (+.) +2 <(33 7%= + # % :1 % - 3 - %, (1 3( ( 7% %# #6 ( 3( %, 2 (+ 7% (# 67% (+: #(6 2 #(, # 67% #: (), #( # 6 17+%(6, 3( % (-( #13 36 ( (33 7%, (. - 2 7#6E , % < 7%= | /( 38 (33 7%, (. + % (2 C : V ! V %(
#(#(% V % #. R 8 7, 2 ##(%(6 (2 C : V ! W %( #(#(% V (#2, n-) % 1 W (#2, k-). & # 67% ( x : : : xn % V y : : : yk % W (#(%(#(%198 7#+ 1#(6 11( ei ej ) (2 C #+%(# ( C = (cij )ij :::k :::n: # %( u 2 V ( i (+ x , 7198 #( x, ( Cu ( (+ yj , 7198 #( y = Cx. 2 (2 #7(6, 3( Pk Cei = cji ej . & 7 ( , %#( # (, # 6711
1
0
=1
=1
0
j
+: 7#% % V W ( C 7(# 1 ( Cb. =1
23
L# D | ( (H ) % V D | ( (H ) % W , ( Cb = D CD : (2.5) (2 C : V ! W 2 (6 7#(6 dim CW 7. #% ( # ( + C ( #, #( +(6, (# 12 C # % #% (+ ( +). L# C = r, ( % V W 9(# ( (+ x = (x : : : xn ) y = (y : : : yk ), 3( % (: /(: ( (2 ( % x 7! y, yi = xi i = 1 : : : r yi = 0 i > r. @+ #%, # ei | 7#+ %(+ ##(+ ( x % V , ei | ##(+ ( y % W , ( ei 7! ei i = 1 : : : r ei 7! 0 i > r. L8 2 #7(6, 3( # 67% /(: ( (2 C #(%(# ( I 0 gr r (2.6) 0 0 gk ; r % ( Ir | 3 ( r- (( 3( - %+: % % r, %+: | n ; r). 2 #7(6, 3( %
:8: (: ( C (2 C %(# < #(-= <6 D= (2.6). <3#( (3+:= (: /( 73(, 3( (k n)-( + C #18#(%19( ( %+2 %( ( n- D ( %+2 %( ( k- D , 3( D CD ( % (2.6). L# V = W , ( 2(# #(#(%+ # 67%(6 % ( 2 #(#(% % ##(+ ( | 1, + % (2+: /(:, 119 | + % : 7:E % %# #13, #(%(# #(#(% ( #(% % #, ( %# % | (, + % (2+: /(:, (, + % : 7:, | # 671(# ( 2 ##( (. C %72#( 7 ( + C #129(# | % (2.5) D = D. # (, 1 #(-1 %1 2 %#( ( 1 /DD (% 7% C # 67% ##(+ ( % V , (. . 1( 7+ C DCD , | /( ( ( 2% 6 1
;1
1
1
1
0
0
1
1
;1
1
;1
24
D+. 4 11 #(%%(6#, , # #(+, /(( % # #%(6 #2 %1 #% 7#6 67 + + (#6, # 1 | 9 /(1 %1 : 73, (+ #( + (6# # 3((. #%7 # #7+ # C : V ! V % (: ( % V 89 %, 3( (6 det C = = det DCD , ( 3( /(( (6 7%#( ( # 671 ( ##(+, (6 ( # (2 C. &/(1 2 7%(6 ( det C # (2 C. 73 37 I (2#(% 7% % V , + 2 ##((6 det(C ; lI), l | 3#% (. 8 ( 7+%(, 3( /( #(6 :(#(3# 3 #(%(#(%198 ( + C E , #( +(6, (2 7%#( (6 ( C 2( 7+%(6# :(#(3# 3 /( 7%. (@ 1( # 7 (;1)n , (. . :(#(3# 3 C 9( det(lI ; C)E % ( #13 #(- 3 #(6 ln .) 2. . F 2 V %( #(#(% V | /( #(#(%, /( ( %9(# + D1 + (+ D+, + D1 ) V . (F1 %1: (: D1 % 7% D1 3# 9(# 3%+ 7.) 3 (1 /( #6 % 3#( #13 <D(3#:=
#(#(%, /(+ V 7+%9( 8 %( %(+ %(, ( /(+ #: V | #( %(, (%(+ %(. V , % V , 2 67%(6# 9+ (, # 12 + %+ 19-( ##(1 ( % V , (%39819 1( 7#1 a : : : an , ( 2 %%#( % V (+, (# #%7+ # /( ( % V : ;1
1
V
! Rn
a
7! (x : : : xn) xi = a(ai) 1
(2.7)
( %, 1 ( %(% # #(%(# %71 ( %( 7 #+%(# % % #(). L# 25
%( a 7%#(+ (x : : : xn ), ( /(( %( P #(69 E #(%(6, 9 %( v = xi ai 1
a(v )
=
X
xi x(ai ) =
X
xi xi :
(, 9 n 3# (x : : : xn ) 2 Rn (# D1 a, ( ( %9(# /( xi , (. . ( a(ai ) = xi . C%+ %(# D1 , P 98 %( v # ( x 73 xi xi . &/(1 (2.7) | #(%(6 ##( ( % V . 1 7#1
#(#(% V #(%(#(%19(? #%7 # /( % # %% # % #198 . R7#+ a : : : an % V a : : : an % V 7+%9(# %7+ ( (-9 1 11), # ai (aj ) = dij %#: i j . 0( 73(, 3( %# (+ (2.7) i- 7# %( ai , i-, %+ 0, i- ( % 1. H3(, ai %(# i- %( ( 7#, (1 #(%(#(%1( ##( ( (2.7). (H + %, 3( 2 7# a : : : an #(#(% V #(%(6 #18#(%1( %7+ 1 7# a : : : an % V .) @(, (%( #(%+ % # #(: (+ (2.7) %9(# ( % 7# a : : : an #(#(% V , %7 7#1 a : : : an #(#(% V . S(3# + D1 + ai % 7%#(+ 1 7%. i- ( xi %( v % 7# a : : : an 7%#( ( v, (. . (##(% D1 ( v) %(# + D1 V . =
1
1
1
1
1
1
1
1
aj = 0 a + : : : + 1 aj + : : : + 0 an = 1
X i
dij ai
( 73 /( D1 %( aj % dij . F%(6, i- ( | /( 7 #(6 ai E %7+ # a : : : an 7# V 7% ( %(% v 2 V , ##(%+ D1 ( v. $ %#-( -1 ai , xi | % #%, (1, 3( 37 xi 1 73(6 (+ -16 (% D#%) %( v, 1
26
( xi D1 ( < %(= (6 1 73 . P L# 2(6 7 #(6 D11 % a = xi ai % #8 %, (1 D1 (2.3) # % % v = Ax, ( #1( %7(6 #( , /( ( %9(# %(+ 7# % V 7]
8]
0a 1 B.C a = (x : : : xn ) @ .. A = xA: 1
1
an
(H#6 + 73 #( 7 %(% ai (3# 1% A, (1 3( 7%19 3#19 <6D1= (1 (3(6 ( (# A. F((+ + -D(% TEX' 2 #2( (: 7%+: 3#: 1%, (+ #- :2 (#.) 3, + 1 67%(6# ( 7 #69. # 2 7(, 3( + 7# A 7D7 +: #(#(% Rn ! V | (2.3) | (, 3, % x 7! xA. &#(#(% V V , 3, 7D+, %8 9+ % n-+ %(+ #(#(%. 3((6, <#((= 7D7 7#6 (. W- 7%#(, 3( # % V %% %% # 7% ( (# 2), ( + #3# 73 h i, 3(+ (3(6 ( #(( #
5 + , xi . ' xi
M ( ), ( | ) ) Tx M dxi ( $ ). ' $, ) dxi $ 3$% Tx M , $ + TxM Tx M = (TxM ) . $ + $ -$ $ , ) , $ dxi. ( Tx M $ $ , $ $ $$). . V = Rn $ , V = Rn , ai | ) - n , % n % n- . 7]
8]
27
7% ( ) % Rn , ( #(#(%+ 7 %7( (+ 7D7 V ! V , #%7+ # /( 7%. , %(1 v 2 V #(%(#(%1( + D1 a(w) = hv wi. L# A = (a : : : an ) | (%+ 7# P % V v = Ax, w = Ay, x y 2 Rn , ( hv wi = xi yi E 73(, a ( (+ xi = xi , (. . % (: #(%(#(%198: ( (2 V ! V #(%(# 7%#(+ (2 Rn ! Rn , %8 x % x . &# (3 ( 2 #%7 # #+ 7% ( ) % Rn , - V ! V | # #+ 7% h i % V E ( 2 +(6, 7D7 (2.3), (%+ 7#, ( ) :( % h i. 3. . ##( 819 #(1 9, % V %#( # 7% 7 + D1 ( D, (2), (. . ( (2
1
0
B: V V ! R
(v w) 7! B(v w)
( 21 1(1 % (6#(: B(v +v w) = B(v w)+B(v w) B(v lw) = lB(v w) (2.8) B(v w +w ) = B(v w )+B(v w ) B(v lw) = lB(v w): 1
2
1
1
2
2
1
2
(: (, #%7+: # 7# A = (A : : : An ), B #(%(# #198 7: # v = Ax w = Ay ((. . # v w 9( (+ x y) P ( B(v w) = bij xi yj = (B x y) = (x By) = x By (2.9) B = (bij ) bij = B(ai aj ): 1
0
0
P
( %+2 bij xi yj #+ 1 /DD (% bij #(#(% #(%(6 %71. C #3((6, 3( % D1198: 7( #+: 7%: B x By #1(6 #( + | 716((+ 2 #(%(#(%198: %(+: ( #(%(#(%198 #( +.) 0( #71
13(# # P
869 (2.8), # % 1(+ B #(%(6 xi ai %#( v P j y aj %#( w. 0
28
R+ D1
v 2 V % D1 a(w)
B 7%( # #(%(6 %(1
= B(v w)
b(w)
= B(w v):
C #+ %79( % (2
V
!V
v 7! a = B(v ) v 7! b = B( v):
(2.10)
@7 (2.8) #71 #1(, 3( | +. 0( (2 #% 9( % ( (6 ( #13, D1 B | #(3+, (. . B(v w) = B(w v). F3# + /( . ($ 6(% : # 73 9( ##(3+ ( 1 ( | 7 +) B, (+: B(v w) = ;B(w v) , (2 (2.10) 739(# 7.) 4#, 3( % (: ( ( 71%(#, 3( % V # 671(# 7# A, %7+ 9]
5 $ , B(v v) = ;B(v v), B(v v) = 0 v. 5 , B , 9]
0 = B(v + w v + w) = B(v v) + B(v w) + B(w v) + B(w w) = = B(v w) + B(w v) . . B . 1 B(v v) = ;B(v v) B(v v) = 0, $ 3, R 2a = 0 $ , a = 0. 1 , 2. 4 (, , mod 2) $ 2, B B(v v) = 0 v. 5 - $ $ , B(v w) = ;B(w v), $ a = ;a, $ . $ , , $3% B(v v) = 0 v. 5 $ $ B .
29
7#1 A % V ) (2 (2.10) 9( #198 %: # v = Ax w = Ay a = xA b = hA B ( % (2.9), P ( xj = bij xi (. . x = B x x = x B (2.11)
hi
=
Pi j
0
0
0
bij xj (. . h = Bx: 0
8 ( 7% (2 (2.10) (. R1 7+%9( : + (2, ## %+ # B #%, #(%(#(% # %. S1 B 7+%(# %+2+, # %+ 9(# #198 /%%(+ 1 11 1#%: 1) (2 v 7! a %(# %+2+, (. . a = 0 (6 v = 0E /( /%%( (1, 3( (2 v 7! a %(# + 7D7 #(#(% V V . 2) (2 v 7! b %(# %+2+E /( /%%( (1, 3( (2 v 7! b %(# + 7D7 #(#(% V V . 3) ( /DD (% B = (bij ) | %+2. 0%%(#(6 1#% 1) { 3) #71 #1( 7 (2.9) (2.11). L# + D1 | #(3+ %+2+, ( 7+%9( ( #%%%+) #+ 7% % V (# % -( #+# ##(%(# <#%=, ##(% + 3#(6 #(1(1+ #(#(% V ) 7 #+%9( 38 % ( 2 %, +3 # 7% % Rn , (. . (v w), (6 # 1( +(6 1. @7%#(, 3( % (+: (:
10]
11]
* ) $ $ , 3$% B . ) $ 3 $ (2.12) n, r , r | 3$% B, . . % B , , V , % B. ' 3 $ 3 . 5 $, $% , . ( V TxM $ M x (. . 3), ) .) 10]
11]
30
( % (v w) =
k X i
xi y i
=1
;
n X i k
xiyi v = Ax w = Ay
(2.12)
= +1
3 3# k 7%#( ( ( %+ #(%(#(%198 ##(+ (. R 7%#( %(# D1% %(3 D+ Q(v) = (v v): %(# <#1 %(%=, % ( (% 1%61 #+#1 #% <#1=) (+ %(+ 1( %#(3(6# # 7 <1#=, 3 3# (%+: 7%#( ( # # % (<7 %(3 D+=). 7+%(# # %(3 D+ #(%(#(%198 D1 (1 # | # 7%). Q + %+2 37 # 7%, #%76 21 (( % (19 #(1 : Q(v w) = 12 (Q(v + w) ; Q(v) ; Q(w)): 0( #%76 7%( 13(6 # %+- 1(%2
% ( ) %1 (2.12) 7 #(%(#(%198 1(%2 Q. L%% # 7% % V , ( %#6 | /( + #(3+ D1 ( ), 98 ( #%#(%, 3( (v v) > 0 %#: v (v v) = 0 (6 v = 0. C D1 | %+2+. 6 # % ## % (2 %( (+ %( v % 1% D1 , ( 13(#, 3(, % 3#(#(, (v v) = 0, (, -1 9, v = 0. &/(1 %% # 7% #(%(6 %(# 3#(+ #13 <8= ( #%%%). 12]
13]
1 ) . 1 $ $ , ;$ ( ) $ , $ min(k n ; k). / + % ( 3 33 %). # 2 , 2. / 2 3 3$%. 12]
13]
31
L# + D1 B %(# %+2+, ( 2 % B(v w) ( %( 8 (2 a 7! v, b 7! w, (+: %1 %1 ## %+ # B + (2 ( % 1( B # 671 %, % | % ## % (c 2). &13(# (+ + D1 B V . R1 7+%9( < D, ( B=. #% /( ( 8 () 7% %(# (( D(, 3( ( # 67% % V ( 7# A, %7 (1 7#1 A % V ) ( c #12( ( B , ( B 7 (2.9). /DD (% B $#(%(6, % -: 2: 73: a = xA b = hA v = Ax w = Ay h = By y = B h
;1
;1
0
B(v w) = a(w) = xy
( 7% #( #( , (. .
c(a b) = B(v w) = xy = xB B
;1
h0
=
0
P x yi),
X bij b
i
xi hj
bbij | /DD (+ ( + B . L# B | %% #%%% # 7%c (2. 0( #71 % 7 #7 (3+: , ( B /DD (:, 2( +(6 7 ##(%. % ( ( 1 /DD (% # 7% % #: V +3 739( 37 (gij ) 7+%9( (3# (7 (/( #(%(6 (7 % ( #+#, /( (# % (7 #3#), ( 1 /DD (% #(%(#(%198 # 7% (<( D+=) % V | 37 (gij ) ( 1#, #( +(6, <+-1= g). ## %+: # #+ 7% (:( + #%%%+) (2 (2.10) #% 9(. (: ( 9( % xi = gij xj (< 1# #=), ( (2 #(6 xi = gij xj (< #=. 73 %(%, %(% : ( #3# ( 2, 3( % (2.11), (6 ( /DD (% # 671 D1 % V | ;1
32
(bij ), (gij ), 71%(# #1% %(98# #). ;%(, 3( (2 C : V ! V #:( + D1 B, # B(v w) = B(Cv Cw) %#: v w 2 V . (: ( (( 6 y 1( ( w) (#9 #1(, 3( (Bx y) = (BCx Cy) %#: x y 2 Rn ( C n ) ( ) | #(( # 7% % Rn (
2 % C n ), (1 B = C BC . L# B | %+2+ D1 , ( (#9 #1(, 3( (det C ) = 1, det C = 1. 3#(#(, (6 (6 #%(6 (#:98 #%%% # 7%) 7% % 1. /( #13 #(- + 7+%9(, 3( 719(# %72#(. $1 %2+ | 7% C, #:98 %+2+ + ##(3+ D1 . # ( #19 6 #18#(% 3( ##(% #(1(1+, ( C 7+%(# # (3# 7%. @7 #7 %, 3( det C = 1. # (3# 7% ( 716(( | 3(6+: 1 # # 2 7(6, 3( det C = 1. 4. . &1#(6 C : V ! W | (2, ( % (: (, (%398: 7# A = (a : : : an ) % V B = (b : : : bm ) % W , #(%(# ( C = (cij )ij :::m :::n . #12%-:# %+- (2 V ! V , 7 (2 ##(% #(% C (+ (2 %( (%( v 2 V # ( x :( % %( w 2 W # ( y)
##(% #(% 7# A ( #+%(# 1( 12 #( B # % C | 7# A :( % %(% (Ca : : : Can ) = (b : : : bm )C = BC P (. . ai :( % Cai = cji bj ) %( 1 (1 2 j 716((1 (# v = Ax, x 2 Rn , ( Cv = ACx). F 2 (2 C : W ! V %( + 0
2
1
1
=1
=1
1
1
33
D1 b W % ( + D1 a = C 3( a(w)
= b(Cv) %#: v 2 V:
b V ,
(2.13)
( (%2 #18#(%% ( a | /( 3% 1(%2, 3( (2 v 7! b(v) %(# + D1 V . % (%#6 #2) 12(# (6 1(%2 #( (2 b 7! C b.) & # 67% % V W (, (%398: 7# a : : : an b : : : bm , %7+: (-9 a : : : an b : : : bm #(%(#(%, (2 C #+%(# (: # b 2 W P i ( (+ h ((.P. b = hi b ), P ( C b ( (+ hC , (. . C b = xi ai , xi = hj cji . $#(%(6, j v = Ax : b(Cv) = h(Cx) (# % #(( 7% ( +-#( h ( 1-#( Cx), 3( % (hC )x, /( 73(, 3( #( hC #(%( (+ D1 C b. (, 3((9 %+3 (:( + % :( 3#(+: #13:) 1 D1%, % ( D11( (# % ( C . 0( D1% (#(# (1 %(1 73, (+ %( 7 #+%9(# % % #( . -: (: /( 73(, 3( %(+ b C b #(%9(# #( h x = C h, #( h x = (C h) = C h . %, % /(: (: #( h , #(%(6, 12(# #% ( 1 C . #( (((6 8 ( #(%+ D(: # C : V ! W | 7D7 %(+: #(#(%, %8 7# (a : : : an ) % % 7# (b : : : bn ) %(, ( C : W ! V %( %7+ (b : : : bn ) 7# (b : : : bn ) %( #(#(% % %7+ (a : : : an ) 7# (a : : : an ) % #(#(%. 0( #71 % 7 (, 3( #+: 2: % /(: 7#: C #(%(# 3 ( , ( 3( 9 b 2 W %( C b ( % 7# (a : : : an ) #(#(% V ( 2 (+, (+ b ( % 7# (b : : : bn ) #(#(% W E % 3#(#(, /( (#(# # bi , (+: 13(#, 3( C bi ( % 7# (a : : : an ) (+ dij .
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
34
L# % V W %% # 7% (:( +
#%%%), ( # 2+ ( 2( (6# 1 5( | (2 C : W ! V , (
(v C w) = (Cv w) %#: v 2 V w 2 W:
(2.14)
F 3#(+ #13 /( + 12 %#(3#6 % (1.9). (, % (: #(%(#(%198: D1 % b = ( w) W a = ( C w) V (2.14) 73(, 3( a = C b ( C (# % ( 2 #+#, 3( % +18 D). & # 67% % V W (: 7#%, % (+: ( ) ( % (2.12), : ( ( x %( C w ( x %( a ( ( y %( w ( h %( b 36 #(. L# # 7% %%, (. . # % (2.12) ( 1#%, ( /(( : #%(# #( (# %9, %8981 #( % #(1E 73(, # C #+%(# ( C , (
x
= hC x = x = (hC ) = C h = C y: 0
0
0
0
0
C 7, % /( #13 # 2+ ( V ! W #+%(# (# % ( . ( % +18 7 + 7 %(-#( x h %(-#( x h , + D(3# 7 - x y, # (6 - (+ 7# + (%+.) L# 2 % (2.12) 9(# 1#+, ( : ( x h x y (6 #(6 #( % % #( %, 7(6 7 1 3#( (. F(%(#(%, : ( D1+ x = C h D1, %+298 x 37 y , (6 (# %(6 ( 1 C , 7(6 7 1 3#( /(%.
1 # (2 V ! V 2 ##(%(6 71 ( % V , #+%19 (%1 # # | (H ) (HR)) ( C . &#(, /( 79(# (+ % V , % %1, 3( ( # 6719(# 7#+ A B, %7+ # 7# A B % V . 0
0
35
(H ) 73: 7 . 1, v = Ax = By, y = Cx. C ai (v )
= i- ( v % 7# A = xi bi (v ) = i- ( v % 7# B = y i (
0b (v) 1 0 a (v ) 1 B@ ... CA = y = Cx = C B@ ... C A n n a (v ) 0b b(v1) 0a 1 B@ ... CA = C B@ ... CA B = C A: 1
1
1
1
bn
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( a(v : : : vk ) = 0. $#(%(6, ( 7 /(: %(%, #2 vi , %(# #(6+:. H% /( # 67%% #(6 a i-1 1(1, 13, 3( a(v : : : vk ) %(# #1 #+:, 1 2 7 (+: i- 1( 1-( 11. , # k > n, ( .k V = 0, (. . #18#(%1( (%6 ( 898# % 0 (2#(%) 7 D+ #( k, k > n 9+ k %(% 7%#+. @ %#-( +%(
7 %(6 # D #( > n, :( (2#(% % 19. S6 /( 31 (%3(, #2(6 /( +%( 7 6-: ##12 ( 2 3, (+ 73+ %+2 %9( #+, # %+ 19. 73 37 Sk #(3#19 1
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1 %#: #(% 3# 1 : : : k. ( 2 % Sk +319 7 9 (2, % #13 f1 : : : kg ! f1 : : : kg: c t 2 Sk i 2 f1 : : : kg
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1
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;1
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, 3( f : V k ! R, v : : : vk 2 V (cf )(v : : : vk ) = f (vc : : : vc k ) : /(: (: ##(3#( k-D+ a (%: 9 (# 7 c ((. . ( #(%, ( ( #( % 3# #(%( #( #(6+ 3#) ca = ;a. @7%#(, 3( 9 #(% %(#
7% (# 7 , 3 3( #(% 3# 2+: (# 7 %# 3( (:( 2( 7%#(6 ( ( #(% c % % 7% (# 7 ), 3( | 3(. &2% 9
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=
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53
1
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. F 3# (3 7 #(#(% Pn %7(6 19 #11 . V = .k V . L /(+ #1(6 #1
Pn
k
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+
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k
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73 #( /( 7+%(# 1 P %(6 # D6+ #1 ak , 3 # # /( ak (6#(. ##3#: % #: DD 6 ( % - 1# (: #1 %7( 8(6# . V + 1. + ## #(69 %# 7%(6 %- 12 %1 #%#(%. & 72 /(: % #% (1( %6 #( + %, 2 + + #((6 #%9 # (, 3( (# % 13%. + (6 :( ((6 % (+ #(+ /( . ((1 9(# % 73+: %- 12 | 73+: (6 D, % ( #+#, 3( % 3 9( 73+ 716((+. &%, %- + . V /( 7+%9(# 7D+ ##(% 7D7, (2#(% . V . V %8 .k V % .k V . : #13: 7+%(#, 3( # a : : : an | 7# V , ( /(+
0
1
1
ai1
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(3.3)
1
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v : : : vk 2 V 1
1
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56
(b
1
^
^ bk )(v :: : vk ) =
b (v ) : : : b (vk ) = : : : : : : : : : : : : : : : : = det(bi (vj ))ij bk (v ) : : : bk (vk )
1
1
1
1
:::k
=1
(3.4)
1
1 | (b
1
^
^ bk )(v : : : vk ) = k1! det(bi(vj ))ij 1
:::k :
=1
(3.5)
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1 #( k + lE %#: /(% c 9( (cf )(v : : : vk l ) = f (vc : : : vc k l ): (: /( 73, 1( #11
+
1
+
1
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+
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57
(a ^ b)(u v) = 12 (a(u)b(v) ; a(v)b(u)) = 12 ab((uu)) ab((vv)) (. . #71 %, 3( % /( %
13(# 3#(+ #13 (3.4), %( | 3#(+ #13 (3.5). #(%(#(% # #7+ %+-, + 1 67%(6# %+ . ((, 3( ## 1, D11 (2.19) 2 7 #(6 (: B=
k X i
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k X
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i
2 ;1
2
2
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=
k X i
(a i
2 ;1
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+
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(. . k! l!. @(, %#( # f (vt : : : vt k l ) % (3.6) %:( k! l! (: 2 #+:. 0( (%1(#, 31 (3.6) (# (6 k! l!. # /( 36 #(#(%, (k + l)! % %( %( (%1(# #(#(%: % (3.6) %:( (k + l)! #+:, %( + (k + l)!, ( #+ #. ( , % 8, ##1(# # 8 (7+ #3#, /(1 3#( 1 (#6 % -. F % %+##6, 3( 7 %# (7 #3# 6-#(% 12 (6 ( 3#(6 3( /( 3#(6 (, ( #7(6, 73(6 %( | 2 7%(6 7%# ( 8 (7 #3#. L# %(6 (6 3# #( ( ( (#(# 7 ), ( 36 ( %(:, +: (2:, +: %(3+: D:, %-: D: %#(%+: 5(: |
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% #% (1, 3( % #( #(#(% D1 (3.4) (6 k!. C 6 2, 3( /(+ (3.3) 719( 7# % .k V . &2 %#, 17+ ( /(+ 29( .k V . $#(%(6, 7+%(#, 3(
a=
X
i <:::
a(ai1 : : : aik )ai1
^
^ aik
(3.10)
1
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1
1
vh =
X jh
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(3.11)
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1
a(v1 : : : vk ) =
X
j :::jk
1
j
j
a(aj1 : : : ajk )v11 : : : vkk :
1
H#6 2+ # jh ( 73 ( 1 n (7%# ( #(6+: ji ). HD#1 -16 i : : : ik , (+: 1 6 i < : : : < ik 6 n. % 3#( (3.10) (# k! #+:, (+: 2#(% fj : : : jk g = fi : : : ik g, /( jh , %8 %, # 2+ ( 1%+) % %7#(. @(# k! # #% # 2(6 i : : : ik % 1 E /( # #+ :(719(# #(% c 2 Sk , , 2 #(% (%3( j : : : jk # j = ic : : : jk = ic k . & /( 1
1
1
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1
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a(aic(1) : : : aic(k) )
= e(c)a(ai : : : aik )
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i <:::
1
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1
60
c2Sk
ic
e(c)v1
(1)
: : : vkic k : ( )
W- 7%#(, 3(
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e(c)v1
(1)
vi : : : vi k = : : : : : : : : : : : ik ik 1
: : : vkic k
1
1
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v
: : : vk
1
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^ aik (v : : : vk )E 73(, ( )
1
1
0 X a(v : : : vk ) = @ 1
i <:::
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1 ^
^ aik A (v : : : vk ): 1
1
= /( % 9+: (v : : : vk ), ( ( #+ (3.10) 7. . +18: ##12: D(3# 7 #198 1(%2: &1#(6 (# nk 3# ai :::ik , 1 6 i : : : ik 6 n, ##(3 7%#8: ( #% ((. . #(% %1: #% 1 a::: (# 7E % 3#(#(, # % # %+, ( #(%(#(%198 a::: = 0). &1#(6 9(# (2 nk 3# vji , i = 1 : : : n, j = 1 : : : k. C 1
1
1
vi : : : vi k i ai :::ik : : : : : : : : : : = ai :::ik v i : : : vkk ik ik 6i <:::
1
1
1
1
1
1
k
1
v
1
: : : vk 61
1
1
(3.12)
# % 71%(# #1% i : : : ik , 7%# 98 73 1 : : : n. +- /( (##6 (1 #139, ai :::ik = = a(ai : : : aik ), a | 7 k-D, ai | 7#+ %(+, vji | (+ %(% vj . 7(6#(% (3.12) # 67% ( (3# #+# /(: 3#. 3#(#(, %2, #(( #+ | %%:1 %71. #((# 7(6, 3( 17+ % (3.3) /(+ .k V 7%#+. &1#(6 ( 7 % 19: 1
1
1
X
i <:::
li1 ::: ik ai1
^
^ aik = 0:
(3.13)
1
H3(, 73 % 3#( (3.13) 9+: v : : : vk % 0. 76 v = aj : : : vk = ajk , j < : : : < jk . C 1
i dj : : : dijk ^
^ aik (v : : : vk ) = det : i:k : : : : : : :ik: : dj : : : djk
1
1
1
1
ai
1
1
(3.14)
1
1
1
L# -16 # ih %#(3(# # 3# j : : : jk , ( % h- #( #(( 1, (6 % 19. H3(, % 1
X
i <:::
li1 :::ik ai1
^
^ aik (aj : : : ajk )
(3.15)
1
1
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2 fj : : : jk g. ( i 1
1
1
1
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62
# 7# fai g % V ,%3 #3(, # 7# fai g % #: V ) #12( Cnk = nk 3#
ai1 :::ik
(1 6 i < : : : < ik 6 n):
= a(ai : : : aik )
1
1
F 1 #(+, %- k-D+ #(#(%+ 7 2 ##(%(6 (+ %(+ (7+ <%(#(= k ((7+ # k 2 #). @, D a # #(%(# (7 ai :::ik , 28]
1
ai1 :::ik
(1 6 i : : : ik 6 n):
= a(ai : : : aik )
1
1
(3 ( +18 D1+, ( 6 3# i : : : ik 1 3+ %7#(9 7 : 1( #% (6 # 1 (% # #13 ai :::ik = 0). F D3# #%#(% 13 ( 1( (7 ai :::ik ##(( % (, 3( | ##(3+ (, 8 %(, (#(3+)E /( 73(, 3( # (6 #( % #, ( 1 ai :::ik 7(# 7. F(%(#(% 21 ##(3+ %(+ (7 %(#( k %- k-D (% ((7 ai :::ik #(%(#(%1( %- k-D X a= ai :::ik (ai ^
^ aik ): 1
1
1
1
1
1
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1
1
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2 <%(= ( %-: D % 8 (7 #3# 2( (6# 3((9 3( ((1+, # # <(7+:= 7 . 1
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63
73 #(#(% %#: %(+: (7% #(#(% V %(#( k 37 Tk (V ) (/( 7 %1: 1 ((6+: 73E %( 1( % 2). % 9
Tk (V ) Tl (V ) ! Tk l(V )
(T S ) 7! T S
+
(7 T S (# #( 2 ( (7% T S : (T S )i :::ik l = Ti :::ik Sik :::ik l : 1
1
+
+1
## (%:
(3.16)
+
(lT + l T ) S = lT S + l T S T (lS + l S ) = lT S + l T S T (S R) = (T S ) R: 0
0
0
0
0
0
0
0
L 7+%9( (7+ 7%. (3.16) # 6719(# -( (+. 1(6#, 3( # 716(( 7%#( ( ( %+ #:. 2 ##(% 1(6#, 3( : 1 ##( ( 7% Ti :::ik Sik :::ik l %+: ( (7% T S %+29(# 37 7% #(+: ( (, /( 2 +(6 D1 (2.16), -1 #139 (%:: #% (, 2: k + l). 8 ((6# 9 (7 D1 . (3.16) /%%( (1, 3( 0 1
0
0
+1
0
+
(T S )(v : : : vk l ) = T (v : : : vk )S (vk : : : vk l ): 1
+
1
+1
+
H#6 (+ 12 # 6719(#, #(6 ( 7 D1 T S ( k + l +: 3%. C 6 2 #7(6, 3(
Tk (V ) = k V := V|
64
:{z: : V }
k
% %1, 3( # (7 7% k
1
1
0
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1
1
1
1
1
k
1
1
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+
1
+1
(cT )(v : : : vk ) = T (vc : : : vc k ) = = Ti :::ik vci : : : vcik k = Tic 1
( )
(1)
1
1
+
(1)
( )
;1 (1)
:::ic
k
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1
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1
1
65
(+ h = ij = ic .) @(, % (: ( (7% #(% c #+%(# (: ;1
(1)
(cT )i :::ik = Tic 1
;1 (1)
:::ic
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:
3, 3#( #3(9(, 3( (cT )i :::ik = Tic :::ic E 1
(1)
(1)
#3# /( %2. 7%8#6 9 %- 12, + %, 3( f 2 .k V , y 2 .l V 81 P > < k! l! c Sk l e(c)c(f y) 1- ^ f^y = P 1 > : (k + l)! c Sk l e(c)c(f y) 2- ^:
2
+
2
+
3#(#(, 7#+: ai 2 V
8 P e(c)c(a
ak ) 1- ^ >
^ a = 1 P > : k! c S e(c)c(a
ak ) 2- ^: 1
2
1
1
2
k
(#9 #1(, 3( f=
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fi1 :::ik ai1
^
^ aik
(3.17)
1
: : : aik
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% ^ #(6
1
f = fi1 :::ik ai1
: : : aik :
(3.18)
$#(%(6, # #(%(6 % (3.17) %+2 ai ^
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1
1
e(c)fi1 :::ik aic(1)
ai k c(
)
= fic :::ic k aic
(1)
(1)
66
( )
ai k E c(
)
/( 7 %# ( #+ % (3.18), 1 (+: #+ #1(6
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^ aik 37 ai : : : aik % k! 7 6-, /(1 /DD (+ ai : : : aik % %+2 f 11( % k! 7 6-. &/(1 %( ^ D (3.17) #(#(% # #(%(6 (7 f k1! fi :::ik g. &#(%9 3((9 1(6#, 3( % (: /DD (% ( D1198: % (3.17) fi :::ik %- 12 #+%(# (: 1
1
1
1
1
1
X e(c)fic :::ic k yic k :::ic k l : (f ^ y)i :::ik l = k!1l! c S 1
+
2
(1)
k l
( )
( +1)
( + )
+
(: <(7+:= ( f y ((. . : ( % Tk (V ) Tl (V )) % ^ D1 f ^ y #(-
(# ( 2, 7 /( (+ #% 9( # /DD ( % (3.17) % 3+: %+2: y f ^ y. %( ^ 13(#, 3( %- 7% T ^ S %1: %(+: ##(3+: (7% fTi :::ik g, fSj :::jl g ( <(7+= (+ 1
1
X (T ^ S )i :::ik l = (k +1 l)! e(c)Tic :::ic k Sic k :::ic k l : c S 1
+
2
(1)
k l
( )
( +1)
( + )
+
##( D+ #(-, (. . n- #( . @(# (6 ##( 3# i : : : in , ( i < i < : : : < in , | , # ##( 1 : : : n. &/(1 % (3.10) (# (6 #, 13(#, 3( %# %- n-D+ 6+ a ^
^ an E /(1 #(#(% .n V 7D R. /( (- 3 #(#(%1 . V . % # #13 (# #%- #((+ 7D7 . V = R, %6 7 . V 9 7
#6 R, 7D7 .n V R (# %+ 7# /( a ^
^an #(#(% .n V . $, 7 (3.4) %, 3(, % -: +3+: 73:, 73 a ^
^ an 1
1
2
1
0
0
0
1
1
67
n %(: vj # ( xij %
a (v ) : : : a (v ) x : : : x : : : : : : : : : : : : : : n: : = : : : : : : : : :n: : n n n n 1
1
1
1 1
1
(3.19) (v ) : : : a (vn ) x : : : xn 0( #(6 (- 5 n- Z(v : :P : vn ), <(1(= %(+ v : : : vn ( ##(( 7 (3 li vi , 0 6 li 6 1 %#: i) 51 n-
Z(a : : : an ), (1( 7#+ %(+ a : : : an . (5+ #3(9(# # 13( ( . a
1
1
1
1
1
1
29]
A priori V 0 ( , V , , , - 0 ). 5
0 | ) . 5 0 $ 3 +. 0 (n- L - % ) , > 0) + ( ) V ( $ + $ 0 ). , 3 $ $ , 0 , , 3$% $+ $ $ $ + $ . ) $ D E(v : : : vn ) = j det Aj D E(a : : : an ) .
$ $ 0 M(v : : : vn ) ( 0 M(a : : : an ) ). $ , , , , n = 1 2 3 ) 3 $ , + , 0 . / 3 $ | ), $+, $ $ , , , $ ) 3 $ +, , n = 2 3 . 1)$ ) $ $ , ) . / %, + $ , $ 0 ( !) $3 $ , 3$% 7! 0 29]
1
1
1
1
68
9, 3( # det(xji ) > 0, ( #3((#, 3( 7#+ v : : : vn a : : : an (%+ %, % (% #13 : ( 73+.) C 7, %- n-D+ | /( + 73+ <((%+) 5+= % V | 7 1% %- n-D+ 9(# ( % V < 5=. D6 % (: ( 2 #7(6, 3( %- n-D+ | /(, # (3#(69 2(, (, ##(%+ 7 ( %(%. ( 8 #(%+ D(. &1#(6 + - ( : ( % V 1 # 869 (H ) # ( C , ( 3( % (: 2, 3( (, 73: 1
1
(a : : : an ) = (b : : : bn )C 1
1
(+ aj % 7# (b : : : bn ) #1(6 /(+ cj : : : cnj j - #( ( + C . C %7+: 7#% (a : : : an ) (b : : : bn ) 1
1
1
1
b1
^
^ bn(a : : : an) = det(cij ) = det C a ^
^ an (a : : : an ) = 1: 1
1
1
0( %( %+%1, 3( b ^
^ bn = det C a ^
^ an (3.20) # #(%(#(%198: ( y = Cx: . - 72 % ( % # 67%#6 ( 11( 8 # 67%(6#) #%#(% (. ( ( #- 6-) 1 72 7 /( 2 + + (#6, (, (, 2 + + (6 (6 det(xji )ij :::n a ^ : : : ^ 1
1
=1
1
$+ $ , ( M(a : : : an )) $ 3 $ . / ) |
( % %, , $ ), , : $ ) ? ( ) , , ) )- 0 , $ 3 $ ? ( , . , +, $ ). 1
69
^ an(v :n: : vn),i vi = (xi : : : xni) #1(6 %(+-#( (/1
1
(+ R ), a | ()%(+ #(( 7# % R . (#9 + + %+%#( %# #%#(% (. C 72 2 + + 7(6 3+. % 1 #19( #(% ( 3, 3( ( 2(6 %%#( 2 +#(, # %- D 2 2(6 (( 3( #(6 %8, # (+ 7(6# 6-). &1#(6 C : V ! W | (2 %(+: #(#(%. F1 ( 2 , ( %( 9 (2.13) # 2 ( C : W ! V , (2 .k W ! .k V , ( + 1 (2 73(6 37 C , #198 7: %( %-99 k-D1 b W % %-99 k -D1 C b V , 19
(C b)(v : : : vk ) = b(Cv : : : Cvk ):
(3.21)
1
1
4#, 3( C b #(%(6 %(# %- k-D V . & k = 1 /( #% ( # (2.13). (, k = 0 9(, 3( 3# l 2 . W = R :( % ( 2 # 3# l, ##(% /( . V . C 7, ( 2 7 C 739(# 73+ (2 (%# (# n + 1 (: (2 .k W ! .k V # k = 0 : : : n). @ %#( C ##(% (2 .k W ! .k V # 6719( 73 .k C E (, 3, #((# ##( #3( (, 3 ( 36. + 1 67%(6# 73 C , % %1, 3( % 2 ( #13 (, 3(
71%(# C , 2 #1(6 (1, 3( #(%1( C . ( (, # <5(6= %# .k W .k V % %- + . W . V , ( 2 %%#( (2
0
0
C
:.W
!. V
(3.22)
#% 98 # +18 #(%(#(%198: +: #+:. F( #((6#(%, 1( (6# k-D # 73+ k.) 70
& k 6= 1 (2 C 7+%9( # 2+ # C, (2, 1 %+ C. C 2 7+%9( <= (2 (3.21). @7 (3.21) %#(%1(, 3( (2 C (9 7 :) | : C (aa + bb) = aC a + bC b (a b 2 R a b 2 .k W ):
(1 1(6# (2, 3( C #:( %- 12: C (a ^ b) = C a ^ C b (a 2 .k W b 2 .l W ) (3.23) (7(, 3( # 67%(6# (2 (3.22), ( 7#6 D119( ( (2, #(%(#(%198 k l k + l). & 7(6#(% #7 #-(6 C , #3(, 3( % D1 (3.21) b +(6 9 k-D W . 21(3+: -: #( a ^ b %-: D a b 7 : + # # 139(# (+
+ (k + l)-D+. @7 %, 3( (2 C %( : % + (k + l)-D+ V , (+ (3 ( 2 # # 139(# 7 C a C b. % /( - 7 -, 12# % (3.23).
7+ <= (2 (3.22) 2 #7(6, 3( C %(# D7 + .W % 1 .V . $#(% C D+ #(- #( #%(# #( 129 3# det C. $#(%(6, P 1#(6 C %( %(+ ai 7# % %(+ bi = Cai = cji aj , ( 3( (
j
C #+%( C ## (R) ( (2 ( )). @# 671 C (3.19), = (cji )
(C (a
1
^
^ an))(a : : : an) = a ^
^ an(b : : : bn) = 1
1
1
= det(cji ) = det C:
a ^
^ an (a : : : an ) = 1, ( .n V , ( 2 n-D ( % la ^
^ an # (+ lE (#9 %, 3( det C (a ^
^ an ) = det C a ^
^ an : 1
1
1
1
1
71
C 6 7(6 # % . 3 x2 1(%2, 3( (6 # (3# 7% % 1. F:+ C + %+2+ ##(3+ D1 B 2 #3((6 %+ %1 (2.19), k | % 7#( n #(#(% V ( # % #13 2 +(6 3(). (: 7# a : : : an % V , %7 # 6711 % V , B #(%(# % % (3.7). & %7% B % %-99 k-9 #( 6 13(# kk #+:, 2 : (+: %(# 7% k #2( % a i ^ a i . F # %1 %+ #2( % 19. #((# k! #+:, 1 (+: %# #2( 73+ , 73(, # (3#(69 #% 9( # #2( 7% 1
2 ;1
(a
1
2
^ a ) ^ (a ^ a ) ^
^ (a k ^ a k ): 2
3
4
2 ;1
2
# /( #2( | 3( (%() #( , (( :
#(% 7 (#. @(,
B k = k! a ^
^ an: 1
(: #(8 D #: B (2 C 73(, 3( C B = B. H3(, C B k = B k . C (a ^
^ an) = det C a ^
^ an, /(1 det C = 1. F198 3# , # ( + 7#, | /( ( 7+% %1( 7% v a %( v 2 V %- k-D+ a V (1 73 | iv a i(v)a). 0( #(6 %- (k ; 1)-D, ( :
1
1
30]
(v a)(v : : : vk ) = a(v v : : : vk ): 1
;1
1
;1
(4#, 3( #(%(6 v a 2 .k V ). & k = 0 , 3( v a = 0. & k = 1 1 0-D+ v a ( 1(%E (6 % ( #+#, 3( v a #(6 3;1
/ $ : (v a)(v : : : vk ) = a(v : : : vk N $ (;1)k . 30]
1
1
;1
;1
72
;1
v ):
# a(v) . L#(#(%, 2 (6 v (2 .V ! .V . & %1( 12, %-, + : 7 #(#(% (7 .k V ) % 1 (% .k V ).
7% <%1(= 12, %(, #%7 # (, 3( 3( <#2(6= v % ( #+# <%#(%(# %1(6= a. 4#, 3( (2 a 7! v a | . ( (, v a v, 3( 7 (2 1( # 67%). #6 2 %(# #%76 %1( 12 # %-: 31]
;1
v (a ^ b) = (v a) ^ b + (;1)k a ^ (v b)
a 2 .k V b 2 .l V :
(3.24)
$2 # %, 3( b : : : bk 2 V , v 2 V 1
v (b
1
^
^
k X k b )= ( i k X
;1)i
;1
b1
^
^ (v
bi )
^
^ bk =
=1
=
i
(;1)i
) $ $ v 3$%
!R
bi (v )b1
^
^ bbi ^
^ bk
=1
31]
V
;1
( C )
(3.25) 2
V 3 a 7! i(v )a = v a = v (a):
, $ i : V ! V (V | + (V ) ), 3 ( $ ). # , i - v 6= 0, a 2 V a(v ) 6= 0, i(v ) 6= 0 ( i(v )a 6= 0)- %, $ $ , , )$ V , V V . 1$ ) 3 ( - 0 , , ), V V , $ V =V .
73
7 <+-= b bi 73(, 3( % #(%(#(%198 3 #2(6 bi (#1(#(%1(. (3.25) 73(, 3(
v : : : vk 2 V 1
b (v) b (v ) : : : b (v ) k det : : : : : : : : : : : : : : : : : : : : : : : : = b ^
^ bk (v v : : : vk bk (v) bk (v ) : : : bk (vk ) k X k i i i k 1
1
1
1
;1
1
1
=
(;1)
i
;1
b
(v) (b
1
^
^ b ^ b ^
^ b )(v : : : vk ;1
i
(;1)i
+1
=1
1
1
;1
;1
1
b (v ) : : : b (v ) : : : : : : : : : : : : : : : : :k: : : : bi (v ) : : : bi (v ) k i b (v ) det i i (vk ) : b ( v ) : : : b : : k: : : : : : : : : : : : k: : : : : : : b (v ) : : : b (vk ) 1
n X
;1
)=
;1
=1
=
1
+1
1
)=
;1
;1
1
+1
;1
1
;1
;1
;1
0( %#(% (% ( 71%(#, 3, 3( % #(%(#(%198: #1: % % # #+: D119( #18#(%198 b bk , b : : : bk b : : : bk ) 13(# 72 % ( %1 #( 1. C 6 2 7(6 (3.24) ( #13, a b %9(# 7% 1-D (% /( #13 %(, 3( a b #1(6 72+ D+): 0
^
^ bk
a = b1
b
+1
= bk
+1
2
1
^
^ bk
l
+
;1
%# bi 2 V :
@# 671 (3.25):
v (b
1
=
^
^ bk
k X i
=1
+
k l l ) = X( i +
+
;1)i
;1
bi (v )b1
^
^ bbi ^
^ bk
+
l=
=1
(;1)i
;1
k l X +
i k
bi (v )b1
(;1)i
;1
^
^ bbi ^
^ bk ^ bk ^
^ bk
bi (v )b1
+1
^
^ bk ^
^ bbi ^
^ bk
= +1
74
l:
+
l+
+
&% #1 % % 3#(, 3%, % (v (b ^
^ bk )) ^ (bk ^
^ bk l ): %( 7 i k + j , j ( 73 ( 1 : : : l, - /(1 #11 % % 1
(;1)k
Xl j
(;1)j (b
=1
;1
1
+1
+
k j ^
^ bk l = ^
^ bk) ^ bk j (v)bk ^
^ bd = (;1)k (b ^
^ bk ) ^ (v (bk ^
^ bl )): +
1
+1
+
+
+1
C #+ 7 13(# % 3#(6 (3.24). #((# (6, 3( 7%6+ P %- P D+ %9(# #1 72+:. L# a = ai , b = bj , ai bj D1 (3.24) 12 7, ( (#9 #1( a b. & 7(6#(% (3.24) + , 3( k + l 6 n, 7(6#(% (3.25) | 3( k 6 n. & k + l > > n + 1, #(%(#(%, k > n + 1 /( D1+ #(%(# (%6+ | % %, % % : 3#(: #(( 1.
k + l = n + 1, #(%(#(%, k > n + 1 % %+: 3#(: #(( 1, #+ % %+: 7(6 %+ 19, % ( % /( #13 D1+ 1(%29(, 3( #1+ 1%+: #+: %+ 19E /(, %8 %, (%6 ( 2( +(6 7). S1 (3.24) ( D11 7%+:, (, # # 67%(6 73 7% 7 D, ( % D(fg) = (Df )g + f (Dg): (3.26) (3 ##(( % 7 (;1)k 1 %( # % % 3#( (3.24). & #(1 %8 %#(3(# . &1#(6 A | 7%6 , % (, +3+: , %%+ 8 % +: (2 A ! A: ( %9 i ((. . (2, %( ( | (2#(% (2) (2 D, 3 %+ (#
C %(, 3( D #(6 # DD % ( # % E %( (2 <(DD %=) % A ((19 ( 7+%9( DD 6 ). L# 2 %9 i %(# (2#(%+ 7% ((. . D1 %+ (# % #% %36 % (3.26)), ( (6 <#= 1#9(. 3, 1#9( (, DD % # #. /( #+# (#) DD % #(6 8 3# (, ( 2, D7. ;D7+, #, 2+ +(6 :- 7%#(+ 3((9, %#(39(# 2 -1. $DD % %#(39(# 2, %# 2 ( 12 . 3# , # (+ 9( 7%( 7 ( 7:, (# #6 % DD %, #+:, -#+:. &# #7 D11 (3.24) 2 %+7(6 #%: (2 v : . V ! . V %(# #+ DD % % . V , #(%(#(%198 %9 (%: # a =
X
ak
ak 2 .k V ( ia =
X
(;1)k ak :
. &1#(6 A | 9 ## (% 9+ . 1(( Ma b] %1: /(% a b (# Ma b] = ab ; ba. $2(, 3( 9 f 2 A (2 A ! A b 7! Ma b] %(# DD % % A. R7# .n V ##(( 7 D (3.3) # k = n ; 1. & 9+: %+: 1 11 3#: i : : : in 2 f1 : : : ng # : (#1(#(%1( % 7 3# f1 : : : ng, #2, k. 17 (#1(#(%198 3#, ( i : : : in , ( 1 6 i < : : : < in 6 n, 73. %6 # 671 <+-1= b 73 (#1(#(%198 2(, 2 #7(6, 3( ;1
1
;1
1
1
;1
;1
ai1
^
^ A ^ ain
;1
=a
1
76
^
^ abk ^
^ an:
/DD ( ai :::in 2 73(6 3 37 ak E ( 1
;1
a=
n X k
ak a
^
^ abk ^
^ an:
1
(3.27)
=1
F%% (3.27) c D1
v (a
1
^
^
n X n a )= ( k
;1)k
;1
vk a
1
^
^ abk ^
^ an
=1
(/( #(6 (3.25) 7#+: D, (+: ak (v) = vk ), : #1981 %+%1. & D#% D 5 w = a ^
^ an %7( 7D7 1
V
;! .n
;1
V
v 7! v
w
(+ % (: ( #+%(# (: # v = Ax = P k = x ak , ( v w ( % (3.27) # ak = (;1)k xk . . 1 DD %, #71 <(+, 3( d(x + y) = dx + dy, /( 73(6 (+( 7#(% 71(6#, 31 2 % DD 7%. #(%(#(% # 1%#6#(69 #%: 7+- +#( - %+%1, 3( DD % | D7 6 , (. . 3( 2 (6 #( D1 d(xy) = dx dy : : : #:#( ( 3# : +# 36 ( 3= (7#6 (% -9: .@. 6, <;9# R1, 69( ;1=, .: 1, 1989.) F #2(6 (3 7 - -( , ( 3( # 1 36 - 7%, DD , (. . % 3#( 8, 7% %1: DD % %(3 89 #%(6 # # %(# #3 %( . +# 3.
+(# 1(6 (%(, , 3( #%76 DD % # 12 2 +(6 3#(+ #13 -( 8 7#(. +#6 ( 12 (2, % 3 #3(, %6), + % (, 3( 1 + ;1
77
7%#( (6 % 8 7#(, ( #%7+% + 19 9 12, | ( #%76, ( ( #( D7 (:( ( + ( (). (, 3( (# 1 ( #%7, %+2+ D1 (3.26), 7, | 1 /(1 D11 8 #( (+(6. 6: 1) (#9( ## , # 7). L# + 7 7 D11 , ( , %(, #71 2 # + %6 %+2 d(xy). 2) ( #(6 ## , #(6 # 7). L# (# 7, 3( (2 #(%1(
7% ## 1-( 81 %1, ( (6 % %1, 3( ( % 7%#(+ %1: 8: ( % | D7+ DD % . (: # A 2(# #% + #(#(% E , # (# (2 L : E ! E # 2(6 2(6 %#9 1 A ( #( %( #(#(%, 1, (. . (, 3(+ #6 <:-= #%76
2 L # 12), ( # % ( #+# 1(6, :( + 13(6 /D7 DD %. &# (#%7+ 1 # 1) % #+, # (+ + 7# | /( # 7% % .k V , D 5, # #(#(%, # 2 .k V . &1#(6 ( ) | %% ( #%%%) # 7% % V . F(%(#(%198 # 7% % V (<(+ + D1 =) (2 73 37 ( ). 7+%(#, % 2 .k V , k = 0 : : : n, (# %% ( #%%%) # 7% ( ), #(#(%+ 7 #%7 # #:+ #+ 7% % V ( ##(% | # #(%(#(%198 #+ 7% % V ). 0( #%76 ##(( % (, 3( 32]
1 %. # % 33 % 3 $ $ + . 32]
78
(f
1
^
^ fk y ^
^ yk ) = det((fi yj ))ij
9+: f : : : fk y : : : yk 2 V : 1
=1
1
1
:::k
(3.28)
(& k = 1 7#6 #7 (6 (, 3( <%= # 7% % . V = V | /( ( # # 7%, ( 12 (# % V . & k = 0 7#6 %8 3 #7E #61 /(+ . #1(6 +3+ 3#, # 7% : 9( +3 7% 3#). 7+%(# (2, 3( /( 1#% # 7% % .k V (# 73. + %% # 7% % .k V # 869 (, (%398: 7#1 (3.3), 7( 2, 3( /( 7% %+ (# (3.28). @7 # #1(, 3( # 7% # 7%#( ( ( (# % 13(#, 3( ( : 7%#( 7% 72+: D, % % 3#( (3.28) (+ D119(, 7( | 3( 9+:, (1 3( 9+ D+ #(%9(# #1+ 72+:).
3 # <7 73=. $ 1#(, 3( % .k V #18#(%1( # 7% :( + + D1 ( ), ( %+ (# (3.28). &671#6 ( 2 7#, 3( %+-, 73 37 (gij ) (3# (7 ((. . (gij ) = (ai aj )E ( (gij ) = (gij ) | ( /DD (% #(%(#(%198 # 7% % V ((. . gij = (ai aj )). F 7% % .k V 73 (# 7% 7#+: /(%, (. . 3# 1
0
;1
gi :::ikj :::jk = (ai ^
^ aik aj ^
^ ajk ) 1 6 i < : : : < ik 6 n 1 6 j < : : : < jk 6 n: 1
1
1
1
1
1
L# + :(, 3(+ %+ #6 (3.28), ( 2 +(6
gi :::ik j :::jk = det((air ajs ))rs
= det(gir js )rs :::k : (3.29) C 7, % V 2( #18#(%%(6 (6 # 7% 2 (6 + D1 , 1
1
:::k
=1
79
=1
( %+ (# (3.28). (: ( () %+2(# #198 7: (f y) =
X
i <:::
gi :::ik j :::jk fi :::ik yj :::jk : 1
(3.30)
1
1
1
1
1
C 6 #(#(% ((6 /( ##12. &671#6 ( 2 7#, ##( + D1 ( ) % lk V , + ## (3.30) # /DD ( (3.29). (
(+ 1 #13 k = 0, 36 ( #( 7% 3#E :%(+%(# (3.30), #(%(#(%198 2#(% #% + + 1#(+.) 4#, 3( /(( D1 #(3+, #, 7%#( ( # 67%+: #( (, %(# %+2+ ( #: # 7% % V 2(6 | 2(6 +) 1%(%( 1#%9 (3.28). B# gi :::ik j :::jk 3( + + -6 i < < : : : < ik j < : : : < jk (% %# #13, (6 (: 73: #% # 67%#6). : ( 6 9+: i : : : jk ## ( 2 D1 (3.29). %+ gi :::ik j :::jk ##(3+ % (6#( i : : : ik j : : : jk , (1 3( #(% %1: #% ir % (, D1198 % (3.29), 9(# #( % #(,
#(% %1: js | % #( . C 6, %2+ 671#6 73, #%7+ # (3.12) ( %+2 ( 72 k-D+ % 7# (3.3)), 13 f = f ^
^fk , y = y ^
^ yk
1
1
1
1
1
1
1
1
1
1
1
f : : : f y : : : y jk ik j i (f y) = gi :::ik j :::jk : : : : : : : : : : : : : : : : : : : : : : = fki : : : fkik ykj : : : ykjk i <:::
1
1
1 1
1 1
1
1
1
1
::: g
1
1
1
g
1
1
1
1
1
1
80
1
gi j f : : : gi jk f i i = : : : : : : : : : : : : : : : : : : yj : : : ykjk = gik j fk : : : gik jk fk gi j fik y : : : gi jikk f yk (f y ) : : : (f yk ) i jk i j = : : : : : : : : : : : : : : : : : : : : : : : = : : : : : : : : : : : : : : : : : : : gik j fkik yj : : : gik jk fkik ykjk (fk y ) : : : (fk yk ) 1 1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
C #+ 7 (3.28). C 6 #, 3( (f y) 72+: f y 7%#( ( # 67%+: (, ( 9+ f y %9(# #1 72+:, ( 7%##(6 (f y) 1#(% 9+: f y. #((# 1(6# % %+2#( ( ). &1#(6 #:+ 7# fai g | (%+, (. . %# gij = (ai aj ) = dij (7 <1#= #93(#, # 7% 2( +(6 #%%%+). C fai g | (%+ 7# % V (%6 (gij ) = (gij ) ). (1 1(6#, 3( (3.3) | (%+ 7# % .k V . 6 # gi :::ik j :::jk 6= 0, ( % 2 #(3 ( det(gir js ) 2 (6# 1% /(, (% 13(# (6 ir = js E (, 2 : 3# i : : : ik % 1 7 3# j : : : jk , ( /( ##(+ 3# | %7#(98, ( %# ih = jh . (, # %# ih = jh , ( % ( det(gir js ) (3+ ( 1 (6 6+ /(+, %+ 1, /(1 gi :::ik j :::jk = 1. @7 #18#(%% % .k V (% 7# #1( %+2#(6 D1 ( ), ( 3( /( #(%(6 # 7%. # #: ( ) % V | 2(6 , ( % % ##12 #93(# 7 <1#=, /(1 % .k V 13(# 2(6 #
7%. &7 (#, 3( # # 671+ 7# | (%+, ( gi :::ik i :::ik = 3#1 (: ih 3( (aih aih ) = ;1: (3.31) 0( #71 % 7 (3.29). &: D 5. +- (3#6, 3( #(#(% .n V 7D R, 3( 7D7 ;1
1
1
1
1
1
1
1
1
81
.n V
^
an
Rn (# %+ 7# /( a ^
^ n 1
#(#(% . V . 8 #13 ( : #% 3((6 7#+ /( 11, /(1 ( #(( 7D7 .n V Rn . &2 (#, # % V %%+ # 7% ( E ( % .n V #198 7 %+(# < 3((6+= 7#+ /( |
^ an %(#
#( D( + 2(# #(#(%. $ #( 3# 7(6#(%. &1#(6 (b : : : bn ) | 1 2(6 (%+ (%+ 7# % V , (b : : : bn ) | %7+ 7#
1
33]
1
1
1
1
1
1
1
1
1
5+ . $ $ %, $ ai (ai ai ) < 0, | (ai ai ) > 0. 1 $ $ % 3 0 w. 33]
82
% V E 1(6#, 3(
a1
^
^ an = b ^
^ bn: 1
@(# % 7% C : V ! V , %8 (a : : : an ) % (b : : : bn ). #%(6 (7 7# (%+ ). &/(1 det C = 1, 7 ( : 7#% %+, ( det C > 0, ( 3( det C = 1. 3 . 4 x2 (3#6, 3( C bi = ai . &/(1 a ^
^ an = C (b ^
^ bn ) = det C b ^
^ bn = b ^
^ bn : ##( ( 6 %19 ##(1 ( # 7# (b : : : bn ) %7+ 7# (b : : : bn ), (% (% 2(6. C (#. (3.20)) b ^
^ bn = det C a ^
^ an = det C w C | ( #(%(#(%198 (H ). (#9 2 + + %+7(6 w 37 #%719 # %( ##( ( D1 b ^
^bn , /( + + %+2 w % (: %( ( ##(+, (1 3( -#6 + (6 det C , C #%7 # (+ ##(. 2 13(6 %+2 w % (: %( ( ##(+, % ( D119( 8 %3, #%7 # %+2 # 7% % %( ( ##( ( ( . (H ) + 73 #(+ (+ 37 xi %+ | 37 yi . F3# 11 % 1: (7 @xi #3# #(6 xi %#( yi . H(, 3( ( @xi | /( ( C 7 (H ). L (6 + 73 3 37 J , ( 3( 1
1
34]
1
1
1
1
1
1
1
1
1
0
0
J = det
@xi 0
@xi
:
, , (ai aj ) = (bi bj ). $ $+ $ % . 34]
83
1(1 2 #:19 ##(1 11 #3((6 (%. 73 det(gij ) 37 g, det(gi j ) | 37 g . = i j gi j = gij @xi @xj 0
0
0
0
@x @x
0
0
0
3( % (3 % 2 7 #(6 (gi j ) = 0
0
@xi 0
@xi
(gij )
@xj @xj
0
( g = J g jg j = jJ j jgj. L# J > 0, ( J = 0
2
2
0
rg
j j
g0 j
j
,
jg j b ^
^ bn = pjgj a ^
^ an:
p
0
1
1
L# ( 6 %7(6 #:19 ##(1 (%, 2(6 (% 1%(%98 #-9 1 ai # 73+ 7 (ai ai ), ( 13(#, 3( w=
p
jg j b ^
^ bn: 1
0
0( 13#6 J > 0, (. . % ##( ( (% 2(6. L# 2 1#(# : (% 2 (%+ ##( ( r g (#(% 299 D1 5), ( J = ; g 0
j j
w=
;pjg j b ^
^ bn:
j j
1
0
H( #((, 3( g % 1 # 12 %#(3#6: ## (3.28), g :::n :::n = g: S 5 w 7%( 1#(%(6 7D7 +: %(+: #(#(% 1
R
1
;! .nV
84
l
7! lw:
#%7 # D 5 %%( ( 7+%+ #% % { B%( ! 8 > 1 : : : n >e # fi : : : i g = f1 : : : ng n > > < i : : : in ei :::in = ((. . # fi : : : in g #(6 ( > >
#(% #%% f1 : : : ng) > :0 % (% #13. 1
1
1
1
@# 671 , 2 #7(6, 3( ai ^
^ ain = ei :::in w (3.32) # #%+ i : : : in %# 73+, ( % : 3#(: /( %#(% #(( 0, # 2 %# 73+, ( 3( fi : : : ing 13(# 7 f1 : : : ng #(% ( #(%, ( 1 ::: n i i n a ^
^ a = e a ^
^ an : 1
1
1
1
1
i : : : in
1
1
F 869 #% % { B%( 2 #198 7 %+7(6 (6 det(vji )ij :::n : =1
vi : : : vi : : : : : : : : : :n: = e vi : : : vin n in i :::in in 1
1
1
(3.33)
1
v
1
1
: : : vn
1
# % 71%(# #1% i : : : in , 7%# 98 73 1 : : : n. S(3# /( #(6 D7% (. @7 2 #( (# /(1 (7 h- #( (# vhih ) %++ /(+ 29(#. &13(# nn 7%, (+ 129(# #(%(#(%198 e::: 7( #+%9(#. & + #93, 3( 7 %1: 7+: #( % 11( %++ /(+, 28 % ( 2 #(E % /( #13 % #% ih #% 9(, #(%(#(%198 ei :::in = 0. C 7, % % 3#( (3.33) #((# (6 n! #+:, 1398:#, 7 7+: #( %+9(# /(+, 28 1
1
85
% 7+: #( :. F(%(#(%198 7 %+ 1:1: :%n7%# #( ( 3(#( 3(#( #(% i : : : i , n /( #(6 % +3 (. (2 (2 ##(6# (3.12), % ( k = n, ai :::in = ei : : : in . % 3#( #1 i : : : in , (+: 1 6 i < : : : < in 6 n, #%(# 1-#(%1 31, % ( %# ij = j .) C 6 #(%(# 3%+ #198 %+2 D+ 5 <% (7 1:=: 1
1
1
1
1
p
= jgjei :::in vi : : : vnin (#1% %(98# #E 7 7%#( ( (, %(# ##( ( (3.34) (% 2(6 ( (6).
w(v1 : : : vk )
1
1
1
p
@7 (3.34) %#(%1(, 3( 3# jgjei :::in 71( #%(7.
2 3# ei :::in %(# (7, #%(7. ( 3(+ <#%# %(#=, # < #%# %(#=. &#(, 3( 13(#, # (6 ei :::in (7 % (2.16). & 7 T:::::: e::: % 3#(6 j jn (2.16) ( % ej :::jn @xj : : : @xjn : & (3.33), /( % 1
1
1
1
@x
1
0 1
@x
0
@x @x @xj : : : @xjn : : : : : : : : : : : : : : @xn @xn : : : @xj @xjn 1
1
0 1
0
0 1
0
L# fj : : : jn g 6= f1 : : : ng, ( 7#6 9(# % %+: #( , (6 % 0. L# fj : : : jn g = f1 : : : ng, (, #(% #( +, 13 0
0
1
0
0
1
@x @x 1 : : : n @x : : : @xn e j : : : jn :@x: :n: : : : : : :@x: :n: : @x : : : @xn 1
0
0
0
1
0
1
0
0
1
1
86
C 7, %#
@x @x : : : @x @xn @xjn : : : jn = ej :::jn : : : : : : : : : : : : : : @x @xn : : : @xn @x @xn 1
@xj1 ej1 :::jn j 0 @x 1
1
0 1
0
1
0
0
0
0
0
1
L# + ##( %3 ei :::in + (7, ( 2 + + 13(6# ej :::jn , 13#6 # 2( @xi det i = J . @(, 1
0 1
0
;1
@x
0
ej10 :::jn0
j
jn
= J ej :::jn @xj : : : @xjn : @x
1
1
0 1
@x
(3.35)
0
#%7 # ( % (7 #3# 1 # (7 ##(%9( (+ #(%+ 5(+. &%, # #(%98 2 ( ##( % V :::ik , 7+%(# (7 (#(69 %# r, nk l 3# Tji :::j l # 9+: %1: ##( ( xi xi /( 3# #%7+ #198 7: +
1
1
0
i
i
j
jl
:::ik r T i :::ik @x : : : @x k @x : : : @x : Tji :::j = J j :::jl @xi jl l @xik j 0 1 0 1
0
;
0
0 1
1
1
1
0
1
@x
0 1
@x
0
C 7, ei :::in | /( (7 (#(6 %# ;1. , + %, 3 # 7% ((3# (7 gij ) ( ei :::in 2 ( #(81 (71, %, #%(71. (%#6 1(1 % #(1, 731, 3( % (: #13 # 869 #% % { B%( %+29(# (+ Mv w]i %( 7% Mv w] %(% v w # ( vi wi . , # 67% (%
2(6 (% ##(+ ( 1
1
Mv w]i = eijk vj wk : (C( D(, 3( #% # i #(( #%:1, # % | #71, #%7 # (, 3( % #(#(% # %%+ #+ 7%87
# 67% (%+: ( #((# 73 21 %( %(, (2 %8 21 (7, 7398# %: 2
2 ( #.) C 6 + ( (3# # #(#(%, # 2 .k V . .k V ##(( 7 +: 7 +: D f, 1( (+: %9(# k %(% v : : : vk . 3 2 ##((6 + 7 + D+ F , 1( (+: %9(# k %(% a : : : ak E ( F 7+%9(# %( #( k, 3 | k- %( 8 3 | k-%(. = % V V D(3# % %+ (3( :( D6 %+2 % ( D(, 3( V = V ), ( %# #7 %+- +: 7 +: k-D: (#(# k- %(. & 2 + 2 2 ##(%(6 k%(+ k-D+, #3( #:+ #(#(% V , V E /(1 1(%2 k-%(: %9(# (6 , D6+ ##(% 1(%2 k-D:. k-%(+ n (2 719( %( #(#(% 7#( Cnk = k , ( 73(# 37 .k V E (# (2
1
1
.k V
.lV ! .k
(F G) 7! F ^ G
lV
+
3 /( 2 #(6 %1 # #, 7 (+: + 1 67%(6# %+E %(+
ai ^
^ aik 1 6 i < : : : < ik 6 n a : : : an | 7# % V 1
1
1
(3.36)
719( 7# % .k V . k- %( 7+%(# 72+ #(+, # %(# %- 7% %(% 7 V . L# #(6 F = v ^
^ vk %(+ vj 9( % 7# 1
88
a : : : an (+ vji , ( F % 7# (3.36) ( (+ 1
vi : : : vi k F i :::ik = : : : : : : : : : : : ik ik 1
1
1
(3.37)
1
v
1
: : : vk
: 3#( +, 21, ( #%
%(6 # 73 k-D #%9 # k- %(. &2 (# % DD 6 (, 21 #(#(% Tx M (#(6 #(#(% 79 M % (3 x) # 2+ 1 #(#(% Tx M (# ( 736 #(3#(6 % (- : #%7 # M . ( 6 k-D 6-, 3 6 k-%(%. C, 3( + #: # 6 7#6 k-D, k-%(: % 3-( 3, 5#(# (, 3( 13(+%9 /( 73 % : 118 . $ # 6 k- %(% ##((
2 %# % (, 3( .k V #(#(%+ 7 7+%(# #(#(%, # 2+ .k V , ( 3( # 869 k- %(% 13(# ( # %# #(#(% (.k V ) , ( (+: /(% | +: D1 % .k V . ( 11( D1%(6 #6 + %.) HD#1 -16 7# a : : : an % V , %76 %7+ 7# a : : : an % V 1 67%(6# 7# (3.3) (3.36) % .k V .k V . C #+ % #: %1: #(#(%: %%(# (+ 1#(%%9(# 7D7+ n n .k V ;! R( k) .k V ;! R( k) (3.38)
1
1
(k-%(1 k-D # #(%(#n () n) ( ) ( (. F3# + #(% # R k R k + 3#, 1%+ n #, (+ ( 73 ( 1 k , k # i : : : ik , 3+ n 1#% 1 6 i < : : : < ik 6 nn.) = R( k) %(# # 2+ #(#(% R( k) , ( # 869 (3.38) + 2
1
1
89
219 f 2 .k V ##(%(6 + D1 .k V , .k V 7+%(# # 2+ .k V #(#(%. , #
f=
X
i <:::
fi1 :::ik ai1
^ : : : aik
X
F=
1
i <:::
1
F i :::ik ai
1
^
^ aik
1
(. . # F i :::ik #1(6 (+ F fi :::ik | (+ f, ( + 1
1
f(F )
=
X
i <:::
fi1 :::ik F i1 :::ik :
(3.39)
1
C 13(# 7D7 .k V
(.k V ) :
(3.40)
& #( # 67%#6 (+. $2, 3( # /(( 7D7 ( : 7%#(. &2 %#, 9+: v : : : vk 2 V 1
^
^ vk ) = f(v : : : vk ): (3.41) $#(%(6, F = v ^
^ vk ( (+ (3.37), f | f(v1
1
1
(+ fi :::ik = f(ai : : : aik ) (#. # (3.10)). & (3.39), 1
f(F )
1
=
X i <:::
vi : : : vi k f(ai : : : aik ) : : : : : : : : : : : ik ik 1
1
1
1
v
1
: : : vk
= 3# f(ai : : : aik ) ##(3+ i : : : ik , ( 7 73. (#8# (3.12), #1(, 3( 1
1
f( F )
= f(ai : : : aik )vi : : : vkik = = f(ai vi : : : aik vkik ) = f(v : : : vk ): 1
1
1
1
1
1
1
C #+ + 7 (3.41), (1 #1(, 3( 73 D1 f 72+: k-%(: 7%#( ( # 671+: (. ( 2+ %( F %(# #1 72+:, ( f(F ) 7%#( ( (. 90
6 # 7D7 (3.40) 7%#( ( (, + % % ##(%(6 #(( (2#(% .k V # (.k V ) . F( +(6, .k V = (.k V ) = (.k V ) . 2, , ( + 7D7 .V = (.V ) (. . 12 %#6, % 6- 12+ #(6 k- %(+, #6 k-D+, k- %(+ + 1 %(6 (6 # # # +: D1 % .k V . &#(%9 3((9 1(6#, 3( 7#+ (3.3) (3.36) %9(# %7+ 7#. 3 (1 /( + # % .k V , 2 %%#( # 7% % .k V . F 1 #(+, # #:(6 7 12 # # 7% % .k V ##(%(6 .k V # 2 .k V #(#(%, ( % .k V (# <(+ + D1 =. 0( % #+: 7% % .k V #% 9(. 4 11 #(%%(6# /( , ( # 7% % .k V (#E 2 3((6 # %( %# ##12. + 7, 3 % %( #(#(% # 7% %7( (+ 7D7 21 /( #(#(% # 2+ 1, 7%#8 ( /( # 7%. #13 %7( (+ 7D7 .k V .k V , (+ % 3 #3( 7%#( ( #: # 7% % V . %(6, 3( /( 7D7 k-D f # /DD ( fi :::ik k - %( F # /DD ( F i :::ik #(%(#(%19( 1 11 ( (6 (,
1
1
F i :::ik =
X
1
gi :::ik j :::jk fj :::jk 1
j <:::<jk
1
1
(3.42)
1
, 3( /%%(, fi1 :::ik
=
X j <:::<jk
gi :::ik j :::jk F j :::jk 1
1
1
1
gi :::ik j :::jk | /DD (+ % %+2 #
7% % .k V % (: (, (%398: 7#1 (3.36). 0( +3+ D1+ 1# #%, 1
1
91
(6 ( 6 <#= | 3#, 3#. (H#6
(# (%#( 7#%.) $1:1 (7 #3# 2 /%%(+ D1+ F i :::ik = gi j : : : gik jk fj :::jk fi :::ik = gi j : : : gik jk F j :::jk ( 71%(# #1% %(98# #). 0%%(#(6 /(: D1 13(# # 869 73, #%7 # (3.12). F18#(%, 3( /(( 7D7, (+ 2 #3((6 <+= 7D7 .V = .V , #:( 9 ^, (. ., % % (: <= 7D7, %(# 7D7 . H#6 7 1 %(6 % /(: (:. 76 (%+ 7# (a : : : an ) % V %7+ 7# a : : : an % .V . &1#(6 # % # 7% | %%. C %# gi :::ik i :::ik = 1, 3 gi :::ik j :::jk = 0,
/(1, % 7 (3.42), %- D f # ( fi :::ik #(%(#(%1( k -%( F # ( 2 (: F i :::ik = fi :::ik : (3.43) 3#(#(, 2 7 7#+: D (3.3) #(%(#(%1( 7#+ %( (3.36) (# ( 2 #). &7% %1: 7#+: D (ai ^
^ aik ) ^ (aj ^
^ ajl ) #(6 16 (# 1 : #(6 8 #2(6 % ah ), , # (3#(69 7, #% 7# D, ( 13(#, # 1#((6 # #(%(6 #+ % %7#(E 7 (# 3(#(69 #(%(#(%198 #(%. 1%6 ( 2 %+3#(# 7% (ai ^
^ aik ) ^ (aj ^
^ ajl ) #(%(#(%198: 7#+: %(%. ( 7 13(# 16, (+ 7#+ (k + l)-D (k + l) %( eah ^
^ ahk l eah ^
^ ahk l 1
1 1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
+
1
92
+
1
# ( 2 hr ( 2 e = 1. 0( D
%( 7 #(%(#(%19( 1 11 ##(% 7D7. $ -6 #2(#, # # 7% %(# 2(6 +. /( #13 %#( (3.43) %- D f # ( fi :::ik #(%(#(%1( k-%( F , (+ ( 1( (3(6# ( ( f 7: 1
F i :::ik = gi :::ik i :::ik fi :::ik : 1
1
1
1
3#(#(, 2 7 7#+: D ai
%(
1
gi :::ik i :::ik ai 1
1
1
^ aik #(%(#(%1(
^
^ aik :
L# 7% %1: 7#+: D % 19, ( 7% %(% % 19. L# 7% %1: 7#+: D #(6 eah1
^
^ ahk l
(3.44)
+
e = 1 ## #71 %+-, ( 7% #(%(#(%198: %(% #(6 eg i1 :::ik i1 :::ik g j1 :::jl j1 :::jl ah1
^
^ ahk l +
21 ( D (3.44) (%3( %( eg h1 :::hk+l h1 :::hk+l ah1
^
^ ahk l : +
/( (( 2 #+ %(, 7 (3.31) (, 3( 3# h : : : hk l #1(6 i : : : ik j : : : jl ((6 #(%+ % -( 1 ), #1(, 3( 1
+
1
gh :::hk 1
1
l h :::hk l
+
1
+
= gi :::ik i :::ik gj :::jlj :::jl : 1
1
1
1
+- + %1( 7% v a = iv a = = i(v)a %( v 2 V %- k-D+ a V . C 6 2 93
3 (6 %1( 7% i(F )f k-%( F 2 .k V l-D+ f 2 .l V , l > k. 0( #(6 %- (l ; k)-D, 98 l ; k %(: v : : : vl k 73
1
(i(F )f)(v : : : vl k ) = f(F ^ v 1
;
1
^
^ vl
;
k ):
;
& l = k 1 i(F )f 12 #((# 1(%E % /( #13 (# % ( #+#, 3( i(F )f #(6 3# F (f) = f(F ). 0( (1( #(1(1+ % V , #(1(1+ %( #(#(%. (( (+ #(+ #%#(% %1( 7%. L# #1 #( %(% F G % #( %- D+ f, ( (i(F )f)(G) = f(F ^ G). $#(%(6, 72 G /( ( 9. 8 2 #13 G 2 #(%(6 #11 72+: %(%. F(%(#(%, 3#( 7+% %#(% #(%9(# % % #1, 3 #+ % % 3#( %+ #+ % % 3#(. $, 12 7%# ( 1#% #( :
i(F ^ G)f = i(G)i(F )f:
(3.45)
L# #1 #( F G 6- #( f, ( % : #(: (3.45) #(( 16. L# /( #1 % #( f, ( (3.45) D(3# #%(# +18 D1 ( 31?) L# #( 6 f ; #( 6 F ; #( 6 G = h > 0
( 9+: v : : : vh 2 V 1
(i(F ^ G)f)(v : : : vh ) = f(F ^ G ^ v ^
^ vh ) = = (i(F )f)(G ^ v ^
^ vh) = (i(G)i(F )f)(v : : : vh ): 1
1
1
1
F198 % #%#(% %+( 98 %6 3#(+ (#. (, ( #(6, (#. L# a 2 .r V b 2 .n r V v 2 V ( a ^ i(v)b = (;1)r i(v)a ^ b: ; +1
;1
94
(3.46)
F( 6 D+ a ^ b % n + 1, ( 3( a ^ b = 0. @# 671
(1 #1( (3.46). $,
# f 2 .k V w 2 .n V F 2 .k V ( f ^ i(F )w = f(F )w:
(3.47)
$#((3 7(6 /( 72+: F ( 31?), (+: 36 ( ( 1(%2: # f 2 .k V w 2 .n V v : : : vk 2 V ( f ^ i(v ^
^ vk )w = f(v : : : vk )w:
1
1
1
%1 (3.45), %19 3#(6 2 #(6
i(vk ) : : : i(v )w: 1
& (3.46) c r = k, a = f, b = i(vk ) : : : i(v )w, v = vk ,
13, 3( # %+2 % ;1
1
(;1)k i(vk )f ^ i(vk ) : : : i(v )w: ;1
;1
1
F% (3.46) c r = k ; 1, a = i(vk )f, b = i(vk ) : : : i(v )w, v = vk , < #= % i(vk ), /( %(# 2(6 (;1)k . &2 /(( ##, # % %# i(vh ), 3( %( %+29 ;2
;1
1
;1
;2
(;1) k
k
:::
( ;1)+( ;2)+
+1
i(v ) : : : i(vk )f ^ w 1
( %%1 (3.45) %
(;1)k k = i(vk ^
^ v )f ^ w = f((;1)k k = vk ^
^ v )w ( ;1) 2
( ;1) 2
1
1
3( ( 12+ 716((, #61 k k ; 1(;: :1): k1k = vk ^
^v = = v ^
^ vk ( %(6, 3( e 1 2 : : : k = (;1)k k = ). ( ;1) 2
1
( ;1) 2
1
95
C 6 2 ( (1 . 71%( # 67% ( D#% # 7% % V #(%(#(%198: #+: 7% % %#: .k V , (2 %+ ( % V . L #% #198 1(%2 #2(# % : 9 k-D+ y 2 .k V #18#(%1( #(% (n ; k)-D y 2 .n kV , ( 35]
;
f
^ y = (f y)w
%#: f 2 .k V :
(3.48)
$7(6#(% ##(( 7 %1: 3#(. - %+:, + %, 3( % 3#(% y 2 %7(6
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(3.56)
@# 671 D11 (3.47) # w ^ F %#( F (#( 6 /( %( 7 % #( k D+ f): f
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f
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1
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1
104
(3.57)
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^ gcih ^
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1
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1
1
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i(w) = i i(aj )w = i(aj )
h
a
^b
=b
i(bj )w = ;aj
(7#6 # 67% (3.24)). &/(1 (#% # 671 (3.24)) i(aj )(w ^ f) = i(aj )w ^ f + w ^ i(aj )f = bj ^ f + w ^ i(aj )f
i(bj )i(aj )(w ^ f) = i(bj )(bj ^ f + w ^ i(aj )f) = = f ; bj ^ i(bj ) ; aj ^ i(aj )f + w ^ i(bj )i(aj )f 3( #1% j (# 13( (3.58) (, 3( P i(bj )i(aj ) = i(w)) ( (3.59). 105
