матан Бесов - весь 2012

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Московский государственный физико-технический университет (МФТИ)

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[НЕСОРТИРОВАННОЕ]

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βε [α, β) :

β , β (βε, β).

βε [α, β) :

f (ϕ(t))ϕ (t) dt

β , β (βε, β).

.pdf

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[ai−1, ai]

[a, b]

f [a, b]


!	b	!	b
	f (x) dx		˜
	f (x) dx		f (x) dx,
	a		a

˜ → f ! [a, b] R

" # f $

	"
%	ab f (x) dx
" b ˜
a f (x) dx # f
8◦

u v

[a, b] &'(

) * +

{ai}k0 [a, b] #

u	v		##			,
[ai−1, ai] &i =			1, . . . , k( -			, u v
ai	&i = 0, . . . , k( *
' 8◦				!
	!	b	k		ai
						u(x)v (x) dx.
	a		u(x)v (x) dx =		ai−1
			i=1

- , '

					!
!	b		k		!	ai
	b			ai		ai
		(x) dx =			−			(x)v(x) dx =
	u(x)v	(x) dx =		u(x)v(x)	−		u	(x)v(x) dx =
	a		i=1	ai−1		ai−1
							!	b
						b		b
							−		(x)v(x) dx.
				= u(x)v(x)			−	u	(x)v(x) dx.
						a		a

* $ #

. /

R3 . "

$ ,

Rn &n 1( 0 , 1

& 2 ( &,

( #

# 3

, E Rn μE

$ # !

4( P " Rn

(a1, b1) × . . . × (an, bn) P [a1, b1] × . . . × [an, bn],

aj bj &j = 1	n( μP =	/n
		j=1(bj − aj )5
6( &	( , E1 E2
E1 E2 μE1 μE25
7( &	( , E1 E2
E1 ∩ E2 = μ(E1 E2) = μE1 + μE2
% 4( 6( μE 0
, E &		(

% 6( 7(

8( & ( , E1 E2

μ(E1 E2) μE1 + μE2

, G R2

G = {(x, y) R2 : a x b, 0 y f (x)}, &+(

f [a, b] f 0 [a, b]

τ = {xi}0iτ a = x0

< x1 < . . . < xiτ

= b mi =

min

f Mi = max

[xi−1,xi]

G (τ ) G (τ )

iτ

G (τ ) =

(xi−1, xi) × (0, mi),

i=1

iτ

G (τ ) =

[xi−1, xi] × [0, Mi].

i=1

G (τ ) G G (τ )

μG (τ ) μG μG (τ ).

y = f (x)

iτ

μG (τ ) =

xi =

τ ,

i=1

iτ

! "#$"

μG (τ ) =

xi = Sτ ,

i=1

Sτ Sτ & ' '

! f τ

(

Sτ μG Sτ .

) |τ | τ

* * Q

	! b
μG = f (x) dx.
+ ! b	a
+ ! b	" ,
μ(int G) =	f (x) dx (int G = G \ ∂G).

" - *

ab f (x) dx [a, b]

- R

Γ = {(r, θ) : r = r(θ), α θ β},
Γ = {(r, θ) : r = r(θ), α θ β},						β
r = r(θ)							α	Γ
[α, β] [0, 2π] G =					0
[α, β] [0, 2π] G =					0
= {(r, θ)	α θ β 0 r r(θ)}						! "#$.
& * *
	τ	=	{αi}0iτ	& [α, β]				mi =
= min	r	Mi	= max	r
[αi−1,αi]			[αi−1,αi]

G (τ ) G (τ )

0iτ

G (τ ) = {(r, θ) : αi < θ < βi, 0 < r < mi},

i=1

0iτ

G (τ ) = {(r, θ) : αi θ βi, 0 r Mi}.

i=1

1	iτ			1	iτ
	mi2	αi = μG (τ ) μG μG (τ ) =			Mi2	αi.
2
	i=1		2
					i=1

) |τ | → 0 *

	!	β
μG =	1	r2(θ) dθ.
	2
		α

D R3

{(x, y, z) R3 : a < x < b, y2 + z2 < R} D

{(x, y, z) R3 : a x b, y2 + z2 R},

μD = πR2(b − a)

[a, b] Ω R3

	!"# $ Ox
	τ	=	{xi}0iτ	% [a, b] mi =
=	min f	Mi	= max	f
	[xi−1,xi]		[xi−1,xi]

0iτ

Ω (τ ) = {(x, y, z) R3 : xi−1 < x < xi, y2 + z2 < m2i },

i=1

0iτ

Ω (τ ) = {(x, y, z) R3 : xi−1 x xi, y2 + z2 Mi2}.

i=1

iτ	iτ
π mi2	xi = μΩ (τ ) μΩ μΩ (τ ) = π Mi2 xi.
i=1	i=1

' ( \|τ \| → 0 ) ) *+ Ω
	! b

μΩ = π	f 2(x) dx.

Γ = {r(t), a t b}

- )

! # )

$ s(t) .

s (t) = |r (t)|.

§
S % Γ &$
! b	!	b
S = s(b) − s(a) =	s (t) dt =	\|r (t)\| dt =

a!a b

=x 2(t) + y 2(t) + z 2(t) dt.

/ Γ = {(x, f (x)) a x b} % +

	! b
S =		1 + (f (x))2 dx.

	a

! " #

[a, b] S % 0

Γ = {(x, f (x)), a x b} $

f $ Ox 0 S !

# )

mes S

τ = {xi}i0τ a = x0 < x1 < . . . < xiτ = b %

[a, b] ( Γ ( Γ(τ )

) (xi, f (xi)) Γ

i = 0, 1, . . . , iτ 0 (

Γ(τ ) $ Ox ) ) S(τ )

' * 0 0

)+ 0 0

. $ .

0 S(τ )

		iτ

		mes S(τ ) = π	(f (xi−1) + f (xi))li,
		i=1
$
li = (xi − xi−1)2 + (f (xi) − f (xi−1))2 =
	&
	&	f (xi) − f (xi−1)	2
		f (xi) − f (xi−1)
=	1 +			xi = 1 + (f (ξi))2 xi, !1#
=	1 +	xi		xi = 1 + (f (ξi))2 xi, !1#
		xi

) ξi (xi−1, xi) ( !2 " 3#

) 0 4$

mes S = lim mes S(τ ),
\|τ \|→0

! b
mes S = 2π	f (x) 1 + (f (x))2 dx.
a

!" στ (f ; ξ1, . . . , ξiτ ) #$ %

# # $ "$ τ

& ξ1

ξiτ ' # # M1 = max |f |

[a,b]

iτ

|f (xi−1) − f (ξi)|

| mes S(τ ) − 2πστ | 2π

i=1

|f (xi) − f (ξi)|

1 + (f (ξi))2

iτ

2π M1|τ | 1 + M12 xi =

i=1 $

(b − a)|τ | → 0 |τ | → 0.

= 2πM1

1 + M12

( ) *

$ |τ | → 0

! b

στ →

1 + (f (x))2

|τ | → 0,

f (x)

# , * [a, b]

(

, * f - [a, b) → R b +∞

# % $" [a, b ] [a, b)

(	! b
	f (x) dx	.
	a

[a, b) / " ) # .

0	! b
! b	! b
f (x) dx lim	f (x) dx,	&

ab →b−0 a

) " )

# . 1

+∞ − 0 +∞

' " "

# $ . &

( # % " #

# % 2 b = +∞ , * f

" # # %

3 " ) # &

2 b < +∞ , * f #

[a, b) # % [a, b]

" ) # &

2 , * f # %

[a, b] " ) # & [a, b)

# 4

"ab f (x) dx , * b . .

5 , * #

[a, b] # % $"

[a, b ] [a, b) # % [a, b] 3 # % [a, b]

" ) # [a, b) $

! " # $

+∞ +∞

∞	" ∞ dx
%	1	xα	"$
α > 1 "$ α 1

[a, b ] [a, b)

' (


	!
		b		< ε. ')(
ε > 0 bε [a, b) : b , b [bε, b)		b	f (x) dx	< ε. ')(
		b

* +"$

' ( $," -,

$ F (x) = ax f (t) dt x → b − 0 .

& , / -

$ F ' ) 0 1

) 2 )( 3$ 1 $ ')(

+"$ ' (

"$ "ab f (x) dx

a [a, b) 4 $ &

/ $ . " $" &"

$& &

+ -, $ "$ b → b − 0

& & $ # $5

◦ 3 ' ( "$ $

! b	! a		! b
	f (x) dx =	f (x) dx +	f (x) dx a [a, b).
a	a		a

6◦

'7 # (

"ab f (x) dx "ab g(x) dx "$

$ λ μ R

! b

(λf (x) + μg(x)) dx = λ

f (x) dx + μ

g(x) dx.

)◦

(

"ab f (x) dx "ab g(x) dx "$ f g [a, b)

! b

9◦

f (x) dx

g(x) dx.

': % , ;7 # ( 3

& [a, b) Φ < $ f

[a, b) $

! b

f (x) dx = Φ(b − 0) − Φ(a),

'9(

" & $ # '9(

0◦ '8 (

3 u v=

[a, b) → R & & & $

& 1$ [a, b ] [a, b) $

−

v dx,

dx = uv

'0(

&" # '0(

2◦

-, &

' ( 3 f &

[a, b) ϕ & $

[α, β) β +∞ 5 a = ϕ(a) ϕ(t) < b =

= lim

ϕ(t) $

! β

t→β−0

! b

f (x) dx =

f [ϕ(t)]ϕ (t) dt.

3 . & " " . # &

"$ "$ $

f (x) dx

bε ϕ(βε)

b , b (bε, b) β , β (βε, β) : ϕ(β ) = b , ϕ(β ) = b .

	!

!				!
		b			β		b , b (bε, b).
		b	f (x) dx =		β	f (ϕ(t))ϕ (t) dt < ε	b , b (bε, b).
		b			β
	" "ab f (x) dx
	# $
% & %
				f 0 [a, b)
"ab f (x) dx
					! b
					! b
			M R :			f (x) dx M b [a, b).

" b

' ( a & % b ) "ab f (x) dx


*		) & %
b		→ b − 0+
"ab	f (x) dx & % b

f g [a, b ] [a, b) 0 f g

[a, b)
,◦ " b
"	a	g(x) dx !
	ab f (x) dx"

§
-◦						" b
						" a f (x) dx !
						ab g(x) dx	"ab g(x) dx
' 1◦							"ab g(x) dx
-
!	b			! b
M R :	f (x) dx g(x) dx M b [a, b).
	a				a
-			"ab f (x) dx
2◦ # "ab g(x) dx

f g
[a, b ]		[a, b) # f > 0 g > 0 [a, b)
$
		lim		f (x)	= k (0, +∞).
		lim			= k (0, +∞).
		x→b−0		g(x)

"ab f (x) dx "ab g(x) dx

'. a [a, b)/

k2 g(x) f (x) 2kg(x) x [a , b) . 2◦

$ 0
!	b	!	b
	g(x) dx		f (x) dx
	a		a

$ a [a, b)

	2$ $ 0 1
0 f g [a, b)
f 0, g 0,	f (x) = O(g(x)) x → b − 0.
	3$	"ab f (x) dx

ab |f (x)| dx

" b

a |f (x)| dx

!
		"ab f (x) dx "
				!
!				!
	b				b	\|f (x)\| dx a b < b < b.
	b		f (x) dx		b	\|f (x)\| dx a b < b < b.
	b				b

# $ "ab f (x) dx % &

'
" b		" b	\|f (x)\| dx
( a	f (x) dx	a
"			)	* %
	ab \|f (x)\| dx			+

"ab f (x) dx , " f & %

$ [a, b ] +

- # $ , " % .( )

	* /$ " &
% % 0 /		∞ sin x		dx
		1	x
$ % &			→ R
	# , " f 1	(a, b]

−∞ a < b < +∞ 2 0%

[a , b] (a, b] * "ab f (x) dx

# -

%$ "ab f (x) dx

	! b	! b
	f (x) dx lim	f (x) dx,

aa →a+0 a

" - / %$

ab f (x) dx 3

§
"	1	dx	α < 1
'	0	α
α 1		x
				(a, b) → R
#			, " f 1

−∞ a < b +∞ 2 0%

[a , b ] (a, b) * "ab f (x) dx

4 %$ " b f (x) dxa

&$
! c			! b
	f (x) dx,		f (x) dx,		5
a			c
c 3 % (a, b)
# - 0 0
! b	!	c	!	b
f (x) dx		f (x) dx +		f (x) dx.	6
a		a		c
7 & % 5
	"ab f (x) dx
* 5			%

c (a, b) - & %

# 6 &

% c a < c < c < b

+
!	c	!	b
	f (x) dx +		f (x) dx =
		!c		! b		! c		! b
a	! c	!c	c	! b		! c		! b
=	f (x) dx+		f (x) dx+		f (x) dx =		f (x) dx+	f (x) dx.
	a		c		c	a		c
				( 2 6			&
		! b				! b
			f (x) dx		lim	f (x) dx,
		a			a →a+0	a
					b →b−0

/$ , "

(a, b) −∞ a < b +∞

c1 ck−1 a = c0 < c1 < . . . < ck−1 < ck = b

f (a, b)

! ci		"i = 1 k −1# $
"ab f (x) dx
c0 c1 ck			"
%	!		ab f (x) dx
&!	k !	ci
b		ci
	f (x) dx	f (x) dx,
a	i=1	ci−1

! ' '

' (ci−1	, ci) ' (
	"ab f (x) dx

' ! )

"*# + !

! , - .

(a, b) (−∞, +∞) '

! "ab f (x) dx

	" x f (t) dt
x0 (a, b) F (x) =
(a, b)	x0

0 '

' !1

! ∞

f (x)g(x) dx. "2#

*◦ f

[a, +∞)

3◦ g

[a, +∞)

,◦ g(x) → 0 x → +∞

"2#

F 4

f 5 f g

[a, b]

! b

f (x)!g(x) dx =					!
a	b		b			b
=				−			(x) dx. "6#
=	F	(x)g(x) dx = F (x)g(x)		−		F (x)g	(x) dx. "6#
	a		a			a

0 ) !

b → +∞ ( /

'				(					M =
= sup \|F \| < +∞					!
!	[a,∞)		!
!	∞		!	∞		∞
	∞			∞		∞
		(x)\| dx M								=
	\|F (x)g	(x)\| dx M		\|g	(x)\| dx = M		g	(x) dx		=

a	a		a
			+∞	= M \|g(a)\|.
		= M g(x) a		= M \|g(a)\|.

& "6# !

b → +∞ + "2# '

[a, +∞)

"2#

	7
	'

f [a, +∞)

F (x) = "ax f (t) dt


	"	∞	f (t) dt
	lim F (x) =
x→+∞		a
! g
	lim g(x) c " g˜ g −c
	x→+∞

[a, ∞) g˜(x) → 0

x → +∞ #$				! ∞
!	∞	!	∞	! ∞
	f (x)g(x) dx =		f (x)˜g(x) dx +	cf (x) dx
	a		a	a

% % % % &

% % ' % (
)	" ∞

# *	sin x		dx %
	1	x

' %

+ ' % * f (x) = sin x g(x) =

=x1 " f F (x) =

=− cos x g

[1, +∞) g(x) → 0 x → ∞

" ∞

sin x

dx %

# k N

, %

! 2kπ | sin x| dx =

2kπ

| sin x|

kπ

2kπ

kπ

k−1

(k+m+1)π | sin x| dx =

π | sin x| dx =

2kπ

2kπ 0

(k+m)π

m=0

∞ | sin x|

0 #

' % 1

§ !

% 3
!		∞	sin x		−	!		∞	cos x
			sin x						cos x	dx.
				dx = cos 1
				dx = cos 1					2
	1		x				1		x

4 %

# % %

5 . % 6

7 h8 [a, b] → R

& ) [a, b]

{ai}k0 a = a0 < a1 < . . . < ak = b

h * (ai−1, ai)

[a, b]

[a, b] ε > 0

[a, b] h = hε

	a	\|f (x) − h(x)\| dx < ε.			&9)
	' # τ = {xi}0iτ				(

		iτ		f (ξi)Δxi (
[a, b]				f (ξi)Δxi (
		i=1
: :
		x (xi−1, xi), i = 1, . . . , iτ ,
	f (ξi)	x (xi−1, xi), i = 1, . . . , iτ ,
	h(x) =	x = xi, i = 0, 1, . . . , iτ ,
	ci	x = xi, i = 0, 1, . . . , iτ ,
ci R "			!
!	b	iτ	!	xi	iτ
	b			xi
	\|f (x) − h(x)\| dx =			\|f (x) − f (ξi)\| dxi	wi(f )Δxi,
	a	i=1		xi−1	i=1

wi(f ) f [xi−1, xi]

ε > 0

|τ | τ

! "#$ % !

f & & h '

# ( )

f [a, b]

ε > 0 !

[a, b] ϕ"

! b

|f (x) − ϕ(x)| dx < ε.

* + ε > 0 , h [a, b] & %

"#$ , % ! [a, b]

& ! ! ϕ &

! b

|h(x) − ϕ(x)| dx < ε.

"-$

. ( / '

h (

(xi−1, xi) τ ) a = x0 < x1 < . . . < < xk = b , ϕi % %

& % [a, b] !/ η 0,		1	\|τ \|
& % [a, b] !/ η 0,		2	\|τ \|
(		2
(
	(xi−1 + η, xi − η),
h(x)	(xi−1 + η, xi − η),
ϕi(x) = 0	x (xi−1, xi),
&	[xi−1, xi−1 + η] [xi − η, xi].

		§
, ϕ =			ϕi \|ϕ\| M
!			k		!
			i=1
	b			k		xi
	b					xi
	\|h(x) − ϕ(x)\| dx =					\|h(x) − ϕi(x)\| dx =
	a	!		i=1		xi−1	!
	k	!	xi−1+η				!	xi
			xi−1+η					xi
=			\|h(x) − ϕi(x)\| dx +					\|h(x) − ϕi(x)\| dx
	i=1	xi−1						xi−η

M 2ηk < ε,

η > 0

"#$ "-$

! b

|f (x) − ϕ(x)| dx

a	! b	! b

|f (x) − h(x)| dx + |h(x) − ϕ(x)| dx < ε + ε = 2ε,

a a

% # - / ! & f

& %

, −∞ a < b +∞ 0 f

% (a, b)

/ {ci}k0 a = c0 < c1 < < . . . < ck = b 1

#◦ f 2 (

[α, β] (a, b) ( / c1

ck−13 " b

-◦ 1 %& a |f (x)| dx

%& %&

1 c0 c1 ck

+ % #4 5 4 -

& 6 1◦

1 "ab f (x) dx

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