матан Бесов - весь 2012

Aik Mjm Njm

α1 a = a1 α2 a = a2

!

!

% & ' ()) Aik Mjm Njm #

' * * ! *+

P

* ! Q ! $

, #

$ ' - ()) #

Aik Mjm Njm ! !. #

& &

/$ $ ! '

* * ! #

$ $ ' ' *#

+ &

n 2

§

$ # ! #

# ! # .!

$

x1 = x2 x t

a(x − x1)(x − x2) = (x − x1)t.

% &" " ! ! " "

! $ ' (

p # $

q # $

p + q # $

)! " " J

)

* + J

a + bt p! +

! I ! $"

) " " I +

!$ ' ! , -' .&

4◦

!

I = R(sin x, cos x) dx

§

! $ '

! !

u = tg x2 , −π < x < π.

)

! $ " ! "

" 0 sinm x cosn x dx m, nQ ! u = sin x u = cos x !

' $

5◦ 0 ! $" !$

I = eαx(β sin βx + α cos βx) + β2C2 =

α2 + β2

= eαx(β sin βx + α cos βx) + C. α2 + β2

6◦

!

" !

§ Rn

' R

! x y

ρ(x, y) 0 ( )

! x y "* "*

&

#◦ ρ(x, y) = 0 x = y+

$◦ ρ(x, y) = ρ(y, x) ( )+

%◦ ρ(x, y) ρ(x, z) + ρ(z, y) ( )

, "" "

- "

. * "

*

ε

'

!

Rn

, n N

Rn

x1 xn / x

. 1◦ 2◦ Rn . 3◦

Rn x + y

λx λ R x = (x1, . . . , xn)

y = (y1, . . . , yn)

x + y (x1 + y1, . . . , xn + yn) x, y Rn, λx (λx1, . . . , λxn) x Rn, λ R.

Rn

x Rn ! " #

$ x − y Rn

x − y = (x1 − y1, . . . , xn − yn), x, y Rn.

i=1

. $ " / n = 2 0 *

, + !

" /$ n N" # +$ $ ! & |x| > 0 |y| > 0"

/ a, b 0

§ Rn

/ * a = xi b = yi

i

/ 3 + 45

|x + y|2 |x|2 + 2|x| |y| + |y|2 = (|x| + |y|)2,

!"

6 0 !

"

. $ 3◦ "

/$ x, y, z Rn x = x − y y = y − z" x + y = = x − z" 0

|x + y | |x | + |y |, " " |x − z| |x − y| + |y − z|,

+$

ρ(x, z) ρ(x, y) + ρ(y, z).

/ Rn - $

ε $ , Rn"

/ ε > 0 ε a

Rn !

Uε(a) {x Rn : |x − a| < ε}.

Uε(a) Rn

ε > 0 a

a Rn

{x(m)}∞m=1 x(m) Rn

{x(m)}∞m=1 a

lim x(m) = a x(m) → a m → ∞

m→∞

lim |x(m) − a| = 0,

m→∞

ε > 0 mε N

|x(m) − a| < ε m mε.

! "

x(m) Uε(a) m mε.

{x(m)}∞1 Rn

a = Rn

k = 1, . . . , n

lim x(km) = ak,

m→∞

x(km) ak x(m) a

#" " "

! & '

R > 0% |x(m)| < R m N

!

* " +, '

& ! " "

k = 1, . . . , n " - -

" ' 0 " 1

(n)

" " {x(mj )}∞j=1 - "

"

§

x Rn

E Rn

ε > 0 : Uε(x) E.

$! E Rn !

'

- ! Rn

ε Uε(a) a Rn

Rn

Uε(a) x Uε(a)

r |x − a| < ε δ ε − r > 0

Uδ (x) Uε(a) y Uδ (x)

%

& x x

U (x)

% ε x '

$ (

U (x) x ' ε > 0 Uε(x) U (x)

()

lim x(m) = a

m→∞

U (a) ε Uε(a)

% *

+

˚ \ { } ˚ \ { }

Uε(x) Uε(x) x , U (x) U (x) x .

a Rn

E Rn

{x(m)}∞ : a = x(m) E m N, lim x(m) = a.

1 m→∞

, int E {x+ x -

E} E

E E {x : x Rn, x - E}

E

0 E

) E = E

E E Rn

E = a -

y(m) E x(m) = y(m) y(m)E y(m) E

y(m) E x(m)

E |x(m) − y(m)| < 12 εm

! {x(m)} x(m) E

0 < 12 εm |y(m) − a| − |x(m) − y(m)| |x(m) − a|

|x(m) − y(m)| + |y(m) − a| < 32 εm.

" a E

# $ %

E = E.

& ' ! G

! F

# F

G ' Rn F \ G

G \ F '

( a Rn '

E Rn Uε(a) ε > 0

E E

∂E E Rn '

) '* E

+ E

E (

∂Uε(a) = {x : |x − a| = ε}, Uε(a) ∩ ∂Uε(a) = ,

∂{x : |x − a| ε} = {x : |x − a| = ε} {x : |x − a| ε}.

#

,◦ E ' ) ) )

E ∩ ∂E = -

.◦ E ) ) )

∂E E-

§

/◦ E Rn ) ) ) ) Rn \ E '

0 3◦

∂E = ∂(Rn \ E)

# E Rn

' %

,◦ ∂E ∂E = ∂E -

.◦ E \ ∂E = int E- /◦ E ∂E = E

)

E Rn '

diam E sup |x − y|.

x,y E

E, F Rn '

dist(E, F ) inf |x − y|.

x E,y F

1 )

Rn '

E F Rn E ∩ F =

dist(E, F ) > 0

# 0 dist(E, F ) = d

2

{x(m)}, {y(m)}, x(m) E, y(m) F m N,

|x(m) − y(m)| → d m → ∞

3 2 ' 4 5 6 ! 7 '

{x(m)} * 2 {x(mk )}∞k=1

{y(mk )}∞k=1 * 2

{y(mkj )}∞j=1

x(mkj ) → a, y(mkj ) → b j → ∞.

E F a E b F

d |a − b|

|a − x(mkj )| + |b − y(mkj )| + |x(mkj ) − y(mkj )|→0 + 0 + d = d

j → ∞ d = |a − b| a = b

E ∩ F = d > 0

!

E " F "

# Γ Rn $

%

Γ = {x(t) : α t β} =

= {(x1(t), . . . , xn(t)) : α t β}, &'(

xi " % % [α, β] ) * (i = 1, . . . , n)

% + & (

x(α) % +

x(β) "

Rn % + %$

+ % GRn + , %- - a b $ + Γ &'( + G &Γ G( +, +

a b &x(α) = a x(β) = b(

. % G G Rn %$

+

# E Rn % ,

+ , + %- $

,-+ X Y &E = X Y X ∩ Y = (

+ E + +, + + $ ! + X + Y &E ∩ ∂X ∩ ∂Y = ( / +

% E + $

, + + ''

/ , ,

K Rn

{Gα}α Ω %Rn K Gα

K Gα {Gα}α Ω

α Ω

{Gαi }mi=1 m N K

§

0 ) *

f : X → R, X Rn, &'(

% ) * 1 % X $

Rn n N # X

% + f 2 f (x) = = f (x1, . . . , xn) + ) * f

x = (x1, . . . , xn) X

f % +

{(x, xn+1) (x1, . . . , xn, xn+1) Rn+1: x X xn+1 = f (x)}

0 ) * + f '◦ x x X3

4◦ x x X3

◦ E E X

) * + f $

E x(0) " + E 2

A % + f x → x(0)

. + - $

+ n = 1

A A ˆ

R R

n = 1

lim f (x) f

x→x(0)

x → x(0)

f

E x(0)

!

#

% $% # # !

' % #

) * + ) * , $% # #

! " - $ f "

( E Rn

E ( f (E) #

B > 0

|f (x)| B x E.

. # / % n = 10 #

' / 0 $ f ( E

# E Rn x(0)