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Московский государственный физико-технический университет (МФТИ)

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[НЕСОРТИРОВАННОЕ]

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матан Бесов - весь 2012

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(x0, y0, z0)

∂f

| grad f | =

∂f

∂e

∂x

∂y

∂z

grad f

e ∂f∂e = | grad f |

	grad f
e =	\| grad f \|

! "

$ " % #

x(0) Rn

f ∂f x(0)

∂xi

∂ ∂f

∂xk ∂xi

k = i


xi xk !			∂2f	(x(0))
xi xk !			∂xk ∂xi	(x(0))
fx	x	(x(0)) k = i
i		k

" #

f xi ! ∂2f (x(0))

∂x2i

fxixi (x(0))

% ! ! &

' f m

x(0)

xi1 xim !				∂mf (x(0))
xi1 xim !				∂x	im	. . .∂x
					im	i1
x(0) &
	∂2f	∂2f
	∂2f	∂2f
			,
			,
	∂x1∂x2	∂x2∂x1

! % !
( ( x y

				x3y
f		x, y					(x, y) = (0, 0),
	(		) =			x2 + y2
					0		(x, y) = (0, 0),

! !

) ! &

& ( (

f							x, y
			∂2f		∂2f
			∂y∂x		∂x∂y
(x0, y0)
	∂2f			∂2f

		(x0, y0) =				(x0, y0).		"+#
	∂y∂x			∂x∂y
,				)!				xf

y f f (x0, y0)

	x x y
y ( \|	x\| \| y\| - !
	y (Δxf (x0, y0))
x(Δy f (x0, y0)) =	y (Δxf (x0, y0))
"	f (x0, y0) −

− f (x0, y0 + y) − f (x0 + x, y0) + f (x0 + x, y0 + y)#

. (

∂f ∂f

( ∂x ∂y (x0, y0)

∂f		(x0	+ θ1	x, y0)Δx =		∂f	(x0, y0	+ θ2	y)Δy.
y					x
	∂x					∂y

y x

∂2f	(x0	+ θ1	x, y0 + θ3	y)Δy	x =
∂y∂x
			=	∂2f	(x0 + θ4 x, y0 + θ2 y)Δx y,
				∂x∂y

0 < θi < 1 i = 1, 2, 3, 4!"

# x y" #

| x| | y| $

$ % &

(x0, y0) '!"

# $( ' &$ )

* n %

& ( m 2"

x(0) Rn

m 2 x(0)

+ & $ " $ * f

" + &$

x(0) % & , &

∂mf

∂xim . . .∂xi1 & ( ) %$

* (k − 1) k %

k 2 k m!"

- k = m & & (

x(0) ' ) *

∂m−2f ∂xim−2 . . .∂xi1 "

- k < m $

% &

∂2g

, g =

∂k−2f

∂x

ik−1

∂x

ik−2

. . .∂x

ik−1

x(0) & '

) * g"

. % *

n = 3 m = 3" $ $ &

		∂3f			∂3f
fzyx	=		fxyz =			" /
		∂x∂y∂z			∂z∂y∂x
fzyx = (fz )yx = (fz )xy = (fzx				)y =

= (fxz )y = (fx)zy = (fx)yz = fxyz .

0 $%

$ * % &

$ &

* "

1 * f & m

) & m

!" 0

* f "

2 $( ''"'"3 ''"'"4 m &

! *

! & % m"

. ) $

" $ * f *

G Rn" -) * &

mα!! Dαf (x) dxα.

i=1 ∂xi

df (x) = n ∂f (x) dxi, x G.

G δ(df (x))

δ(df (x))

dxi
	n
			∂f
δ(df (x)) =		δ	∂f		(x) dxi			=
δ(df (x)) =	i=1	δ	∂xi		(x) dxi			=
	i=1

n		n	∂2f					n	∂2f
			∂2f						∂2f
=							(x)δxj	dxi =		(x) dxi δxj .
=		∂x		∂x			(x)δxj	dxi =		(x) dxi δxj .
		∂x		∂x		i			∂x ∂x
i=1	j=1		j			i		i,j=1	j i

f x δxj = dxj "j = 1, . . . , n#

f x $ d2f (x)

		n

		∂2f
d2f δ(df ) δxi=dxi	=			dxi dxj .

(i=1, ..., n)			∂xi∂xj
		i,j=1

m − 1

f x

m f x dmf (x)

δ(dm−1f (x)) δxi=dxi

(i=1, ..., n)

% n	∂mf

dmf (x) =		(x) dxi1	. . .dxim .

i1, ..., im=1	∂xi1 . . .∂xim

f m

x m − 1

& x &

' (

) $

* & $ n ' '

α = (α1, . . . , αn), αi N0 (i = 1, . . . , n),

\|α\| =			αi + &
, α! = α1! . . .αn! dx	α		i=1α1	αn	α	f =
		= dx1		. . .dxn D

∂|α|

=∂xα1 1 . . .∂xαnn f

-. ' $' f n '

m & x m −1

& &

dmf (x) =

|α|=m

- f ' ' x y

m		∂mf
		∂mf			dxm−kdyk.
dmf = Cmk	∂x	m−k	∂y	k	dxm−kdyk.
k=0

) . m = 2 )

& (x, y) f

	∂2f			∂2f		∂2f
d2f =		dx2	+ 2		dx dy +		dy2
	∂x2			∂x∂y		∂y2

. &

x(0) Rn

m f Uδ (x(0))

. ) "

# , m

Uδ (x(0)) / ϕ0 [0, 1] → R

ϕ(t) = f (x(0) + t x), x = x − x(0).

ϕ [0, 1]

m−1

ϕ(1) =

(k)

(0) +

(m)

(θ),

0 < θ < 1.

k=0

" ϕ

m−1

∂kf (x(0))

xi1 . . . xik +

f (x) =

i1, ..., ik =1 ∂xi1 . . .∂xik

k=0

+rm−1(Δx), #$%

rm(Δx) =

∂mf (x(0) + θ

xi1 . . . xim .

∂xi1 . . .∂xim

, ..., im=1

& & f

'( )*

'( !

f (x	(0)	1 α					(0)		α
	+	x) = \|α\| m−1		D	f (x			)(Δx)		+
			α!
		1					α		(0)	α
		+ \|α\|=m					D		f (x	+ θ x)(Δx)	. #+%
						α!

, &

f m Uδ (x(0))

| x| < δ

(0)

f (x

+ x) = |α| m

f (x

)(Δx)

+ rm(Δx),

#-%

α!

Dαf (x(0)

x) − Dαf (x(0)) (Δx)α =

rm(Δx) =

+ θ

|α|=m α!

ε(Δx) → 0

= ε(Δx)|

x → 0.

, . &

f m

δ x(0) Rn

f | x| < δ

#+% #-% / &

(*' *

m N0

+ o(|

)

f (x) =

aα(Δx)

x → 0,

|α| m

+ o(|

f (x) =

bα(Δx)

)

x → 0.

|α| m

aα = bα α |α| m

/ *

& 0 ' *

cα(Δx)

+ o(|

)

0 =

x → 0

|α| m

cα = 0 α! |α| m

1 y = (y1, . . . , yn) =

x = ty

cαyα tk + o(|t|m) t → 0.

0 =

k=0

|α|=k

n = 1

cαy	α	= 0	k = 0, . . . , m,	y =
					0.

|α|=k

β = (β1, . . . , βn)

|β| = k

	∂\|β\|
0 =		cαyα = β!cβ .

∂yβ1 . . .∂yβn

1 n |α|=k

cβ = 0 β : |β| = k, k = 0, . . . , m,

f m

x(0)
	(0)		x\|	m
f (x		α			) x
	+ x) =	aα(Δx) + o(\|				→ 0.

|α| m

! " #

$ %

& # ' ( ' ) ex2+y2 * * (0, 0) +

o((x2 + y2)2) (x, y) → (0, 0)

eu = 1 + u + u2 + o(|u|2) u → 0. 2

, u = x2 + y2 ex2+y2 = 1 + + x2 + y2 + 12 x4 + x2y2 + 12 y4 + o((x2 + y2)2)

- # # ' )' (

, X Rn Y Rm ,

X Y #

(x, y).

X × Y {(x, y) : x X, y Y } Rn+m.

ε x(0) Rn

Qε(x(0)) {x Rn : |xi − x(0)i | < ε, i = 1, . . . , n}.

, n = 1 Qε(x0) = Uε(x0)

(δ, ε) (x(0), y(0))

Rn+m x(0) Rn y(0) Rm / #

Qδ,ε(x(0), y(0)) = Qδ (x(0)) × Qε(y(0)).

- '

(x, y) ' +

F (x, y) = 0, $

F 0 ' ) * * x y #

1 ) f . X → R X R +

F (x, f (x)) = 0 x X.

2 # # E R2 $ y = f (x) -.

F (x, y) = 0 y = f (x)


		x2 + y2 − 1 = 0,		!
"			" [−1, 1] # #
$"% &
	f1(x) = 1 − x2, f2(x) = − 1 − x2.

' ( !

" " " "

Qδ,ε(x0, y0) " (x0, y0) " # "

! & x20 + y02 − 1 = 0

y0 Qδ,ε(x0, y0)

0	x0	x
		1

) !

x0 > 0 y0 > 0 *

# δ > 0 ε > 0 Qδ,ε(x0, y0)

x2 + y2 − 1 = 0 y = f1(x).

+ !

Qδ,ε(x0, y0) , $" $"% f1

Qδ,ε(x0, y0) - $"

$" $"% f1 Qδ (x0) δ > 0

ε $" f1

" # (a, b) Qδ (x0) ε > 0 "

. " (x0, y0) " (x0, y0) = (1, 0)

" " / " Qδ,ε(1, 0) !

# y

! Qδ,ε(1, 0)

$" " " $"% y = f (x)

0 #

# $"% F " #

y "

" Qδ,ε(x0, y0)

◦ F U (x0, y0)

(x0, y0)

!◦ F (x0, y0) = 0

1◦ Fy (x0, y0) = 0 Fy (x0, y0)

! ! Qδ,ε(x0, y0) = = Qδ (x0) × Qε(y0) (x0, y0)

F (x, y) = 0 y = f (x),

f : Qδ (x0) → Qε(y0)

Qδ (x0) f (x0) = y0

2◦ F (x0, y0) f


f (x0) = −	Fx	(x0, y0)		.
	Fy	(x0, y0)
" # $ % F					F
		x				y
U (x0, y0) % f (x)		= −				(x, f (x))
			Fx			(x, f (x))
			Fx			(x, f (x))
			Fy			(x, f (x))

Qδ (x0)

3◦

Fy U (x0, y0)

3◦

σ ε > 0

Qσ,ε(x0, y0) F

Fy ! " #

Fy > 0 Qσ,ε(x0, y0) $ F (x, y) %

x Qσ (x0) y

[y0 − ε, y0 + ε]

& ' ( F (x0, y0) = 0)

F (x0, y0 − ε) < 0, F (x0, y + ε) > 0.

* F (x, y0 − ε) F (x, y + ε)

x Qσ (x0) ( '

! ) # δ (0, σ]

F (x, y0 − ε) < 0, F (x, y0 + ε) > 0 x Qδ (x0).

+ x Qδ (x0)

F (x , y0 − ε) < 0 F (x , y0 + ε) > 0

,- %

F (x , y) # y (y0−ε, y0+ε) F (x , y ) = = 0 . y

F (x , y) & y = f (x ) .

f / Qδ (x0) → Qε(y0)

f (x0) = y0

F (x, y) = 0 y = f (x) Qδ,ε(x0, y0).

0 -

F (x, f (x)) = 0 x Qδ (x0).

1 f Qδ (x0) 2#

x0 # !

y
y0 + ε		F > 0
y0 + ε				0
			=	0
y0		F	=				)
		F					)
						x
					(
				=	f
y0 − ε			y	=
			y
	F < 0
	F < 0
0	x0 − δ	x0			x0 + δ			x
	" 34 3 4

! ε > 0 %

# % ε > 0

δ = δ(ε) > 0 f (Qδ (x0)) Qε(y0) =

=Qε(f (x0))

x 5 Qδ (x0) y =

=f (x ) & 3

3◦ 3 '

! (x0, y0) (x , y ) 6 %

f x

% 4◦ 7

F 33 3 3

F (x0 + x, y0 + y) − F (x0, y0) =
	= Fx(x0, y0)Δx + Fy (x0, y0)Δy+
	+ε1(Δx,	y)Δx + ε2(Δx, y)Δy, ( )
εi(Δx,	y) → 0 (Δx,	y) → (0, 0) i = 1, 2	8
\| x\| y = f (x0 +			x) −
− f (x0) =	f (x0) y0 +	y = f (x0 + x)
. ( )
0 = Fx(x0, y0)Δx + Fy (x0, y0)Δy+
	+ε1(Δx, y)Δx + ε2(Δx,		y)Δy.

(x, y)

\|			x\| \|		y\| = \|		f \|

		(x0	, y0) + ε1(Δx,	y)			(x0, y0)
y = −	Fx				x = −	Fx		x + o(Δx)
		(x0, y0) + ε2(Δx,		y)			(x0, y0)
	Fy					Fy

x → 0 f

f (x0) = −Fx(x0, y0) = −Fx(x0, f (x0)) . Fy (x0, y0) Fy (x0, f (x0))

(x0, y0) Qδ,ε(x0, y0)

x Qδ (x0)

		(x, f (x))
f (x) = −	Fx		x Qδ (x0).
		(x, f (x))
	Fy

! " #

$ % & #

% #

f : Qδ (x0) → Qε(y0).

' ( ) ε δ #

ε = δ * ! ) " !

F (x, y) = 0 y = f (x)

Qδ (x0, y0) = Qδ (x0)×Qδ (y0) R2 # F (x, f (x)) = = 0 x Qδ (x0) #

% ' + % "

% ,- % " "

" F (x1, . . . , xn, y) = 0 .

# n = 1

x = (x1, . . . , xn), x(0) = (x(0)1 , . . . , x(0)n ),

(x, y) = (x1, . . . , xn, y), (x(0), y0) = (x(0)1 , . . . , x(0)n , y0), F (x, y) = F (x1, . . . , xn, y).

F x y

= (x1, . . . , xn, y)

%◦ F U (x(0), y0)

(x(0), y0)

0◦ F (x(0), y0) = 0

1◦ Fy (x(0), y0) = 0 Fy (x(0), y0) ! " Qδ (x(0), y0)

(x(0), y0)

F (x, y) = 0	y = f (x),
	Qδ (x(0)) → R
f :	Qδ (x(0)) → R

Qδ (x(0)) f (x(0)) = y0 F (x, f (x)) = 0 x

Qδ (x(0))

#
2◦ F (x(0)					, y0) f
x(0) i = 1, . . . , n
	∂f		F	(x(0)	, y0)
		(x(0)) = −	xi	(0)	.

∂xi			Fy (x , y0)

# $ % &

F U (x(0), y0) i = 1, . . . , n

&

	∂f	(x) = −	Fxi (x, f (x))

	∂xi		Fy (x, f (x))
Qδ (x(0))
/
%

F (x, y) = 0

m m

{uj (t)}jm=1

" m #$

t =

= (t1, . . . , tm)

∂u1

(t)

. . .

(t)

∂(u1, . . . , um)

∂t

(t)

. . .

∂(t1, . . . , tm)

∂um

(t)

. . .

(t)

∂t1

∂tm

& '

x = (x1(

( xn)( y =

= (y1( (	ym) = (˜y, ym)( y˜ = (y1( ( ym−1)( (x, y) = (x1(
( xn( y1(	( ym) Rn+m( (x, ym) = (x1( ( xn( ym)(
F (x, y) = F (x1( ( xn( y1( ( ym)


)◦	Fj (x, y) = Fj (x1, . . . , xn, y1, . . . , ym) j										=
	= 1, . . . , m
	U (x(0), y(0)) (x(0), y(0))
*◦	Fj (x(0), y(0)) = 0 j = 1, . . . , m
+◦
+◦		∂(F1, . . . , Fm)
	J =							= 0
	J =	∂(y1, . . . , ym)		(0)		,y	(0)	= 0
			(x			,y		)
Qε(x(0), y(0))

	{Fj (x, y) = 0}jm=1				{yj = fj (x)}jm=1,
(f1, . . . , fm)! Qε(x(0))						→ Rm			fj
Qε(x(0))									fj (x(0)) = yj(0)	j =

= 1, . . . , m "

Fj (x, f1(x), . . . , fm(x)) = 0 x Qε(x(0)) (j = 1, . . . , m). ,)-

§
" # % #$	m

(.) {Fi(x, y) = 0}jm=1.

/ m = 1 )* ) *

)* ) ) / (

m − 1 ( # ( m

J 0

$( ( ( # ( 0 0 $ #

% (

∂(F1, . . . , Fm−1)

Jm−1 = ∂(y1, . . . , ym−1) (x(0),y(0)) = 0.

1 #$ !		m − 1
	,.- y1(
( ym−1	2 , )* ) *

)* ) )-( η > 0

$ #$

ϕj : Qη (x(0), ym(0)) → R, ϕj (x(0), ym(0)) = yj(0)

(j = 1, . . . , m − 1)

# ( ,.- ,3- , ,.- ,3- 0#- Qη (x(0), y(0)) U (x(0), y(0))(

	{yj = ϕj (x, ym)}jm=1−1,
(3)
	Fm(x, y) = 0.
/ 0
Fj (x, ϕ1	(x, ym), . . . , ϕm−1	(x, ym), ym) = 0
	(j = 1, . . . , m − 1)		,*-
	(j = 1, . . . , m − 1)

(x, ym) Qη (x(0), ym(0))

Qη (x(0), y(0))


( )	{yj = ϕj (x, ym)}jm=1−1,
( )	Φ(x, ym) = 0

Φ(x, ym) = Fm(x, ϕ1(x, ym), . . . , ϕm−1(x, ym), ym),

Φ(x(0), ym(0)) = Fm(x(0), y(0)) = 0.

! "# Φ !! #$

Qη (x(0), ym(0)) " " # !$ ! # % ! "# & Φ(x(0), ym(0)) = 0

'	∂Φ(x(0), ym(0))	= 0

	∂ym

() ( ) * + "

∂Φ

∂Fm ∂ϕ1

+ . . . +

∂Fm ∂ϕm−1

∂Fm

. -

∂ym

∂y1

∂ym

∂ym−1

∂ym

!! # $

) ym .

∂Fj

∂ϕ1

∂Fj

∂ϕm−1

∂Fj

+ . . . +

= 0,

∂y1

∂ym

∂ym−1

∂ym

j = 1, . . . , m − 1.

J =

∂(F1, . . . , Fm)

∂(y1, . . . , ym) (x(0),y(0))

∂F1

. . .

∂y

m−1

∂y

m−1

. . .

. . . . . . . . .

. . .

. . . . . .

= 0.

∂Fm−1

∂y1

. . .

∂ym−1

∂ym

∂y1

. . .

∂ym−1

∂Fm

. . .

∂Fm

. . .

∂Fm ∂Φ

∂y1

∂ym−1

∂ym

∂y1

∂ym−1

∂ym

1 +

" # % .* %

∂ϕj −

# % ∂ym j = 1, . . . , m 1 $

- / 2

∂Φ (x(0), y(0)) = 0.

∂ym

3 $

ym & () ( )

" & " " & " Qε(x(0), ym(0))

Qη (x(0), ym(0))

Φ(x, ym) = 0 ym = fm(x),

! "# fm4 Qε(x(0)) → R !! #$

fm(x(0)) = ym(0)

	Φ(x, fm(x)) = 0		x Qε(x(0)).	5
6 7 8 Qε(x(0), y(0))

(7)	{yj = ϕj (x, ym)}jm=1−1,		(8) {yj = fj (x)}jm=1,
	ym = fm(x);
		(i = 1, . . . , m − 1).
	fj (x) = ϕj (x, fm(x))	(i = 1, . . . , m − 1).		9

: ; 8 Qε(x(0), y(0)) 1 + ! "# fj 4 Qε(x(0)) → R !! #

fj (x(0)) = yj(0) j = 1, . . . , m

6 ( . ) 9 5 6 "

, < !! # (

xk & % " % &

∂F		m		∂f
		∂F
j	+		j	i	= 0, j = 1, . . . , m,
∂x			∂y	∂x
k		i=1	i	k

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