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Московский государственный физико-технический университет (МФТИ)

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

матан Бесов - весь 2012

.pdf

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03.06.2015

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y − x = 0

!
√				√					√					√			√		√
	6		,				6	, −	6				−	6	, −		6	,	6	;
	6		,	6				, −	3				−	6	, −		6	,	3	;
	6			6					3					6			6		3
" λ1 = −					√						√
								6	λ2 =			6
						12			λ2 =	12
						12				12
# $ $ %
$ $
d2L = −2λ1(dx2				+ dy2					+ dz2) + 2z dx dy + 2y dx dz+
+2x dy dz = ±		√								+ dz2 −4 dx dy + 2 dx dz + 2 dy dz)
		√		6	(dx2				+ dy2
				6

6
&																						%
'																						(
dx = 1, dy = dz = 0 dx = dy = 1, dz = 0)

* d				2L		&& dx, dy, dz d2L
+ (,)!							x dx + y dy + z dz = 0,
							x dx + y dy + z dz = 0,									.					(-)
									dx + dy + dz = 0							.					(-)
									dx + dy + dz = 0
'						$ $ x = y

(dx, dy, dz) (-) (dx, −dx, 0)
	√

* " d2L = ± 6dx2 ' ( %

) & %

											√			√		, −	√
	. ,											6	,		6			6
	. ,										6		,	6			3
											6			6			3

−	√		, −		√		,	√
		6				6			6	/	%
	6			6				3		/	%
	6			6				3

& f $ $ 0 &

√√

f "$ $ −	6	6
f "$ $ −	18	18
	18	18

τ [a, b] %

τ = {xi}iiτ=0

a = x0 < x1 < . . . < xiτ −1 < xiτ = b

1 ' [xi−1, xi]

τ xi xi − xi−1 |τ | max xi %

1 i iτ

# + τ

τ + τ τ τ

' + τ + %

+ + %

2◦ τ1 τ2 τ2 τ3 τ1 τ33 4◦ +$ τ1 τ2 τ ! τ τ1 τ τ2

* 5

τ + '

+ τ1 + τ2

* %

[a, b]

( ) & f

τ = {xi}0iτ

/ +

+ [xi−1, xi] %

ξi

−1 x

% +

ξi %

b x

		26 2
		iτ
		iτ

		Sτ (f ; ξ1, . . . , ξiτ ) f (ξi)Δxi,
		i=1

& f

|τ |→0

f f (ξi)Δxi

[xi−1, xi] f (ξi)

! !

f [a, b] I #

$ # ε > 0 δ = δ(ε) > 0

iτ
iτ
		< ε
	f (ξi)Δxi − I	< ε

i=1

# ! τ # |τ | < δ #

! ξ		1		" " " ξ
!	b	1		iτ " %

f (x) dx"

& # f ' #

[a, b] ( [a, b])"

* +

! b

	f (x) dx lim Sτ (f ; ξ1, . . . , ξiτ ),
a	\|τ \|→0

(ε, δ)! , ('

- + .

# #

0 # /

+ !

1 " - f

[a, b]" -+ [a, b]" 1 τ

Sτ (f ) = Sτ (f ; ξ1, . . . , ξiτ ) = f (ξk)Δxk +
	f (ξi)Δxi, (2)

1 i iτ i=k

[xk−1, xk] τ f

" 3

ξi ! k" 4

# (2) + .

# / ξk" 3 #

τ Sτ (f ) +

. # ( |Sτ (f )| > |1|)

# ! " %

# ( ) lim Sτ (f )"

5 f [a, b]"

6 !

/ + $

1,	x ,
f : [0, 1] → R, f (x) =
0,	x .

1 ' τ Sτ (f ) = = 1 Sτ (f ) = 0

3 1 !

[0, 1]"

- f

[a, b]" / '

ω(f ; [a, b]) sup	\|f (x ) − f (x )\| = sup f − inf f.
x ,x [a,b]	[a,b]	[a,b]

f [a, b]

τ = {xi}i0τ ωi(f ) = ω(f ; [xi−1, xi])

f [a, b]

iτ

ε > 0 δ = δ(ε) > 0 :	ωi(f )Δxi < ε τ : \|τ \| < δ.
	i=1

! "	iτ
	iτ
lim
lim	ωi(f )Δxi = 0,	#

|τ |→0 i=1

$%$ $ % % $% $ (ε, δ) & $

	! $					' !
f [a, b] !					"ab f (x) dx = I (
	ε > 0 δ = δ(ε) > 0 : \|Sτ (f ) − I\| < ε
	τ : \|τ \| < δ, ξ1, . . . , ξiτ .
	) ε δ τ			' ! ξ	ξ
				i	i * $ +
$ [xi−1, xi] + ωi(f ) 2(f (ξi) − f (ξi )) (
iτ		iτ
	ωi(f )Δxi 2	(f (ξi) − f (ξi ))Δxi
i=1		i=1		)\| + 2\|I − Sτ (f ; ξ , . . . , ξ )\| < 4ε.
	2\|I − Sτ (f ; ξ , . . . , ξ			)\| + 2\|I − Sτ (f ; ξ , . . . , ξ )\| < 4ε.
		1	iτ		1	iτ
,- +			iτ
			iτ
				ωi(f )Δxi = 0.
		lim		ωi(f )Δxi = 0.
		\|τ \|→0 i=1
			' + τ = {xi}ik=0 τ =

= {xj }kj=0 τ

|Sτ (f ; ξ1, . . . , ξk) − Sτ (f ; ξ1 , . . . , ξk )|

k ω(f ; [xi−1, xi])Δxi. .

i=1

' ! xi = x		i = 0	iτ xi−1			= x	< . . . <
< xji	ji					ji−1
< xji	= xi

	ji				ji
	ji				ji

f (ξi)Δxi −		f (ξ )Δx			ωi(f )Δx = ωi(f )Δxi.
		j	j			j
	j=ji−1+1			j=ji−1+1

τ τ [a, b]

τ ! τ τ τ τ " # #

# $ τ τ #

τ #

k	k
ω(f ; [xi−1, xi])Δxi	+ ω(f ; [xi−1, xi ])Δxi .	%
i=1	i=1
& ' % #
ε > 0 δ = δ(ε) > 0 : \|Sτ (f ) − Sτ (f )\| < ε,
	\|τ \|, \|τ \| < δ.	(

& ) $ ( * # +

# +

, * -+

# ) +

{τn}∞ |τn| → 0 n → ∞ . /+ n=1 $

τn = {x(in)}ki=0n #

ξ1(n)	ξ(n)	Sτn (f )	0 +
	kn 0
# {Sτn (f )}n∞=1			+

- $ , *

( ε > 0 nε! |Sτn (f ) − Sτm (f )| < ε

n, m nε 1 , * +

{Sτn (f )}∞n=1 )

I = lim Sτn	(f ; ξ1(n), . . . , ξ	(n)).
n→∞		kn

kn	iτ
\|Sτn (f ) − Sτ (f )\| ω(f ; [xi(−n)1, xi(n)])Δxi(n) +	ωi(f )Δxi.
i=1	i=1

n → ∞

	iτ
	iτ
	\|I − Sτ (f )\| ωi(f )Δxi.

i=1

lim Sτ (f ) = I.

|τ |→0

!" #$ %

& ' ( )$& ) ( % *

+ " ( & ) (,

$ '

		& -" f '
# [a, b] & τ = {xi}0iτ			. #) [a, b] &
mi inf		f (x),	Mi sup f (x).
	xi−1 x xi				xi−1 x xi
/ ' $
	iτ				iτ

Sτ (f ) mi xi,					(f ) Mi xi
				Sτ
	i=1				i=1

-" f 0 #)

1
			Sτ (f ) Sτ (f )			(f )
			Sτ (f ) Sτ (f )		Sτ	(f )

) % ' & % $ * Sτ (f )
# &
				iτ
		(f ) − Sτ (f ) =
				ωi(f )Δxi.
	Sτ			ωi(f )Δxi.

2 0& ' ( -#

& % ' -"

3 ) 4

[a, b]

ε > 0 δ = δ(ε) > 0 : Sτ (f ) − Sτ (f ) < ε τ : |τ | < δ.

$& ' % $

' &$ 3 ) " &$ -" %

2 0& $ 5 &

f [a, b]

iτ

ε > 0 τ : ωi(f )Δxi < ε.

i=1

6 ' & $ $$ -" %

3 # & 3 , )

& -" f # # [a, b] / '

|τ | < δ

iτ	iτ

ωi(f )Δxi =	(f (xi) − f (xi−1))Δxi
i=1	i=1
	iτ
	\|τ \| (f (xi) − f (xi−1)) = \|τ \|(f (b) − f (a)),
	i=1

5 $ 7 & f '

' $ 5

i=1

[a, b]

! " a c < d b

ε > 0 δ = δ(ε) > 0 : ω(f ; [c, d]) ε, d − c < δ.

# ! ε > 0 $

τ [a, b] % |τ | < δ

iτ	iτ
ωi(f )Δxi ε	xi = ε(b − a).
i=1	i=1

& f

[a, b]

[a, b] (a, b)

[a, b]

	\|f ( x)\| M x [a, b]

& ε			0,	b − a	%$ '

$ τ [a, b]			2
iτ
	ωi(f )Δxi =			ωi(f )Δxi +
i=1	[xi−1	,xi] [a+ε,b−ε]

		+			ωi(f )Δxi = Σ + Σ .
		[xi−1,xi] [a+ε,b−ε]
(" ! "		Σ 4M (ε + \|τ \|).

f ! !

) [a + ε, b − ε]!

* δ = δ(ε) > 0 ! "

Σ < ε |τ | < δ.

+ " ! " δ ε |τ | < δ ε

iτ

ωi(f )Δxi = Σ + Σ < 8M ε + ε = (8M + 1)ε.

, " ! " - .

f [a, b]

/ f 0 [a, b] → R

[a, b]! * $

τ = {ai}i0τ ! " %$ i = 1, . . . , iτ f$ 1 [ai−1, ai]! $ '

1 2 *

$3 3 4 ,

! " %$ i = 1, . . . , iτ

◦ f (ai−1, ai)5

◦ * % " f (ai−1 + 0)! f (ai − 0)

2 ! 6!

◦ f [a, b]

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

τ = {xi } 7 '

$ [a , b ] $ τ = {xi}

[a, b] % |τ | = |τ |

ωi (f )Δxi ωi(f )Δxi,

1 i iτ 1 i iτ

ωi (f ) = ω(f ; [xi−1, xi ])

1 " - f '

[a, b]. -6 . # !

1 " & '

f [a , b ]

◦ -8 '

. a < c < b! f

[a, c] [c, b] f [a, b]! "

! b	! c	! b
f (x) dx =	f (x) dx +	f (x) dx.	-.

i=1

[a, c] [c, b] |f (x)| M x [a, b]

τ = {xi}i0τ

[a, b] τc [a, b]

τ c !" τ c τ #

τc τc [a, c] [c, b]

$% & τc

' & & ( Sτ (f ) Sτc (f ) Sτc (f ) & )

& & & & $ !" &

Sτ (f ) *

Sτ (f ) − Sτc (f ) − Sτc (f ) = 0, c τ.

+ $ c τ c (xi0−1, xi0 ) ξi0 [xi0−1, xi0 ]

ξ [xi0−1, c] ξ [c, xi0 ]
Sτ (f ) − Sτ	(f ) − Sτ	(f ) =
c	c
	= f (ξi0 )Δxi0 − f (ξ )(c − xi0−1) − f (ξ )(xi0 − c).
, !				2M xi0	2M \|τ \|.
Sτ (f ) − Sτ		(f ) − Sτ (f )		2M xi0	2M \|τ \|.
	c		c

- |τ | ! & .

Sτ (f ) →	! c	Sτ (f ) →		! b
Sτ (f ) →	f (x) dx,	Sτ (f ) →		f (x) dx,
c	a	c		c
	a	! b		c
!

	lim Sτ (f )		f (x) dx
	\|τ \|→0	a		"
&		/#		"	0
0 /		$		aa f (x) dx	0
"ba f (x) dx − "ab f (x) dx a < b *						/#

! $ a b c

1 f $" .

§
2◦ 3 # + 1	f g

& [a, b] λ, μ R 1 λf + μg $

[a, b] %

! b	! b	! b
(λf (x) + μg(x)) dx = λ	f (x) dx + μ	g(x) dx.
a	a	a

& )

|τ | → 0 !"

&) (

4◦ + 1 f g & [a, b] )

f g $ [a, b]

	0 5 Δ(f g)(x0) = (f g)(x0 +
+	x) − (f g)(x0) = f (x0 + x)g(x0 + x) − f (x0)g(x0 +
+	x) + f (x0)g(x0 + x) − f (x0)g(x0) = f (x0)g(x0 + x) +

+ f (x0)Δg(x0)

6 1 1 3 11

) 1

, ! 1 f g

[a, b]

ω(f g; [c, d]) M ω(f ; [c, d]) + M ω(g; [c, d]),

[c, d] [a, b] |f | |g| M [a, b]

iτ	iτ	iτ
ωi(f g)Δxi M	ωi(f )Δxi + M	ωi(g)Δxi.
i=1	i=1	i=1

- |τ | !

f g

[a, b]

7◦ 1 f [a, b] inf f > 0

[a,b]

* 1 f1 [a, b]

ωi f1 ωi(f )

◦ f g [a, b] f g

[a, b]	! b	! b
	! b	! b
	f (x) dx	g(x) dx.
	a	a

|τ | → 0

	!
Sτ (f ; ξ1, . . . , ξiτ ) Sτ (g; ξ1, . . . , ξiτ ).
"◦ # f		[a, b]
\|f \| [a, b]			! b
!	b		! b

			\|f (x)\| dx.
	f (x) dx		\|f (x)\| dx.	$%&
a			a

' |f |

|f (ξ )| − |f (ξ )| |f (ξ ) − f (ξ )|,

ωi(|f |) ωi(f )

iτ	iτ
ωi(\|f \|)Δxi	ωi(f )Δxi.
i=1	i=1

( $%&

)* + !

Sτ (f ; ξ1, . . . , ξiτ ) Sτ (|f |; ξ1, . . . , ξiτ ).

, % ' |f | [a, b]

- . + f [a, b] /

ψ! [0, 1] → R

1 x ,

ψ(x) =

−1 x .

0◦ $' 1 2

f [a, b] f

f 3

f [a, b]

! b	! b
f (x) dx =	f (x) dx.
a	a

ϕ =

=f − f [a, b] "ab ϕ(x) dx = 0 ϕ

N max |ϕ| = M

[a,b]

|Sτ (ϕ)| M 2N |τ |

- + 4 |τ | → → 0

56 % 7

+ 56 % 6 + 2◦

[a, b] f

f 0 x0 [a, b] f (x0) > 0

! b

f (x) dx > 0.

f (x0) = d > 0 +

- [a , b ] [a, b] b − a > 0 f

8 5◦

! b

! a

f (x) dx =

f (x) dx +

f (x) dx

(b − a ) > 0.

0 +

dx + 0 =

f g [a, b] m f M [a, b],

g [a, b]

		! b	! b
μ [m, M ] :		! b	! b
μ [m, M ] :		f (x)g(x) dx = μ		g(x) dx.
		a		a

f [a, b]				! b
	! b			! b
ξ (a, b) :		f (x)g(x) dx = f (ξ)		g(x) dx.
	a			a

g 0 [a, b]
	mg(x) f (x)g(x) M g(x), x [a, b].
6◦				! b
	! b	! b		! b
m	g(x) dx	f (x)g(x) dx M		g(x) dx.
	a	a		a

!			"ab g(x) dx = 0			!
"ab f (x)g(x) dx = 0 ! μ " # $
!		"
		ab g(x) dx > 0 ! " !
				" b f (x)g(x) dx
	" b	m		a "			M.
				ab g(x) dx


% μ =	a "f (x)g(x) dx				& "
	b
	a	g(x) dx
' " ( " " ) $
* !			"ab g(x) dx > 0 +! m = min f M =
							[a,b]

= max f " " " # *& $

[a,b]

! , m < μ < M μ = m μ = M % " &

" " # ! " ! * -.

ξ (a, b), f (ξ) = μ

+ ! μ = m μ = M "

0 " " " M -. f

! ξ (a, b) / " ! " ξ

" ! μ = M f (x) < M

x (a, b) # " ! / ! 1 " !

) "*

%& / ! 2* !

[M − f (x)]g(x) dx = 0.

		a	! b−ε
ε	0,	b − a

	2
! b−ε
0 =	[M − f (x)]g(x) dx αε			g(x) dx,
a+ε	[M − f (x)] > 0		a+ε
αε = min
[a+ε,b−ε]
	! b−ε			b − a

G(ε)		g(x) dx = 0 ε	0,		.	3

	a+ε		2
4 " " !
\|G(0)\| = \|G(0) − G(ε)\| 2ε sup g.
			[a,b]

/ " & 3 ε → 0 + 0
! " !	! b

G(0) =	g(x) dx = 0,
	a
! ! #

-. f " [a, b]

[a, b] -.
! x
F (x) = f (t) dt, a x b,	5
a

/ " " " ) ! μ = M

* "

[a, b] F [a, b]

x0 x0 + x [a, b]

F (x0 + x)−F (x0) =	! x0+Δx	! x0	! x0+Δx
F (x0 + x)−F (x0) =	f (t) dt−	f (t) dt =	f (t) dt.
	a	a	x0

[a, b] f

M R

|f (t)| M t [a, b].

|F (x0 + x) − F (x0)| M | x| → 0 x → 0,

[a,"b] x0 [a, b] F (x) =

=ax f (t) dt x0

F (x0) = f (x0) "

F (x0) x0			= a x0 = b

#$						F (x0)
						x
$% f (x0) x0 +					x [a, b]
	F (x0)		!		x0+Δx
		− f (x0) =		1	[f (t) − f (x0)] dt.
				x
	x				x0

ε > 0 $ f x0

δ = δ(ε) > 0 : |f (t) −f (x0)| < ε, t [a, b], |t −x0| < δ.

x| < δ x0 +

x [a, b]

+Δx

F (x0)

− f (x0)

|f (t)

− f (x0)| dt

x0+Δx

1 dt = ε.

F (x0) → f (x0) x0 + x [a, b], x → 0, x

f [a, b] [a, b]

	!	b
G(x) =		f (t) dt, x [a, b],
		x

$
	! b
	! b
	G(x) = f (t) dt − F (x),
	a

G $ [a, b] ' ( f $

x0 [a, b] ) G (x0)

G (x0) = −F (x0) = −f (x0). *

+ , a, b ! ! (

% !

a b

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

$ " x

a, b x x0 $ * x a, b x x0

" x F (

F (x) = − x 0 f (t) dt

f [a, b]Φ ! " #

! b

f (x) dx = Φ(b) − Φ(a).

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

f [a, b]


!	F (x) = Φ(x) + C, a x b.

	x

f (t)dt = F (x) − F (a) = (Φ(x) + C) − (Φ(a) + C) =

= Φ(x) − Φ(a), a x b.

x = b !"#

$% & '( )

% * + ,

+ , , ) %

+ - % +

, ) +

	f ,
[a, b] [a, b]	F	) %
"ab f (x) dx = F (b) − F (a)
§

ϕ [α, β]

f ϕ([α, β]) a ϕ(α) b ϕ(β)

! b	! β
f (x) dx =	f (ϕ(t))ϕ (t) dt.	!.#
a	α

Φ /

f ϕ([α, β]) 0 , Φ(ϕ) /

f (ϕ)ϕ [α, β]) (Φ(ϕ)) (t) = f (ϕ(t))ϕ (t)) , t = α, β

! 1 1 . % #

- 2 & '(

) % ! - % a b#

! b

f (x) dx = Φ(b) − Φ(a),

! β

f (ϕ(t))ϕ (t) dt = Φ(ϕ(β)) − Φ(ϕ(α)) = Φ(b) − Φ(a).

3 4 4 -

) % ϕ

[α, β]) ϕ = 0 [α, β]) ϕ(α) = a) ϕ(β) = b)

5 , , !.#

u v

[a, b]				!
!	b			!	b
	b	b			b
			−			(x)v(x) dx,
	u(x)v	(x) dx = u(x)v(x)	−		u	(x)v(x) dx,	!6#

a	a	a

b
	= u(b)v(b) − u(a)v(a)
u(x)v(x)	= u(b)v(b) − u(a)v(a)
a

u(x)v (x) = (u(x)v(x)) − u (x)v(x),

) %		! b	!
!	b
	u(x)v (x) dx =		(u(x)v(x)) dx −

a x b,

u (x)v(x) dx.

+ ) % & '(

! b

(u(x)v(x)) dx = u(b)v(b) − u(a)v(a).

f * [a, b] → R

# [a, b])

[a, b] 5 {ai}k0

[a, b]) f -

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