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Московский государственный физико-технический университет (МФТИ)

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[НЕСОРТИРОВАННОЕ]

Файл:

матан Бесов - весь 2012

.pdf

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03.06.2015

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∂fi
∂xk
	∂f1	∂fm
	∂x	∂x
	k	k


f : G → Rm, G Rn

G Rm

f (x) Rm

f (x) = (f1(x), . . . , fm(x)), x G, !

" f #

G $ %

f1(x), . . . , fm(x) : G → R,	&
$
'
f (E) = {y Rm : y = f (x), x E}, E G,

E f (G) (

f −1(D) = {x Rn : f (x) D}, D Rm,

( D

) "

x(0) G

U (f (x(0))) U (x(0)) : f (G ∩ U (x(0))) U (f (x(0))).

* + x(0) ( G

, G ∩ U (x(0))

U(x(0))

x(0)G x(0)

-. # n = 1 m =

) "

-" f

G D Rm

f −1(D) x(0) f −1(D)

f (x(0)) D ' D #

f (x(0)) / # ! 0

U (x(0)) G f (U (x(0))) D

U (x(0)) f −1(D)

' x(0) f −1(D)

f −1(D) #

f −1(D)

" . .

" f f

G x(0) G D = U (f (x(0))) (

f (x(0)) 1 . f −1(D) (

0 x(0) f −1(D) (

x(0) )" 2 U (x(0))

f (U (x(0))) = f (f −1(D)) D = U (f (x(0))).

# ! " f

x(0) 3 " x(0) G "

f G

) " "

x(0) G

m = n

∂f1

∂(f1, . . . , fn)

∂x

(x)

. . .

∂x

(x)

J(x) =

(x)

. . .

∂(x1, . . . , xn)

∂fn

(x)

. . .

(x)

∂x

" # " # #

$ n = 1 J(x) = 0 #

# # %

" # %

& " y = x2

G = (−1, 1) % y (0) = 0 # # #

$ n 2 #

" # # %

" # n = 2 "

	x = r cos ϕ,
#	y = r sin ϕ

	G = {(r, ϕ) : 1 < r < 2, 0 < ϕ < 4π}
{(x, y)' 1 < x2 + y2 < 4}
(		# # %

(r0, ϕ0) G " ) * )

Uδ (r0, ϕ0) "

" # ) # # #

& " )+ #

, % # # # +

G Rn

f G → Rn G

J = 0 G

f (G) Rn

- x(0) G y(0) = f (x(0))

! U (x(0)) U (y(0))

◦ f " " #

U (x(0)) ↔ U (y(0))$

-◦ g : U (y(0)) → U (x(0)) f

U (y(0))$

◦ g # U (y(0))

. # " f ' G → D # # #

' G ↔ D " f −1' D ↔ G # # )+ '

f −1(y) = x, f (x) = y (y D, x G).

			-
. y(0) f (G) x(0)					G y(0)		= f (x(0))
/ %
	Fi(x, y) fi(x1, . . . , xn) − yi = 0,				i = 1, . . . , n.
0

∂(F1, . . . , Fn)				∂(f1, . . . , fn)

		(x(0),y(0))	=			(x(0))	= 0.
	∂(x1, . . . , xn)			∂(x1, . . . , xn)

. # 1 %

U (y(0)) × V (x(0)) 1 # 1

	{yi = fi(x)}1n {xi = gi(y)}1n,	2
	y = f (x) x = g(y),	2
	y = f (x) x = g(y),
	g(y) = (g1(y), . . . , gn(y)) "
	g : U (y(0)) → V (x(0)) G
	U (y(0))
	3
	U (y(0)) f (G),

y(0) f (G)

f (G)

U (y(0))

!" y = f (g(y)) # y U (y(0))

$ #

yi = fi(g1(y), . . . , gn(y)), i = 1, . . . , n, y U (y(0)).

% && ' ( # # yj #

n
				∂yi		1	# j = i,

∂fi ∂gk			=		=
							# j = i.
	∂xk	∂yj		∂yj		0
k=1

) # # *

	∂(f1, . . . , fn) ∂(g1, . . . , gn)				= 1

	∂(x1, . . . , xn) ∂(y1, . . . , yn)
	∂(x1, . . . , xn) ∂(y1, . . . , yn)				+"

# x = g(y) V (x(0)), y U (y(0)).

, ∂(g1, . . . , gn) = 0 U (y(0))

∂(y1, . . . , yn)

U (x(0)) g(U (y(0))),

. x(0)

. x(0) /

f .

U (x(0)) ↔ U (y(0))

# n = 1 y = x20 (−1, 1) → [0, 1)

G Rn

f G → Rn

%G Rn f 0 G → → Rn .

f (G)

y(1) y(2) # f (G)

x(1) x(2) G f (x(1)) = y(1) f (x(2)) = y(2) Γ = {x(t)0 α t β}

Rn Γ G # x(1) Γ x(2) ' Γ 1 f (Γ) = {f (x(t))0 α t β} f (G) y(1) = = f (x(1)) f (Γ) y(2) = f (x(2)) ' f (Γ)

! f

x(0) Rn " x(0)

# $ f

(0)	(0)	˚ (0)
U (x	) : f (x ) f (x)	x U (x ).

% & & !

x(0) # $

' # !$

# !$ f

# $

% x(0) #

$ f &

x(0) f #

f x(0)

	∂f	(x(0))	∂f	(x(0)) = 0
	∂xi		∂xi

( ! ! i =

=1 ) ϕ x1*


§
ϕ(x1) f (x1 x2(0)		xn(0)) +
x1(0)	" ,
0 = ϕ (x1(0)) =			∂f
				(x(0)).
			∂x
		1
		" x(0)

f f x(0) df (x(0)) = 0

- .

f x(0) f

x(0) x(0)

- / !

& ! &

f (x) = x3

(

& !

1	n
	n

A(ξ) = A(ξ1, . . . , ξn) aij ξiξj		#.$
	i,j=1
#aij = aji i, j = 1 n$
$ A(ξ) > 0 ξ =
$ A(ξ) > 0 ξ =		0
/$ A(ξ) < 0 ξ =
/$ A(ξ) < 0 ξ =		0

$ & !

A(ξ) #.$

! μ > 0

A(ξ) μ|ξ|2 ξ Rn.

#2$

|ξ| = 0

|ξ| > 0 |ξ|2

η = |ξξ|

μ min A(η) > 0.

η Rn,

|η|=1

! " #$

# % & A(η) #

% S {η Rn' |η| = 1} () #

# η S $

μ = A(η ) > 0

* # %% & % & f

n	∂2f
	∂2f
d2f (x(0)) =			(x(0)) dxi dxj ,	+
d2f (x(0)) =	∂x ∂x		(x(0)) dxi dxj ,	+
i,j=1	i	j

() # ( % d2f (x(0)) = A(dx) = A(dx1, . . . , dxn)

, dx1 dxn

x(0) Rn

d2f (x(0)) f

x(0)

f ! d2f (x(0))

" x(0) f #

- $ * % & f % . # & #

x(0) % / + '

f (x(0)) = f (x(0) + x) − f (x(0)) =

∂2f (x(0))

xj + ε(Δx)| x|2,

i,j=1

∂xi∂xj

ε(Δx) → 0

x → 0

" 1 # % (

x(0)

2 &

d2f (x(0)) + *

3 # % #

x|2, μ > 0.

f (x(0))

μ + ε(Δx)

ε(Δx) → 0

x →

0 ) δ > 0

x : 0 < | x| < δ.

|ε(Δx)| <

x : 0 < | x| < δ.

f (x(0))

x|2 > 0

- f (x(0)) > 0 x(0) #

% & f

5 d2f (x(0))

% + & 3 #

# % # x(0) 2 % & f

% d2f (x(0)) +

3 # # % # 6

) ( ξ ξ Rn A(ξ ) < 0

A(ξ ) > 0 η =	\|ξξ			η =	\|ξξ
			\|			\|

α = A(η ) < 0, β = A(η ) > 0, |η | = 1, |η | = 1.

x = tη | x| = t t > 0 . 0

f (x(0) + x) − f (x(0)) =			α + ε(tη )
		1		t2	α	t2	< 0
	2			t2	4	t2	< 0
	2				4

t = | x|

x = tη t > 0

	x) − f (x(0)) =			β + ε(tη )
f (x(0) +			1		t2	β	t2	> 0
f (x(0) +		2			t2	4	t2	> 0
		2				4

t = | x|

U (x(0)) f (x) − f (x(0)) x U (x(0))

! " #

x(0) $ % # ! f

x(0) Rn

f x(0)

d2f (x(0)) 0 dx Rn d2f (x(0)) 0 dx Rn

& % d2f (x(0))

' ! x(0) % ! f

d2f (x(0)) 0 d2f (x(0)) 0

( $ #

% ! f x(0) #

f % ) $

% !

% ! (

(


				a12
1 = a11 > 0,		a11		a12	> 0, . . . ,
1 = a11 > 0,	2	=	a21	a22	> 0, . . . ,
			a21	a22


a	11	. . .	a
	11		1n
n = . . .		. . .	. . .	> 0.

a		. . .	a
n1			nn

, % A(ξ) +

! (

% −A(ξ) (

" ! (

% +

(−1)k k > 0 k = 1, . . . , n.

"% ' #

- % !

f ! !

(x0, y0)

fx(x0, y0) = fy (x0, y0) = 0.

" (x0, y0)

fxxfyy − fxy 2 > 0,

(x0, y0)

# fxx(x0, y0) > 0 $

fxx(x0, y0) < 0" f " (x0, y0)

fxxfyy − fxy 2 < 0,

(x0, y0) f

" (x0, y0)

fxxfyy − fxy 2 = 0,

(x0, y0) f $

* , ( #

% ' " 1 > > 0 1 < 0 2 > 0 , ( '

dy = t dx fxx(x0, y0) = 0 dx = = t dy fyy (x0, y0) = 0

d2f (x0, y0) =

= fxx(x0, y0) dx2 + 2fxy (x0, y0) dx dy + fyy (x0, y0) dy2

dx2 dy2

t !

" # #$

t ! !

" %

& ! fxx = fyy = 0 fxy = 0


2				2
2			, y0)t dx	2
d	f (x0, y0)	= 2fxy (x0	, y0)t dx

dy=t dx

t = −1

t = 1" ' #

( ) (x0, y0) #

f "

(x0, y0) = (0, 0) +

f (x, y) = x4 + y4, g(x, y) = x4 − y4.

( $ $ #

(0, 0) , #

f $ g

# "

G Rn

f ϕ1 ϕm (1 m < n) E {x x G ϕj (x) = 0

§
1 j m}
{ϕj (x) = 0}jm=1	!"

#$ x(0) E

! %

f & !"

δ > 0 : f (x	(0)	) f (x) (f (x	(0)	) < f (x)) x	˚	(0)
					E ∩ Uδ (x ).

' ($ $ (

! ( ( ( ( ) ! ( ( ( )

*$ ( + %

,-. , %-. / , & !" . %

, !" .

G = R2 f (x1, x2) = x21 + x22 m = 1

ϕ1(x1, x2) = x1 + x2 − 1 0- $ ( ) %

f x1 + x2 − 1 = 0

0 - ϕ1(x1, x2) = 0 f (x1, x2) = f (x1, 1 −x1) = 2x12 −
− 2x1 + 1 = 2 x1 −	1	2 +	1	1	;	1
	2		2 1 $	2		2

$ - ( ( f

ϕ1 = 0

+ -2 $ $ +f ϕ1+
+∂ϕj +
ϕm G rang +		+	= m
	∂xi

G, x(0) E 3 ($ 4


	∂(ϕ1, . . . , ϕm)
$	∂(x1	, . . . , xm)		(0)	= 0 # (
			x

& - 4 (

Q(xm(0)+1, . . . , xn(0)) × Q(x1(0), . . . , xm(0)) - !"
!1 (
{xj = μj (xm+1, . . . , xn)}jm=1 ,	(1 )

μj

Q(x(0)m+1, . . . , x(0)n )

{ϕj (μ1(xm+1, . . . , xn), μ2(), . . . ,

μm(), xm+1, . . . , xn) = 0}mj=1.

Φ(xm+1, . . . , xn)

f (μ1(xm+1, . . . , xn), μ2(), . . . , μm(), xm+1, . . . , xn).

x(0)

! f "

(x(0)m+1, . . . , x(0)n ) !

Φ# $ % & ' (

! ' (

! &

# ' ! &

μ1 # # # μm

& #

! ) 3 '


	n				m
	∂ϕj		dxi = 0			,	)
		∂x	dxi = 0			,	)
	i=1	i			j=1
					j=1
		n			m
		n		∂μ
				∂μ			(3 )
dxj =				j	dxi	,
dxj =					dxi	,
				∂xi

i=m+1

j=1

! & x(0)# *

% ' ' dxm+1, . . . , dxn +

)

, + 3 (

# * ( + 3 )

& + 3 )

' ( #

-& dx1, . . . , dxn % % &

) 3 #

$ x(0) E &

f "

(0)

(x(0))

∂f (x

)

dxi = 0.

i=1

∂xi

x(0)

" f

/& # 0 1x(0) = (x(0)1 , . . . , x(0)m x(0)m+1 # # # x(0)n ) 2 ! f " 3

[(x(0)m+1, . . . , x(0)n ) 2 % !

Φ3 [dΦ(xm(0)+1, . . . , xn(0)) =

(0)

∂f (x

(0)

)

∂f (x )

0 =

dxi

i=1

∂xi

= ∂f (x

df (x(0))

i=1

) dxi

% & #

(0)

i=1 ∂xi

& % &

x(0) E

f "

λ1 λm

! x(0)

L(x) f (x) − λj ϕj (x).

j=1

" λj

m + 1

∂ϕ

i=1

∂xi dxi = 0

)

∂f

i=1

∂xi

dxi = 0

x(0) !

[x(0) " f (1)]

[df = 0] [(3) (df = 0)]

[rang (3 ) = rang (3) = m]

∂f

λ1, . . . , λm R :

, . . . ,

∂x

∂ϕj

λj

, . . . ,

grad f =

λj grad ϕj

∂xn

j=1

∂x1

j=1

[grad L = 0] [dL = 0].

x(0) f

#$% L

& ' f ' ϕ1, . . . , ϕm

(

x(0)' ) x(0) "

f #$%' * )

( E + δ > 0 ' x = (x1, . . . , xm' xm+1' ' xn) E ∩ Uδ (x(0))'

Φ(xm+1	, . . . , xn) = f (x)\| 1 =
			m

		f (x) −	λj ϕj (x)				.
	=	f (x) −	λj ϕj (x)		L(x)		.
			j=1	1		1
			j=1

+ ( ' (x(0)m+1, . . . , x(0)n )

Φ' d2Φ ,

d2L
- dΦ'				d2Φ	(xm(0)+1, . . . , xn(0))'
xm+1'	' xn .
	n
	n
		∂L		(0)	(0)	(0)
		∂L		(0)	(0)	(0)
dΦ =			dxi	, dΦ(xm+1, . . . , xn		) = dL(x )	1 = 0.
dΦ =		∂x	dxi	, dΦ(xm+1, . . . , xn		) = dL(x )	1 = 0.
	i=1	i	1

/ ' (xm(0)+1, . . . , xn(0))

(0)

∂

L(x

)

(0)

Φ(xm+1, . . . , xn

) =

dxi dxk

∂x

i,k=1

(0)

∂L(x

) 2

∂

L(x

)

(0)

dxidxk d L(x ).

∂x

∂x ∂x

i=1

i,k=1

! ' ) ,

f #$%'

) , Φ'

d2Φ 0

' , (

				n
					∂	2	L(x	(0)	)
	2		(0)			2		(0)
	2		(0)
d		L(x )								dxidxk.
d		L(x )		i,k=1	∂xi∂xk					dxidxk.
				i,k=1	∂xi∂xk

f ϕ1 ϕm

x(0) E L

$◦ d2L(x(0)) > 0(< 0) |dx| > 0 x(0) !

" # f #$%$

1◦ 2 (0) # | | (0)

d L(x ) > 0 "< 0 dx > 0 x

" # f #$%$

◦ d2L(x(0)) f

◦ d2L(x(0)) x(0) f

f ϕ1 ϕm 1 m < n

G Rn
+	+
+	∂ϕj +
rang +		+ = m G E = {x	x G ϕj (x) = 0
	∂xi
1 j m} ! " # $
f % ! &
1◦. ' & & ( #
		m

		L(x) f (x) − λj ϕj (x).
		j=1
2◦. ) ! " ( #
* E $ " #			"

# $ + & n + m

	∂			n
,
			L(x) = 0	,
		∂xi
		{ϕj (x) = 0}1m1
n + m % ! x1 x2 xn λ1 λ2

λm - , , " ( #

!	% "
. " {ϕj (x) = 0}1m
		∂	L(x) = 0	m
/ %
		∂λ		1
3◦.		j
	, , " x(0)				( #
		, f			ϕ1 ϕm

& d2L

/ d2L & 0

# $

4◦. ) ! % " f " ! # $

) , 1 " # $

f (x, y, z) = xyz x2 + y2 + z2 = 1 x + y + z = 0 2 ϕ1(x, y, z) = x2 + y2 + z2 − 1 ϕ2(x, y, z) = x + y + z

- " G %

G = (x, y, z) R3 : \|ϕj (x, y, z)\| <	1	, j = 1, 2 .
G = (x, y, z) R3 : \|ϕj (x, y, z)\| <	2	, j = 1, 2 .
	2

( #

L(x, y, z) = xyz − λ1(x2 + y2 + z2 − 1) − λ2(x + y + z)

, 1 " &*

% + ,
Lx ≡ yz − 2λ1x − λ2 = 0,
Lx ≡ yz − 2λ1x − λ2 = 0,



Ly ≡ xz − 2λ1y − λ2 = 0,
Ly ≡ xz − 2λ1y − λ2 = 0,
Lz ≡ xy − 2λ1z − λ2 = 0,
Lz ≡ xy − 2λ1z − λ2 = 0,		.
2 2 2
2 2 2
x + y + z − 1 = 0,
x + y + z − 1 = 0,

	x + y + z = 0

' # "

yz + xz + xy − 3λ2 = 0. 3

) 2(yz + xz + xy) = (x + y + z)2 − (x2 + y2 + z2) = 0 − 1 % 3 " λ2 = −16

4 % ! ! ,

(y − x)(z + 2λ1) = 0 5 # " " *1

(z − y)(x + 2λ1) = 0, (x − z)(y + 2λ1) = 0.

6% $! 1! , ! !

, % "

(y − x)(z − y)(x − z) = 0.

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