6 downloads
121 Views
357KB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
!!"
# $ % #$ & ## $ ' ( #) *# # +
, - # - # - #) #)-$ . & $ DZ #) *# # 0$ 1 $ $ # $
2 . 3$4$ DZ , -% 5 -2 67 89 !!" $
! " # $%%&
7
DZ
! " # $ % ! &% '! & & ( $ % ! &% " " )(( $ ( $ % ! * &% + , - $& . & ( $ % ! * &% , + ' & ( / 0 1 2 & . & - , R
n
"
DZ
8$ ) # )$ 8$8$ 4 $ 8$8$8$ 5 $ 8$8$ $ $ 8$8$7$ # #
8$8$"$ :)# $ 8$8$;$ . )# $ 8$8$9$ < # # # $ 8$ $ ) #)$ 8$ $8$ = ) #)$ 8$ $ $ = ) # )$ 8$ $7$ DZ ,$ 8$7$ > ) #)$ 8$7$8$ = $ 8$7$ $ : ) $ 8$7$7$ %# 3 $ 8$7$"$ # $ 8$"$ # ) #)$ 8$"$8$ ? $ 8$"$ $ # $ 8$"$7$ # $ 8$"$"$ $ 8$"$;$ # $ 8$"$9$ DZ ,$ 8$"$@$ $ 8$;$ DZ ) ) #)$ R
n
R
n
R
n
R
R
n
n
; 8$9$ # % $ 8$@$ A B# # ) #)$ 8$@$8$ = $ 8$@$ $ . #$ 8$@$7$ $ 8$C$ > $ 8$C$8$ > $ 8$C$ $ D $ 8$C$7$ > $ 8$E$ 0 B# #$
F ) # $ A GH B 1 A , B $ 1 # ) ) $ 3 k H 'k, . . . , k +2 k ∈ , ' s+2 |k| GH k I . . . I k , k2 k J GH k J . . . k J2 x H 'x, . . . , x +2 x ∈ 2 x GH x . . . x 2 f 2 g # D ⊂ x # D2 f 'x+ f 'x+ ∼ g 'x+ x → x ⇔ KLM H 8N g 'x+ f 'x+ H O'g 'x++ x → x ⇔ ∃ C > ! ∃ r > ! ∀ x ∈ B 'x , r+ ∩ D |f 'x+| ≤ g 'x+N f 'x+ H !N f 'x+ H o'g 'x++ x → x ⇔ KLM g 'x+ f 'x+ g 'x+ x → x ⇔ f 'x+ H O'g 'x++, g 'x+ H O'f 'x++. 4 x ∈ 2 x H 'x, . . . , x + Z
s
Z
j
s
s
s
j
k
R
k½
ks s
R
x→x¼
x→x¼
R
n
n
n
9
!" # $
R
R
n
n
§ § ! 33 X I G X × X → X × G R × X → X !
I H I I' I +H' I +I ' I + H I ' I +H I O G I O H O I x H x, G I '−x+ H O, 8 H
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.
x, y, z ∈ R λ, µ ∈ R" # O −x x" - 33 # ! " $ !
"
DZ 3 DZ ,
#) P) 'x, . . . , x +$ Q # #2
# 2 $$ x I y GH 'x I y, . . . , x I y +, λx GH 'λx , . . . , λx + ) x ∈ 2 y ∈ 2 λ ∈ $ 2 O GH '!, . . . , !+ n R
n
n
n
R
n
R
n
n
R
¾
n
x y x, y λx ×λ, x
@ R
n
GH '−x, . . . , −x + 2 x GH 'x, . . . , x +$ −x
n
n
♠
% & "! & &$
§ § § ! 33 % ρ G X × X → R x y z X ! & '" ρ'x, y+ ≥ ! (" ρ'x, y+ H ! ⇔ x H y )" ρ'x, y+ H ρ'y, x+ *" ρ'x, y+ ≤ ρ'x, z+ I ρ'y, z + " ! 'X, ρ+ ! + " , 'X, ρ+ - X " . " ! 33 DZ X " % '., .+ G X × X → R x y z X λ ∈ R ! & '" 'x, y+ H 'y, x+ (" 'x I y, z + H 'x, z+ I 'y, z + )" 'λx, y+ H λ'x, y+" *" 'x, x+ ! 0" 'x, x+ H ! ⇔ x H !"
C DZ 3 = '., .+2 'x, y+ GH
$ R
n
n
j
♠
xj yj
1
! 33 % . G X → R
x y z X λ ∈ R ! & '" x ≥ ! (" x H ! ⇔ x H O )" λx H |λ|x *" x I y ≤ x I y " - 33 $ x y 'X, .+ | x − y | x − y.
- 33 1 -23 , '., .+
+ X
' +
' +' +
'8+
| x, y | ≤ x, x y, y
x y X " # '
x y "" ! λ ∈ R µ ∈ R |λ| I |µ| H ! λx I µy H θ" - 33 DZ x
GH 'x, x+ H /
⎛ ⎝
n
xj ⎠
j ⎛
' + GH x − y H
ρ x, y
⎞/
⎝
,
'x
n
j
j
x ∈ Rn y ∈ Rn" 4
Rn"
+
⎞/
− yj ⎠
.
' +
ρ ., .
E 4 3 "#$ $% & $ ! !
ρ'., .+
R
n
$
- 33 DZ xp
GH
.
⎧ /p ⎪ n ⎪ ⎪ p ⎨ |xj | , j ⎪ ⎪ ⎪ |xj |, ⎩ j ,...,n
MRS
$ %
8 ≤ p < I∞ p H I∞
ρp'x, y+ GH x − yp x y Rn" 4 .p ρp'., .+ Rn" 4 3 ρp'., .+ ' Rn ! $ ∀ p, q ∈
T8, I∞U
∃C>
!
∀ x, y ∈ Rn
8 ρ 'x, y+
C
p
' +
ρq x, y
' +
Cρp x, y .
!' &( ' & ! ! "! & &$
§ §
DZ # X # ρ'., .+$
! 3 3
' + GH {y ∈ X G ρ'x, y+ < r}
B x, r
' + GH {y ∈ X G ρ'x, y+
B x, r
r}
& & X 5 x r > !" ! 3 3
V ' + GH {y ∈ X G ρ'x, y+ < r} \ {x}
B x, r
& X 5 x r > !"
8! ! 3 3
' + GH {y ∈ X G ρ'x, y+ H r}
S x, r
( X 5 x r > !" ! 3 3 DZ D X x ∈ D ! - B 'x, r+ B 'x, r+ ⊂ D" ! 3 3 4 x X ! ) D ⊂ X ! - B 'x, r+ ⊂ X " ! 3 3" D ⊂ X "
DZ 3 3 1 #2
# $
♠
! 3 3, D ⊂ X
6 ! "
DZ 3 = W ) )
#2 B , $
♠
! 3 3+ DZ
*%# " ! 3 31 DZ D ! ! - B 'x, r+ D ⊂ B 'x, r+"
DZ 3 DZ 2 #
## $
♠
- 3 3 % D
∃C>
!G
⊂
∀ x ∈ D x < C.
R
n
88 ) *!+'! &, "! +!$& ,! & - $
R
n
§ § §
DZ # X # ρ'., .+ 'x + ⊂ X $ ! 33 7 'x + + $ x ∈ X $ $ % 'x + ∀ ε > ! ∃ n H n 'ε+ G ∀ k > n ρ'x , x+ < ε. , 'x + + $, - KLM x H x x → x n → I∞" 4 3 DZ $ % % 'x + + $ x $ %
k
k
k
k
k
k
k
n→ ∞
k
$ $
∀ε>
!
∃ n
H n'ε+ G
' +
∀ k > n xk ∈ B x, ε .
- 33 , 'x + + k
" ! 33 7 'xk+ ! " $ 'xk+ ∃C>
!G
∀ n xk < C.
- 33 # !
" - 33 % 'xk+ ⊂ Rn + ""
!
H 8, . . . , n |x | < C. - 33 DZ 'x + ⊂ x H 'x + x H 'x, . . . , x + ∈ ∃C>
!
k j
∀k ∀j
k
n
R
n
! &
R
n
k
k j
n j
8 '" xk − xp → ! n → I∞ p ∈ T8, I∞U" (" k→KLM∞ xjk H xj j H 8, . . . , n" $ Rn
" . & - *!+'! & "! +!$& ,! & - $
R
n
§ § ! 3 3 DZ 'xk+ ⊂ Rn $ % %# . & ( $ % $ % %# Rn ∀ ε>
!
∃ n
H n'ε+ G
∀ k > n ∀ s ∈ N xk s − xk < ε.
- 3 3 8 ! 9 " - 3 3 # 9 "
! 3 3 X
9 " - 3 3 $ 'xk+ ⊂ Rn 9 ! & '" 'xk + 9
'Rn, ρp+ p ∈ T8, I∞U (" 'xk+ 9 "" + ! & ∀ε> ∀s∈N
!
H ' +G
∃ n n ε ∀ k > n k s − xk | < ε. ∀j , . . . , n |xj j
H8
5 3 3 DZ R
n
ρp'., .+ p ∈ T8, I∞U" 5 3 3 # n k
'x + ⊂ R ! ! "
87 / 0' &1 !'" &1 "!+'!2 &$ $
R
n
§/ // §
DZ # X # ρ'., .+$
! 3"3 DZ D ⊂ X " 4 x ∈ X
$ % ! ) D ! 'xk + ⊂ D \ {x} ! x X " 4 x ∈ D'⊂ X + D" - 3"3 DZ D ⊂ X " 4 x ∈ X D ∀ε>
!D
◦
' + H ∅.
B x, ε
! 3"3 DZ
" - 3"3 , D ⊂ X X \ D " , D ⊂ X X \ D " DZ " ! 3"3 DZ D X 'xk+ D ! x ∈ X " D " - 3"3 , D ⊂ X D ⊂ D " 6 3"3 DZ Rn " 5 3"3 : - Rn "
8"
! " #" ) " + 3 #- !, * " ' 1*
§ §/ / 0$! 1 23 ! 33 DZ D ⊂ Rn E ⊂ Rm" 7 f G D → E + D
E f x ∈ D
H f 'x+ H f 'x, . . . , x + H H 'f'x, . . . , x +, . . . , f 'x, . . . , x ++. D GH D *%# $ f E GH E *%# ! f " # m H 8 f 9 5 y
n
n
m
n
f
f
" ! 33 DZ
G G 8 5 h GH g ◦ f D E h'x+ GH g'f 'x++ x ∈ D " ! 33 DZ f G D → E D ⊂ D " % f G D → E f 'x+ GH f 'x+ x ∈ D f Df → Ef , g Dg → Eg , Ef ⊂ Dg . f
g
f
f
|D
f
f
|D
f
f D ⊂ Df "
)% " + " + 3 # !, * " ' 1*
§ / §/ / // 0$!
214
DZ f G D '⊂ + → E '⊂ + x ∈ # D $ f
R
n
f
R
n
f
R
m
8; ! 33 7! ! ! 89: 7 f A ∈ Rm
!
x → x
H ' + !G
∀ε> ∃ δ δ ε > ∀ x ∈ Df , < x − x < δ ⇒ f x − A < ε.
!
'+
, - x→x KLM
´¼µ
! 33 7! ! ! ;:
7 f A ∈ Rm
x → x
' + H A"
f x
' +
' + , - KLM f 'x+ H A" ∀ xk ⊂ Df \ {x }, xk → x ⇒ f xk → A. x→x´¼µ
6 33 % 1 -
7 " 4 3 DZ m H 8 $ / / + $ $ $ ( 5 %+ +
. & 6 - 33 , !
" f ! 33 7 x ! - B 'x , r+ C > ! f 'x+ < C x ∈ B 'x, r+" - 33 , x → x " KLM f 'x+ H A H ! - 33 , f 9 5 ! x→x ´¼µ
' + H sign A. - 33 8! f H 'f, . . . , f + ∃ε>
!G
'
+
∀ x ∈ B x , ε ∩ Df sign f x
m
x → x ! 9 5 fj " *"'"' x → x" 1
KLM f 'x+ H ' KLM f'x+, . . . , KLM f 'x++.
x→x´¼µ
x→x´¼µ
x→x´¼µ
m
89 ) DZ + 3 # "! "$ 5
§/ / 0$! 214 ! 3 3 n; lh ⊂ Rn x h ∈ R h H 8 lh
GH {x I tx G t ∈
R, t
!}.
! 3 3 DZ f G D '⊂ + → f
R
n
R
m
" DZ lh ⊂ Rn
x h ∈ Rn h H 8" 7 f A x → x lh t→KLM f 'x I tx+ !
A" - 3 3 < ! f
! " 3 " 4 3 DZ n > 8 )$ * $$ ,
*, 7 ( 5# f G R → R x
∈ Df
' + GH
f x, y
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x y , x x y , x
!
I
I y H ! I y H !
.
8 ( 5 $ ! '!, !+ $ % #* ! ! ' ! $ ! '!, !+
$ % ## & ! " #" . " +
§ §/ // 0$! 4/13 3/ 3 13
DZ f G D '⊂ + → $ f
R
n
R
m
8@ ! 33 DZ x ∈ D " 7 f
f
x Df x→x KLM f 'x+ H f 'x+" ´¼µ
! 33 % f
" 4 3 DZ m H 8 $ + $ $ ( 5 %+ +
! #$ - 33 = f x ∈ Df 9 5 fj " *"'"' x j H 8, . . . , m" .% $!- &$ " 1$1* 3 #-
§ §/ / - 33 DZ 9 5 f G D'⊂ Rn+ → R g G D'⊂ Rn+ → R x ∈ D" , 9 5 f g x 9 5 f I g f g " , g'x + H ! 9 5 fg x"
- 33 DZ f G D '⊂ + → E '⊂ + R
n
R
n
G G Ef '⊂ Rn+ → Eg '⊂ Rn+" DZ f x ∈ Df g y H f 'x +" 4 5 g ◦ f x" f
f
. ! '1 - 6&
§ § ! 3 3 7 9 5 f G D'⊂ Rn+ → R L ⊂ D ∃C>
!G
'+
∀ x ∈ L |f x | < C.
8C # 9 5 D" 6 3 3 7* < 9 : > 5 n + R " 6 3 3 7* < 9 : DZ n 9 5 f D ⊂ R " 8! xj ∈ D j H !, 8
' + H XYZ{f 'x+ G x ∈ D}, ' + H L[\ {f 'x+ G x ∈ D}.
f x f x
.) $!' ( " 1$! &,
§ § 0$! 313 ! 33 % f G Df '⊂ Rn+ → Rm
!
H '+ ! + '+ '+ - 33 , f G D '⊂ + → ∀ε> ∃δ δ ε > ∀ x, y ∈ Df , x − y < δ ⇒ f x − f y < ε. f
R
n
R
n
" 6 33 1 DZ f n D ⊂ R " 4 "
' (&& ! & ! " #" / 7 &1 "!8$!+1
§/ / § 0$! 19
DZ f G D '⊂ + → x 1
# D $ f
f
R
n
R
8E ! "33 > k k H 8, . . . , n" ?
9 5 fk
' + GH f 'x, . . . , x, x, x, . . . , x+,
fk x
k−
k
n
+ x k " , 9 5 99 5 xk fx 'x k + 9 5 f x ∂f 'x + xk fk 'x+" ∂x k
/% 33 # '1 3 #
§/ / / § 0$!
1 ! "33 # ]x H ']x, . . . , ]xn+ , ( 5 f x x I]x ∈ Df " DZ , ( 5 f x ! ! 9 5 ]x
]f 'x+ H ]f 'x, ]x+ GH f 'x I ]x+ − f 'x+.
- "33 %
f
x
' I ]x+ H f 'x+ I o'8+, x I ]x ∈ D , ]x → !. ! "33 DZ 9 5 f G D '⊂ + → x D " 7 9 5 f 99 5 x ! A H 'A, . . . , A + f x
f
f
R
n
R
f
n
' I ]x+ H f 'x+ I A ]x I o']x+, x I ]x ∈ D , ]x → !. n
f x
j
j
j
f
7 9 5 99 5 99 5 "
! - "33 , 9 5 f 99 5 x " - "33 , 9 5 f 99 5
x
'x+ ]x I o']x+, ' I ]x+ H f 'x+ I ∂f∂x x I ]x ∈ D , ]x → !. n
f x
j
j
j
f
/ ! &&! 1 !$( +33 # '! & 3 #
§/ // § 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 $! 213
8 6 "3 3 DZ
@" 9 5 x H x't+ y H y't+ + t ∈ R 99 5 tA @@" > 5 u H u'x, y+ + 'x't+, y't ++ 99 5 " 4 9 5 v't+ H u'x't+, y't++ 99 5 t
' + H ∂u'x't+, y't+ dx't+ I ∂u'x't+, y't+ dy't+ . ∂x dt ∂y dt
dv t dt
5 "3 3 DZ
@" 9 5 x H x't, t+ y H y't, t+ + t H 't , t + ∈ R 99 5 tA @@" 9 5 u H u'x, y+ + 'x't+, y't ++ 99 5 " 4 9 5 v't, t+ H u'x't , t+, y't, t++ 99 5 t
' + H ∂u'x't +, y't+ ∂x't+ I ∂u'x't +, y't+ ∂y't+ , k H 8, ∂x ∂t ∂y ∂t
∂v t ∂tk
k
5 "3 3 DZ
.
k
@" 9 5 xk H xk 't+ k H 8, . . . , n + t ∈ Rm 99 5 tA @@" 9 5 u H u'x+ + x ∈ Rn x H 'x't+, . . . , xn't++ 99 5 " 4 9 5 v't+ H u'x't+, . . . , xn't++ 99 5 t
' +H
∂v t ∂tk
' ' ++ ' + H 8, . . . , m.
∂u x t ∂xj t ,k ∂xj ∂tk j n
5 "3 3 DZ
@" 9 5 xk H xk 't+ k H 8, . . . , n + t ∈ Rm 99 5 tA @@" 9 5 u H u'x+ + x ∈ Rn x H 'x't+, . . . , xn't++ 99 5 " 4 m n
' +H v't+ H u'x't++" dv t
' ' ++ dt H
∂u x t ∂tj j
j
' + dx ,
∂u x ∂xj j
j
// DZ!8$!+( "! "$ 5
§/ /2 § 0$! /1/ ! "3"3 DZ 9 5 f x l H lh " *")"' fl
' + f 'x I th+ − f 'x + . ' +H GH KLM t x
∂f x ∂h
t→
- "3"3 DZ 9 5 f 99 5 x" 4
9 5 f x
' +H
∂f x ∂h
' +
∂u x hj . ∂xj j n
/9 + & 3 #
§/ /2 § ! "3,3 DZ 9 5 f 99 5 x " 7 9 5 f x
' + H ∇f 'x+ GH
grad f x
⎛
' +
' +
∂f x ∂f x ⎝ ,..., ∂x ∂xn
⎞
⎠.
7 - "3,3 DZ 9 5 f 99 5 x" 4
lh x
' + H 'grad f 'x+, h+ H ^_X'ϕ+ grad f 'x+,
∂f x ∂h
ϕ h 9 5 f x " 5 "3,3 7 9 5 " 1 grad f 'x + H O h ∂f - " , grad f 'x + H O ∂h ∂f ∂h H ! h" /: DZ!8$!+1 +33 #1 $1 6* "!(+ !$ 3 # !, * " ' 1*
§ §9 9 9 0$!
194 1 3 /1 9 ! "3+3 DZ k H 'k, . . . , ks+ kj ∈ {8, . . . , n}" B -
' + 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
⎛
k½
'+
⎞
∂sf x ⎝ ⎠ ∂xk½ ∂xk¾ . . . ∂xks·½
H
s
'+
∂fk¾ ,...,ks·½ x . ∂xk½
" B s fks'x+" 6 "3+3 DZ 9 5 f G B 'x, r+'⊂ R+ → R ∂f ∂f f f ∂x ∂x∂ ∂x ∂x∂ ∂x " DZ ! ∂x
∂ f ∂x ∂x
∂ f ∂x ∂x
x" 4
' + H ∂f 'x+ . '+ ∂x ∂x DZ 3 : P , f G → 2 #, P ) ) 2 ' + $ 3 # ## , 2 # , # x − y xy , 'x, y + H '!, !+ f 'x, y + GH x I y 'x, y+ H '!, !+ !, ∂ f x ∂x ∂x
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" $99 5 dsf 'x+ & '" , s H 8 " D"*"'A (" ds f 'x+ GH d'dsf 'x++ s ∈ N s > 8" - "3+3 , x H 'x, . . . , xn+ f
n
l
' +H
df x
J J
'+
l ∂ |k| f x dxk , k ∂x k k½ ,...,kn k
dx H 'dx, . . . , dn+ + k H 'k, . . . , kn+ |k| H l" , 9 - &
∂ dx ∂x
I ... I
∂ dxn ∂xn
l
GH
' +H I ... I # n H ∂ f 'x+ d f 'x+ H C ∂x ∂x l
df x
l
∂ dx ∂x l
∂ dxn ∂xn
k l
k
l
J J
l ∂ |k| k dx , k k k½ ,...,kn k ∂x
l
'+
f x.
k l−k l−k dx dx .
k
/;
§ / §2 2 2 0$!
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
