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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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!! b

Q dy

Q(x(t), y(t), z(t))y (t) dt,

Γa

!! b

R dz

R(x(t), y(t), z(t))z (t) dt.

Γa

"◦ #$ $ % &

	' ( P (x(t)& y(t)& z(t))& Q(x(t)& y(t)& z(t))& R(x(t)&
	y(t)& z(t)) ' ( t
	) [a, b]
"* & a = (P, Q, R) Γ&
	Γ P dx + Q dy + R dz
+◦ *

!) !


	P dx + Q dy + R dz =		[P cos α + Q cos β + R cos γ] ds. ,
Γ		Γ
	#$ $ % ) x (t)& y (t)&
z (t) -
	x =	dx	\|r \| = cos α\|r \|,
		ds
			\|r \| = cos β\|r \|,
	y =	dy
		ds
			\|r \| = cos γ\|r \|,
	z =	dz
		ds

. ) )$ ) t&

+/ " + %◦ 0 )

) ( '

(

# ) & $

t (τ ) > 0 ) t = = t(τ )& ) ( "

1 )

τ 2◦ 3 ) ( Γ

$ ) #$ $ )$ , 4

& $ $ )

( * $ , cos α& cos β& cos γ& ) & 1 &

$ )

& , $ )

!		◦	5 A = (x(a), y(a), z(a))& B = (x(b), y(b), z(b))&
			!	!
	dx = x(b) − x(a), dy = y(b) − y(a), dz = z(b) − z(a).


AB			AB	AB
		◦	0 &

$ 3$ 6

		3 Γ 7 $ " & a < c < b&
Γ1 = {r(t) : a t c}, Γ2 = {r(t) : c t b}.
8	!	!		!
	(a, dr) =		(a, dr) +	(a, dr),
	Γ		Γ1	Γ2

9 ) $ %

) 1 )

Γ = {r(t) a t b}

! " # a = a0 < a1 < . . . < < ak = b# Γi = {r(t) ai−1 t ai} !i = 1# # k"

! "

!k !

F ds

F ds,

Γi=1 Γi

!k !

(a, dr)

(a, dr),

Γi=1 Γi

R3 Γ = {r(t)


a t b}# τ = {ti}ii=0τ		[a, b]# \|τ \| =
= max(ti − ti−1)	Λτ

& rˆ(ti)#

Γ ' ( Λτ %

! % &

i = 1# # iτ # rˆ(a)
# rˆ(b)	)"

Γ = {r(t) a t b}

R3 τ = {ti}iiτ=0

[a, b] Λτ Γ

R3 Γ Λτ !

"! |τ |

E " " " #$ P Q R %

	!	!
lim		P dx + Q dy + R dz = P dx + Q dy + R dz.
\|τ \|→0	Λτ	Γ

* & Q ≡


≡ R ≡ 0 % Ai = rˆ(ti)# Ai−1Ai = {r(t) ti−1		t ti}#

Ai−1Ai

Ai−1 ) Ai ε > 0 +

r = r(t) [a, b]# δ = δ(ε) > 0

# τ # \|τ \| < δ# Λτ	E#
\|r(t) −r(ti)\| < ε t [ti−1, ti], i = 1, . . . , iτ ,	!,"

Ai−1Ai Ai−1Ai % E ∩Uε(Ai−1)

- η > 0 + #

# P E#

ε = ε(η) > 0 #

|P (M ) − P (Ai)| < η,

M E ∩ Uε(Ai−1), i = 1, . . . , iτ . !."

/ # |τ | < δ# δ = δ(ε) ε = ε(η)#

( ) !,"# !." )

P dx −

P dx =

Ai−1Ai

Ai−!1Ai

Ai−1Ai

AiAi−1

(P (x, y, z) − P (Ai)) dx +

P (Ai) dx,

Ai−1Ai

AiAi−1

Ai−1AiAi−1Ai #

Ai−1Ai

5◦

3◦ §

|i| < η2(s(ti) − s(ti−1)),

s(t) ! " #$ Γ #$ %

& '

!	!			iτ
				iτ
	P dx −					< 2ηS,
	P dx −	P dx	=		i	< 2ηS,

Λτ		Γ		i=1

S ! Γ

( " ' η > 0 " # )

' " ) * % # # "

R3 ( $ # ) ) ' "

$ $ $ " ( + #

, ' + " ' # *

z = 0 - $ Γ # + " "

Γ = {(x(t), y(t)) : a t b},

" " "$

Γ F (x, y) ds

. Γ

" a(x, y) = P (x, y)ı + Q(x, y)j

	P dx + Q dy = (a, dr).
Γ	Γ

( #

#$ " #

' D ! "$ , ' R2 " ' Γ ! "

Γ ∂D /

Γ , # '

D , ' Γ+ "

) " # " 0 , ) 1$ ' , D %$ ( " #

Γ , # '

D , ' # # Γ−

. 0 ∂D , D


" " " *$ "
Γi ∂D =			%k
Γi ∂D =			i=1 Γi )
" ) ' ' D ∂D , # , '
# # ∂D	+	∂D		+	=	%k	+
# # ∂D		∂D			=	i=1	Γi


	O	x
		2 3
, ' D %#
Oy #
D = {(x, y)		: a < x < b, ϕ(x) < y < ψ(x)},

4 0 ϕ ψ " " 4 4 0 # [a, b] ϕ < ψ (a, b)

, ' D %#

Ox #

D = {(x, y) : ϕ(y) < x < ψ(y), c < y < d},

ϕ ψ

[c, d] ϕ < ψ (c, d)

! D

{D }I
	%		i i=1
		I
"◦
		i=1 Di D#
$◦	Di ∩ Dj = i = j#
%◦	%I

		i=1 Di = D#
&◦	(∂Di ∩ ∂Dj ) ∩ D i = j

' ' (

)

* + ,

∂D

Γi ∂D =

i=1 Γi

! D ∂D

i=1

Γi "

a(x, y)

P (x, y)ı + Q(x, y)j $# %

∂P

∂y

∂Q

∂P

∂Q

∂x

∂y ∂x

D !

& '

∂Q

−

∂P

dx dy =

P dx + Q dy =

(a, dr). (%)

∂x

∂y

∂D+

-. ' +

D '

/ + ' ( ' ")

- (%) Q ≡ 0

!! ∂P		dx dy = − !		P dx,	(&)
	∂y
D			∂D	+

P ≡ 0

(%) 0

" , 1 (&) D

Oy (") ( $2 %$)

= ψ ( x

)

y = ϕ(x)

3 $2 %$

4 56

!! ∂P

dx dy =

! b ! ψ(x) ∂P

dy dx =

∂y

a ϕ(x)

∂y

=[P (x, ψ(x)) − P (x, ϕ(x))] dx =

!a !

=	P (x, y) dx −			P (x, y) dx =
DC	= −	!		AB	!		!
			P dx − P dx = −				P dx,
		CD					∂D+
					AB
(&)				"
"
		BC	P dx = 0 DA			P dx = 0

!	= {(ϕ(y), y) : c y d},
Γ1	= {(ϕ(y), y) : c y d},
Γ2 = {(ψ(y), y) : c y d}		"
Γ2 = {(ψ(y), y) : c y d}

# ! $ % {ci}ki=0 % [c, d] & ' ϕ ψ ! (

% [ci−1
% [ci−1	, ci] ) * % D
		%k
		%k
D =			i=1 Di Di '

Di {(x, y) : ϕ(y) < x < ψ(y), ci−1 < y < ci},
( % !+ #
Ox
y
d = c4	x		x
c3	=	D	=
c2	=		=

	ϕ		ψ
c1	ϕ		ψ
c1	y(		y(
c = c0	y(		y(
	)		)

O			x
		, -. .

) %

!!	∂P	dx dy = − !	P dx, i = 1, . . . , k.
	∂y	dx dy = − !	P dx, i = 1, . . . , k.
Di	∂y		∂D+
			i

/ ( ! 0%

k	!!				!!
			∂P	dx dy =		∂P	dx dy.
			∂P			∂P
i=1		Di	∂y		D	∂y
		Di			D

1 (

k	!		!
	−	P dx = −	P dx,
	−	P dx = −	P dx,
i=1	∂Di		∂D+

( !+

! ∂Di+ ∂Di++1 + %

{(x, y) : ϕ(ci) x ψ(ci), y = ci}

% ( # ! !

#2 ' $

% & %

. D

Ox #2 ψ − − ϕ 3h0 h0 > 0 [c, d] ) 0 < h < h0

Dh {(x, y) : ϕ(y) + h < x < ψ(y) − h, c < y < d} D, Γ1h = {(ϕ(y) + h, y) : c y d},

Γ2h = {(ψ(y) − h, y) : c y d}.

)

Λ1hτ = {(ϕτ (y) + h, y) : c y d}, Λ2hτ = {(ψτ (y) − h, y) : c y d}

! ! ! Γ1h Γ2h

! 2 # % τ % [c, d] %

y § 3 . 4 |τ | % τ

)

Dh,τ {(x, y) : ϕτ (y) + h < x < ψτ (y) − h, c < y < d} D.

1 %

!!	dx dy = −	!
∂P		P dx.

Dh,τ ∂y
Dh,τ ∂y		∂Dh,τ+

\|τ \|
!!	∂P	dx dy = − !		P dx.
		dx dy = − !		P dx.
Dh	∂y		∂Dh+
M = max{max \|ϕ \| max \|ψ \|}
			[c,d]	[c,d]

y = c y = d

x = ϕ(y) + h − M |τ | x = ϕ(y) + h +

+ M (τ ) x = ψ(y) − h − M |τ | x = ψ(y) − h + M (τ )

2M |τ |(d − c)

∂P

dx dy −

∂y

dx dy

∂y

dx dy

∂y

(D \D

h,τ

) (D \D )

h,τ

∂P

max

∂y

4M |τ |(d − c) → 0

(|τ | → 0).

P dx −

! D

P dx → 0 (|τ | → 0, i = 1, 2)

Λihτ

Γih

""D

!" ! #

$ h → 0 %

∂P

dx dy

∂y

∂P

dx dy max

μ(D\Dh) = max

h(d−c).

D\D ∂y

∂y

& ' (

% %

∂D+ P dx h → 0 ) * +

% %
	!		!
		P dx → P dx h → 0 (i = 1, 2),		,
	Γih		Γi
% % h → 0
!	ϕ(c)+h	!
	ϕ(c)+h		ψ(c)
	+		\|P (x, c)\| dx+

ϕ(c) ψ(c)−h

! ϕ(d)+h ! ψ(d)

i = 1 y

Γ1 Γ1h

P D


		!			!d
!		!			!d
	P dx −				\|P (ϕ(y), y) − P (ϕ(y) + h, y)\| \|ϕ (y)\| dy
	P dx −			P dx	\|P (ϕ(y), y) − P (ϕ(y) + h, y)\| \|ϕ (y)\| dy

Γ		Γ			c
1			1h

ω(h; P ; D) max |ϕ |(d − c) → 0 h → 0.

[c,d]

! i = 2

" # $ %

& ' $ " & D '

Ox () ' *

+ , ( ( ϕ(c) = ψ(c)

ϕ(d) = ψ(d) - ε > 0

Dε {(x, y) : ϕ(y) < x < ψ(y), c + ε < y < d − ε}.

. & Dε % $ / 0 & ' ε → 0

+ ' $ " & 1 2

# D #

1 ' {D }I
			i	i=1
3$ & # ' ' Di4
!!	∂P	dx dy = − !	P dx (i = 1, . . . , I)	5

Di	∂y		∂Di+

# 6 7 '

( # '

	!!			!!
I	!!			!!
		∂P	dx dy =		∂P	dx dy.
		∂P			∂P		8
i=1	Di	∂y		D	∂y		8
	Di			D

- # 1 ' 5 0

ϕ(d)

+ ψ(d)−h

|P (x, d)| dx → 0.

∂Di+ = ∂ Di+ ∂ Di+,

∂ Di = D ∩ ∂Di ∂ Di = ∂D ∩ ∂Di

∂D
%I		Di = ∂D	i
i=1	∂	Di = ∂D

j = i Eij ∂ Di ∩ ∂ Dj

! "

# $ %

# Di &

Dj $ ' (

#)$ ( ∂Di+∂Dj+ # Eij $ " &

I	!		I	!			!
		P dx =				P dx =		P dx. #*+$
		P dx =				P dx =		P dx. #*+$
i=1	∂D	+	i=1	∂	D	+	∂D	+
	∂D	i		∂	D	i	∂D
		i				i

," #-$ " #*+$ #.$ , * #/ #0$$ &

D " "

(

( 1

2& 3 " * %

/ " 14 &

∂D

Γi ∂D = = %Ii=1 Γi !

3 " , &

D " ( &

" ' ( D

" " 3 "

" &

*& 5 ∂D = = %Ii=1 Γi 6 &

δ > 0 ! &

" (	( Γi Γj #i = j$ 1
! ∂D " % &
	Γi ( 4 " '

7 ∂D

( (

( 1 " ( &

# %l /$ 8

i=1 Qi ( 6 "

4 dist(Qi, Qj ) >

> δ i = j Qi
%l

i=1 Qi " % 9 "

Qi ∂D 1 1

" ( ( 14 ( Qi

14 ( ' ( ( &

1 1

6 Qi

" ' (

Qi
Qi	(1 i k),
Di = D ∩ int Qi	(1 i k),

1 " & " # Qi$

	D ∂D \ int	%l

:& ; ∂		i=1 Qi &

( ( &

( ( ( 3 &

∂ D

Γ = {r(t) : a t b}

τ = (cos α, sin α)

α = α(t) τ

Ox τ

cos α sin α ! " t

# !$ ! [a, b]	{tj }jj=0
! "
Γ(j) {r(t) : tj−1 t tj },	j = 1, . . . ,	j	%

| tg α| < 2 [tj−1, tj ]

& !

| ctg α| < 2 [tj−1, tj ]

& !

' ! ! [a, b] (

! " & cos α sin α ) ! (

!" x (

* ! (

& " +

$ + ""

&, ) $ (

" "" % &

' {(cos θ, sin θ)- 0 θ 2π}

! " . $ ! /-

(cos θ, sin θ) : π4 θ 2π + π4

' rˆ(tj ) 0 j j −1 " ! (

! !

' " (

θ = 14 π 34 π 54 π 74 π

' rˆ(tj−1) rˆ(tj ) Γ(j) ! % !

" / (

/ &

" Γ(j) ! % (

! " {P		ji	}ij
	%	ji	i=1 (
-	%
◦	ij
◦	i=1 Pji Γ(j)0
%◦	Pji ∩ int Pjk = i = k0
1◦	Dji D ∩ int Pji 1 i ij " " "
2◦	&0
2◦	" ! / Γ(j) "

3 / " +

, +

Γ(j) ! % !

" x ! 3 Γ(j) ! %

Γ(j) = {(x, ψ(x)) : x x x }, |ψ | 2 [x , x ].

τ = {xi}i0j ! ! [x , x ] (

! [xi−1, xi] Pji " (

/ " Ox [xi−1, xi] / "
	xi−1 + xi	, ψ	xi−1 + xi		"
				2
2

3 ! + |τ | ! " τ ! diam Pji !" (

4 1◦ 2◦ 3◦ 5 " (

" 4◦ rˆ(tj−1) rˆ(tj )

" " " Pj1 Pjij (

Oy " " +

12 | tg α| 2 [x0, x1]

[xij −1, xij ] 14 |ψ | 4

" #

# # # $

" # %

# {Pji}iij=1 1◦ 2◦ 3◦4◦◦ 5◦ # & '

'( ) *

4◦◦ rˆ(tj−1) rˆ(tj )

+ # #

Γ(j)

# #

5◦ int Pji #

# Qk

Pji # # ∂ D

{P	}m
j	j=1 # '( )
(
		l
		l		m
		0 Qi	0 Pj ∂D.
		i=1		j=1

" (

# Qi Pj - ) + )

# )

Rk 1 k r ' D '

)( # Qi

Pj # Rk D . Rk

' D R2 \ D '(

(x , y ) (∂D) ∩ int Rk / Rk

)( ' '' # Qi

Pj (x , y ) 0

' Rk / D ∩ int Rk = int Rk 1

)

	l		m		r
D	l	0	m	0	r
D	0 Q	0	0 P	0	0 R	k	,
	i		j			k
	i=1		j=1		k=1

# l + m + r # ' )(

D ∩ int Pj D ∩ int Rk

' ) D ∩ int Qi )

) ' ) )

) 2

3 & 4 " '

# ' & 5 ' 62 ) * #

) #

& # 0 ) 7 4

( ) ( ' # #

" #

8 0 # " (0, x)

−

(−y, 0) (P (x, y), Q(x, y))

∂Q

−

∂P

= 1 & 4

∂x

∂y

μD = !

x dy =

−y dx + x dy = − !

y dx. 9

∂D+

- ' # #

) # )


a = a1ı + a2j + 0k,

b = b1ı + b2j + 0k

			b1
	a1		b1
a × b =		a2	b2	(ı ×j)


							b1
				sgn	a1		b1
				sgn		a2	b2

x ı


a b !

		b1
a1		b1	> 0 (< 0)
	a2	b2	> 0 (< 0)
	a2	b2

a b Oxy

" #

$% &

' ( )

x = x(u, v),

F :

y = y(u, v)

( G Ouv ) * +*

( %

Oxy " #,

γ1 = {(u, v) : u = u1(t), v = v1(t)}, γ2 = {(u, v) : u = u2(t), v = v2(t)}, Fγ1 = {(x, y) : x = x1(t), y = y1(t)}, Fγ2 = {(x, y) : x = x2(t), y = y2(t)},

x1(t) x(u1(t), v1(t)), y1(t) y(u1(t), v1(t)), x2(t) x(u2(t), v2(t)), y2(t) y(u2(t), v2(t)).

Fγ2

γ2

γ1

Fγ1

u O

- &

% γ1 % γ2

( Fγ1 Fγ2 . /

+* % !

dx1		dx2

	dt	dt

	dy1	dy2
	dy1	dy2
	dt	dt

xuu1

+ xv v1

xuu2

+ xvv2

yuu1

+ yv v1

yuu2

+ yv v2

∂(x, y)

xv u1

∂(u, v)

u2 . v2

$ (' %&

γ1 γ2 " %# Fγ1 Fγ2 " %# - / &

J(u, v) ∂∂((x,u, yv)) > 0 "< 0# &

% ( )

" #

$% % γ1 %+ & ' ( D &

) G $% γ1 ) % &

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