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x y y x, x y z x y z, λ µ x λx µx, λx y λx λy, ∃ ∈X ∀x∈X x ∀ x∈X ∃ −x∈X x x x.
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!
k j
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n
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n
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n
k
k j
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8"
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E f x ∈ D
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n
n
m
n
f
f
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+
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n
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m
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f
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- 33 DZ f G D '⊂ + → E '⊂ + R
n
R
n
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f
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' + H XYZ{f 'x+ G x ∈ D}, ' + H L[\ {f 'x+ G x ∈ D}.
f x f x
.) $!' ( " 1$! &,
§ § 0$! 313 ! 33 % f G Df '⊂ Rn+ → Rm
!
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R
n
R
n
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DZ f G D '⊂ + → x 1
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f
R
n
R
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fk x
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k
n
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f
x
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f
f
R
n
R
f
n
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f x
j
j
j
f
7 9 5 99 5 99 5 "
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x
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f x
j
j
j
f
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§/ // § 6 "3 3 DZ 9 5 f x x " 4 9 5 f 99 5 x " /) 33 # 3 # " $! ! "!(+
§/ / § 0$! 1/ ! "33 DZ 9 5 f 99 5 x " $99 5 9 5 f x
' + ' +H ' + GH : dx H 'dx, . . . , dx + ∈ df x
df x , dx
n
R
∂f x dxj . ∂xj j n
n
99 5 "
/. 33 # '! &, !2!- 3 #
§/ / /9 § / 0 $! 213
8 6 "3 3 DZ
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dv t dt
5 "3 3 DZ
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' + H ∂u'x't +, y't+ ∂x't+ I ∂u'x't +, y't+ ∂y't+ , k H 8, ∂x ∂t ∂y ∂t
∂v t ∂tk
k
5 "3 3 DZ
.
k
@" 9 5 xk H xk 't+ k H 8, . . . , n + t ∈ Rm 99 5 tA @@" 9 5 u H u'x+ + x ∈ Rn x H 'x't+, . . . , xn't++ 99 5 " 4 9 5 v't+ H u'x't+, . . . , xn't++ 99 5 t
' +H
∂v t ∂tk
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∂u x t ∂xj t ,k ∂xj ∂tk j n
5 "3 3 DZ
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' +H v't+ H u'x't++" dv t
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∂u x t ∂tj j
j
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j
// DZ!8$!+( "! "$ 5
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lh x
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ϕ h 9 5 f x " 5 "3,3 7 9 5 " 1 grad f 'x + H O h ∂f - " , grad f 'x + H O ∂h ∂f ∂h H ! h" /: DZ!8$!+1 +33 #1 $1 6* "!(+ !$ 3 # !, * " ' 1*
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' + H ∂x∂ .f.'.x∂x+
s x
fk
s
k½
ks
9 5 f & C" , k H 'k+
' + GH f 'x+. +
x
fk
CC" ,
H 'k, k, . . . , k ' + ∂ f 'x+ GH ∂
k
s fk¾ ,...,ks·½ x s
∂xk½ ∂xk¾ . . . ∂xks·½
s
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s
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∂ f ∂x ∂x
∂ f ∂x ∂x
x" 4
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R
R
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
♠
5 "3+3 DZ 9 5 f - B 'x, r+ ∈
R
n
+ s x" 4 k s k H 'k , . . . , ks + fk 'x + kj " 4 σ G {8, . . . , s} → {8, . . . , s}
' + H ∂ f 'x+ . ∂x . . . ∂x 4 3 DZ % $ ( 5 f G B 'x, r+ ∈ → #, ! $ % $ $ s $ : & B 'x , r+ ! x $ ∂ f 'x+ ∂ f 'x+ ∂ f 'x+ GH H ∂x ∂x . . . ∂x . . . ∂x . . . ∂x ∂x . . . ∂x ∂ s f x ∂xk½ . . . ∂xks
s
kσ´½µ
kσ´sµ
R
|k|
|k|
k
k½
n
R
|k|
k½
n
n
$ kn $+ $ % %$ k k H 'k, . . . , kn+ |k| s ; $ %$ < k
; ! "3+3 DZ 9 5
- B 'x , r+ ∈ R + s x" $99 5 dsf 'x+ & '" , s H 8 " D"*"'A (" ds f 'x+ GH d'dsf 'x++ s ∈ N s > 8" - "3+3 , x H 'x, . . . , xn+ f
n
l
' +H
df x
J J
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l ∂ |k| f x dxk , k ∂x k k½ ,...,kn k
dx H 'dx, . . . , dn+ + k H 'k, . . . , kn+ |k| H l" , 9 - &
∂ dx ∂x
I ... I
∂ dxn ∂xn
l
GH
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df x
l
∂ dx ∂x l
∂ dxn ∂xn
k l
k
l
J J
l ∂ |k| k dx , k k k½ ,...,kn k ∂x
l
'+
f x.
k l−k l−k dx dx .
k
/;
§ / §2 2 2 0$!
41 3
DZ f & B'x, r+ ⊂ # B s$ ! "313 DZ x H x I ]x ]x < r" ∂ f 'x + P ']x , ]x + GH C ]x]x ∂x ∂x R
s
s
j
s j
s
j
j
s−j
s−j
9 4 s 9 5 f x" ? Rs']x, ]x+ GH f 'x, x+ − Ps']x, ]x+ ! ! ( ( 5 f ! x" - "313 DZ P 'x+ H kk ,...,n ck xk x ∈ Rn " ,|k| s n , P 'x+ H ! x ∈ R P 'x+ ≡ !" - "313 ,
½
' I ]y+ H
f x
k k½ ,k¾ ,|k|s
ck xk
I o']x+, ]x < r, ]x → !. s
8 ∂ f 'x +
! c H kJ ∂y k H 'k, k+ k I k m" 6 "313 $ ]x ∈ ]x < r ! ζ ∈ '!, 8+ ( |k|
k
k
R
! ! ( = )&
I ]x+ H j8J ]x ∂x∂ I ]x ∂x∂ f 'x, x+ I I m8J ]x ∂x∂ I ]x ∂x∂ f 'x I ζ ]x, x I ζ ]x+
' I]
f x
m−
x, x
j
j
l
m ( ! ! ( DZ &
' I]
f x
x, x
I ]x+ H j8J ]x ∂x∂ I ]x ∂x∂ f 'x, x+ I Io']x +, ]x < r, ]x → !. m
j
j
m
4 3 4 $ * *, ( 5 , * ! $ + + > %
]x < r , ζ ∈ '!, 8+ !
' I ]x+ H
f x
]x
@ H j8J ]x ∂x∂ I . . . I ]x ∂x∂ f 'x+ I I m8J ]x ∂x∂ I . . . I ]x ∂x∂ f 'x I ζ ]x+ H 8 ∂ f 'x+ ]x I H kJ ∂x 8 ∂ f 'x I ζ ]x+ I kJ ]x . ∂x m−
j
n
j
m
n
n
n
|k|
k
k
k∈Nn ,|k|m−
|k|
k∈Nn ,|k|
k
k
m
? $ $ m
;( ! ! ( DZ <
' I ]x+ H H j8J ]x ∂x∂ I . . . I ]x ∂x∂ f 'x+ I o']x +, ]x < r, ]x → !. f x
m
j
m
n
j
n
) * !# + # & ! " #" 9 " + (
§ §4 4
DZ f G D '⊂ + → x 1
# D $ R
f
n
R
f
! ,33 4 x ! % 9 5 f ∃r>
!G
'
+ ' + f 'x+.
∀ x ∈ B x , r f x
, ! < x ! % 9 5 f "
C ! ,33 4 x ! % 9 5 f ∃r>
!G
'
+ ' + f 'x+.
∀ x ∈ B x , r f x
, ! > x ! % 9 5 f " ! ,33 7 x 9 5 f % ' " 7 x 9 5 f % ' " 6 ,33 DZ 9 5 f x " , 9 5 ∂f 'x + f H ! j H 8, . . . , n ∂xj df 'x+ H !" 9% $+& 1 3!'1
§ §4 4 ! ,33 5 A H 'ai,j +ni,j ! ! ai,j H aj,i i, j H 8, . . . , n" ! ,33 DZ 5 A
H 'a +
n i,j i,j .
> 5 P G Rn → R ! &
' + GH
P x
n i,j
ai,j xi xj , x
H 'x, . . . , x + ∈ n
R
n
$ ! ( " 5 A 5 9 P "
E 4 3 ? ( 5 f $
! x ∈ Rn ' ! ! $
$ $(( 5 $ $ ( 5 ! x ∈ Rn ) % $ ! # ( dx H 'dx , . . . , dxn + - ,33 , P G Rn → R 9 P 'λx+ H λP 'x+ λ ∈ R x ∈ Rn " ! ,33 1 9 P G Rn → R '" ) % P 'x+ > ! x ∈ Rn" (" 5 % P 'x+ < ! x ∈ Rn" )" 5 % P 'x+ ! x ∈ n R " *" ) % P 'x+ ! x ∈ n R " 1 9
$ : 5 " , 9 + + $ : " , 9 5 + " - ,33 # 9 " - ,33 , 9 P G Rn → R
ML[{P 'x+ G x H 8} > !. , 9 P G → 5 MRS{P 'x+ G x H 8} < !. , 9 P G → + XYZ{P 'x+ G x H 8} > !, L[\ {P 'x+ G x H 8} < !. R
n
R
R
n
R
7! 6 ,33 7 5$ : DZ n H
9 P G R → R 5 A H 'ai,j +i,j" '" 1 9 a, > ! a,a, − a, > !" (" 1 9 5 a, < ! a,a, − a, > !" )" , a,a, − a, < ! 9 P " 9 ! &&! 1 !$(
§ §4 4 0$! 1 3
DZ f #
x $
x ∈ R
6 ,3 3 DZ x 9 5 f ai,j
GH
' +
∂ f x . ∂xi ∂xj
'" , a, > ! a,a, − a, > ! x 9 5 f " (" , a, < ! a,a, − a, > ! x 9 5 f " )" , a,a, − a, < ! x 9 5 f "
, %# & - : ($! 8+1 3 #
/ § / §/3 /3 0$!
912 2144
78 2 F G D →
DZ D ⊂ 'x, y+ ∈ D 2 F 'x, y+ H !$ ! +33 # 'x, y+ 9 5 F + y 9 5 x H 'x, . . . , x + ! ε > ! δ > ! 9 5 ϕ G B 'x, ε+ → R
n
R
n
R
& '" y H ϕ'x++A (" 'x, ϕ'x++ ∈ B ''x, y+, δ+A )" F 'x, ϕ'x++ H ! x H 'x, . . . , xn+ ∈ B 'x, ε+" - +33 DZ n H " DZ ! 'x − ε, x I ε+ × 'y − δ, y I δ+
'x − ε, x I ε+ × 'y − δ, y I δ+ ⊂ D
9 5 F ! '" 9 5 F A (" x ∈ 'x −ε, x Iε+ 9 5 Fx G 'y−δ, yIδ+ → R Fx 'y + GH F 'x, y + " 4 !
'x − ε , x I ε + × 'y − δ , y I δ + ⊂ 'x − ε, x I ε+ × 'y − δ, y I δ+ x ∈ 'x −ε , x I ε + F 'x, y+ H ! - y H ϕ'x+" > 5 ϕ 'x − ε , x I ε +" $ 'x, y+
y 9 5 x" - +33 DZ !
'x − ε, x I ε+ × 'y − δ, y I δ+ ⊂ D
9 5 F ! &
7 '" 9 5 F ∂F A ∂y ("
'
+ H !"
∂F x , y ∂y
4 'x, y+ y 9 5 y H ϕ'x+ x" ,
'x, y+ ! ∂F ∂x 9 5 ϕ 99 5 ! x
' +H
ϕ x
' '
∂F x , y ∂x − ∂F x , y ∂y
+ +
" - +33 DZ n > " DZ ! δ > ! & '" 9 5 F ∂F - B ''x , y+, ε+A ∂y ("
'
+ H !"
∂F x , y ∂y
4 'x, y+ y 9 5 y H ϕ'x+ x ∈ B 'x , ε+" , 'x, y+ ! ∂F 9 5 ϕ ∂xj ! x
' +H
∂ϕ x ∂xj
"
' '
∂F x , y ∂xj − ∂F x , y ∂y
+ +
77 :% &# = ! !&!2 -
/ § §/3 /3/ ! +33 DZ
`'t+ H 'ϕ't, . . . , t +, . . . , ϕ 't, . . . , t ++, 9 5 ϕ j H 8, . . . , m t H 't , . . . , t +" 5 ∂ϕ 't + ∂ϕ 't + ... ∂t ∂t ∂ϕ 't + ... ... ... GH ∂t ∂ϕ 't + ∂ϕ 't + ... m
n
m
j
m
⎛
⎛ ⎝
⎞
j
⎠
k
j
,...,n k
,...,m
⎞
n
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
n
∂tm
∂tm
5 E ` t" , n H m
' '
+ ' + GH abc +
∂ ϕ , . . . , ϕn t ∂ t , . . . , tn
' +
⎛
∂ϕj t ⎝ ∂tk
⎞ ⎠
j ,...,nk ,...,m
` t" - +33 DZ '" F H 'F, . . . , Fm+ + t ∈ Rm ! A (" G H 'G, . . . , Gm+ + x ∈ Rm x H F 't + ! " 4 H H 'H, . . . , Hm+ H G ◦ F 99 5 t
'
+ ' + H ∂'F, . . . , F + 't+ ∂'G, . . . , G + 'x+. + ∂ 't , . . . , t + ∂ 'x , . . . , x +
∂ H , . . . , Hm t ∂ t , . . . , tm
'
m
m
m
m
7" - +33 DZ F
H 'F, . . . , F + G D '⊂ + → D '⊂ +.
m
R
m
R
m
DZ '" D
(" ∂∂''Ft,, .. .. .. ,, Ft m++ 'x+ H ! D" m 4 ! G
H 'G, . . . , G + G D m
→ D
F '" F 'G'x++ H x x ∈ D (" G'F 'x++ H x x ∈ D )" G ! ! D
' '
+ ' + H ∂'F, . . . , F + 'G'x++ + ∂ 't , . . . , t +
⎞ −
⎛
∂ G , . . . , Gm x ∂ x , . . . , xm
m
⎝
⎠
m
x ∈ D"
: ($! 8+1 !&!2 (
/ § / § §/3 /3 /3 /3 0$! /31/3
DZ D ∈ 2 D ∈ 2 F H 'F , . . . , F + G D × D → 'x, y+ ∈ D × D 2 F 'x, y+ H O$ ! +3 3 7 F H 'F, . . . , F + + 'x , y+ y H 'y, . . . , y + ! x H 'x, . . . , x + ! ε > ! δ > ! ϕ G B 'x, ε+ →
R
n
m
R
m
R
m
m
m
n
R
m
7; '" y H ϕ'x ++A (" 'x, ϕ'x++ ∈ B ''x, y+, δ+ x ∈ B 'x, ε+A )" F 'x, ϕ'x++ H ! x H 'x, . . . , xn+ ∈ B 'x, ε+" DZ y H 'ϕ'x+, . . . , ϕm'x++ - ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
+H! +H!
' '
F x , . . . , xn , y , . . . , ym ... Fm x , . . . , xn , y, . . . , ym
'7+
x ∈ B 'x, ε+" 6 +3 3 DZ 9 5 Fj 'x, y+
' '
+' +
+ H !.
∂ F , . . . , Fm x ,y ∂ y , . . . , ym
4 ! ε > ! δ > ! x ∈ B 'x , ε+ ) -
H ϕ'x+ ∈ B'y, δ+. $ 'x, y+ F + y ! x H 'x , . . . , x +" y
n
. / # + #
/ §/ §/ 0$! /1 4
DZ D ⊂ 2 F H'F, . . . , F + G D → ` G D → $ DZ x ∈ D 2 F 'x + H O$ .# 2 2 F x #$ DZ 2 λ grad F 'x + I . . . I λ grad F 'x + H O 2 λ H ! , j $ R
R
m
n
m
R
j
m
j
m
79 ! 1303 7 9 5 ` x
% %
' + H F 'x, . . . , x + H !, 8 j m, 9 5 ` L GH {x H 'x , . . . , x + ∈ D G F 'x+ H O} Fj x
j
n
'"+
n
x " DZ 0 ∃ε>
'
+
`'x+ < `'x+
';+
'
+
`'x+ > `'x+
'9+
!G
∀ x ∈ B x , ε ∩ L
!G
∀ x ∈ B x , ε ∩ L
D ∃ε>
5 9 5 ` x * 9 5 x *" 6 1303 DZ 9 5 ` Fj x " , x 9 5 ` * grad
`'x+, grad F'x+, . . . grad F 'x+ m
" $ ! λj grad
`'x+ I λ grad F'x+ I . . . I λ
! 1303 > 5
m grad Fm
`'x, λ+ H `'x, . . . , x , λ, . . . , λ + GH `'x+ I n
9 5 ; "
m
n j
'x+ H O. '+
λ j Fj x
'@+
7@ - 1303 DZ x 9 5
` *" 4 ! λ ∈ 'x , λ + R
m
9 5 ; F" - 1303 DZ L
⎧ ⎨
GH 'dx, . . . , dx + G n
⎩
' + I . . . I ∂F 'x+ dx H !, 8 ∂x
∂Fj x dx ∂x
j
n
n
j
⎫ ⎬
m⎭ .
4 d`'x+|L 99 5 9 5 ` L 9 ! 'dxj , . . . , dxj + js ∈ {8, . . . , n} " - 1303 DZ 'x, λ+ 9 5 ; F" , d `'x +|L 5 9 'dxj , . . . , dxj + js ∈ {8, . . . , n} " ! x 9 5 ` *" , d`'x+|L 9 'dxj , . . . , dxj + js ∈ {8, . . . , n} x " ½
k
*
½
k
½
k
T8U d F$5$2 : 3$5$2 : F$ e$ #
\\ 2 $ ->
-2 $ $ $ # $
$ @8E $ 8E@E $ ' , +N T U . A$ $ . # # \\ 3 2 $ -fK\R-$ $ $ 7C8 $ 8EEC $ ' , +N T7U # F$DZ$ : # # # \\ 2 $ ->
-2 $ $ $ # $
$ 9 " $ 8EE! $ ' , +N
