матан Бесов - весь 2012

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Московский государственный физико-технический университет (МФТИ)

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[НЕСОРТИРОВАННОЕ]

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β = β(t0) !

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Γ = {rˆ(t) a t b}

Γi = {rˆ(t) ti−1 t ti} a = t0 < t1 < < . . . < tk = b Γi

x y z

Γ = {r(t) a t b} ! τ = {ti}iiτ=0

" [a, b] a = t0 < t1 < < . . . < tiτ = b

! rˆ(ti−1) rˆ(ti) " i = 1iτ " Γ

# Λτ

iτ
	\|r(ti) −r(ti−1)\|.
SΛτ =	\|r(ti) −r(ti−1)\|.
i=1
Γ "
SΓ sup SΛτ .
	τ
$ Γ "
#	SΓ < +∞

% #

Γ = {r(t) a t b} &

c (a, b) "

Γ = {r(t) : a t c}, Γ = {r(t) : c t b}.

" Γ

Γ = {r(t) a t b}

|r(b) −r(a)| SΓ max |r (t)|(b − a).

a t b

' ( ) |r (t)| "

[a, b] #

τ = {ti}ii=1τ	& [a, b]
* + , - -
	iτ
\|r(b) −r(a)\| SΛτ =
\|r(b) −r(a)\| SΛτ =	\|r(ti) −r(ti−1)\|
	i=1

	iτ
max \|r (t)\|
max \|r (t)\|	(ti − ti−1) = max \|r (t)\|(b − a).
a t b	a t b
	i=1

. τ

Γ = {r(t) a t b}

s = s(t) (a, rˆ(a))

! ! !

+ y

+ z

ds2 = |dr|2 = dx2 + dy2 + dz2.

= s(t) &

Γt = {r(u) : a u t}, a t b,

t0 <

< t0 +

t b

" +

{r(t): t0 t t0 +

t} "

s(t0) = s(t0 +

t) − s(t0)

|r(t0 + t) −r(t0)|

max

|r (t)|

t0 t t0+Δt

t → 0 + 0 + s+(t0) = |r (t0)|

/ + s−(t0) = = |r (t0)|

s (t0) = |r (t0)|

s (t) s(t) [a, b]

Γ = {r(s) 0 s SΓ}

	dr
s	dr	= 1
s	ds	= 1

			dr
			dr	= 1

! "

# $ "

= 1

= (cos α, cos β, cos γ),

cos2 α + cos2 β + cos2 γ = 1,

α, β, γ

ds !"

#$ %

& # Ox Oy Oz

dx	= cos α,	dy	= cos β,	dz	= cos γ,

ds		ds		ds

' #

dr ds

&$ r

|r(t)| = C t U (t0),

$ r (t0) ' r (t0) r(t0)

( ( )) & &

(r(t),r(t)) = |r(t)|2 = C2

(r (t0)r,(t0)) + (r(t0)r, (t0)) = 2(r (t0)r,(t0)) = 0.

* Γ & & & &

Γ= {r(s) : 0 s S},

s & ' s0 [0, S] *

			dr				2
%			dr		dt		d r
%
t	ds U (s0) ∩ [0, S] ds					=	ds2
s0

+ k = k(s0)		dt	(s0) &			#
+ k = k(s0)		ds	(s0) &			#
Γ (s0, rˆ(s0))
, s0 = 0 s0 = S &

- #				k(s0)

k(s0) & & & # #

# ! s $ .


	t	# \| t\|				= \|t(s0 +
+ s) − t(s0)\| /
				s , %
	s)
t(s0 +	s)	t(s0) % /
ϕ = ϕ(Δs)
			ϕ
\|	t\| = 2 sin		2 ϕ		s → 0 + 0

! &

+ s)

$ +

t(s0

t(s0) % #

2 sin

ϕ(Δs)

k = lim

lim

s→0

s→0+0

. R =

+∞ &

(s0, rˆ(s0))

Γ = {r(t) a t b}

t =

= r

d r

− s r

s 3

ds2

|s r − s r |

k =

d r

! " k

ds t

d r

k =

× t =

×r

− s r

|r ×r |

k =

|r |3

x 2 + y 2 + z 2

k =

d r

= 0

|n|

= n,k

= 1,

(t,n) = 0.

! " # "

$ #$ "

% $ n

Γ = {r(s) 0 s S} " ! k = dds2r2 = 0 & ' &$

r = r(s0 + s) −r(s0) =

dr(s0)

+ 1 d2r(s0)

s →

s 2

(Δ )

((Δ )

)

ds2

s → 0.

r = (Δs)t +

k(Δs) n +o((Δs)

( " $

" $ $

n " ) o((Δs)2)

! "

) ) $ " $

)) ' r

$ "

)!$ " ) o((Δs)2)

)!$ $ " $ $ Γ = {r(t) a t b} $

k = 0 ( " " rˆ(t0)

s r

− s r

t = s ds

s 3

" r s = |r | = 0

)!$

y − y0

z − z0

(r −r0,r (t0),r (t0)) = 0,

x − x0

x (t0)

y (t0)

z (t0)

= 0,

y (t0)

x (t0)

z (t0)

r0 = r(t0) = (x0, y0, z0) = (x(t0), y(t0), z(t0))

k = 0

R = k1 n

r #

1 d2rρ = r + Rn = r + k2 ds2 .

% &'(

− s r

= |r

| = x 2 + y 2 + z 2,

ρ = r +

s 3

, s

x x

+ y y + z z

+ y

+ z

) ρ = ρ(t)

$ Γ

* Γ +

) Γ Γ = {(x(t), y(t), 0), a t b}

Γ = {(x(t), y(t)) : a t b}.

- Γ .

r Γ xOy

r r n

/ α #

		dα
	dt
t =ı cos α +j sin α,		= (−ı sin α +j cos α)		,
	ds		ds

dα

k =

= | −ı sin α +j cos α|

= ±

|x y − x y |

k =

+ y

3/2

)

. $ k > 0

			2
		ξ = x + R2		d x		,
				ds2
ρ = (ξ, η),			2 d2y
		η = y + R			,
				ds2
ξ = x − y	x 2 + y 2		x 2 + y 2
	x y − x y	, η = y + x		x y − x y			.

0 Γ 1

11$ 1$ y = f (x)


ds2 = dx2 + dy2 = dx2 + f 2(x) dx2, ds = 1 + f 2 dx,
k =	\|y \|				,
k =		2	)	3/2	,
(1 + y			)
						1 + y 2
						1 + y 2
				ξ = x −		y	y	,
k > 0,						1 + y 2
				η = y +		y	.

/ Γ = {r(t), a t b} #

11$

rˆ(t0) Γ

(t0, rˆ(t0))

Γ (t0, rˆ(t0)) ! 2

t(t0) n(t0) β(t0)

rˆ(t0)

t #

$ n

& Γ ' (( ) #

		n∂			= −κn.
dt	= n,k	n∂		dβ
ds	= n,k	∂s	= −kt + κβ,	ds

* ( + ," % - ( + (( )

β = [nt,]

dβ

ds =

,n + t, ds

= [kn,n] +

t, ds

= t, ds .

dβ

. ds t ,

−κn

dβ

κ (−∞, +∞)% * / /(( ) κ

Γ (t0, rˆ(t0))% 0

|κ| |κ| #

! , #

s #

% 1 ! #

2 ( + #

( + %

a, b ! ! % % !

[a, b] ! [a, b) !

(a, b] ! (a, b)% * / #

! ! %

* ( ) f F

a, b % + ) F f a, b

F = f a, b % * / a a, b

b a, b F (a) F (b) #

* F ' ! f a, b % 3 F + C

C ' ! fa, b % 2 (F (x) + C) = F (x) = f (x)%

2 ! F Φ ' #

! ( ) f a, b Φ(x) = F (x) + C

C ' % 2

(F (x) − Φ(x)) = f (x) − f (x) = 0.

3 ( 4 #

F (x) − Φ(x) = C.

. ) ( )

! ,

% * / ( ) f

! ! #

% 1 ! !

( ) f ! f (x) dx%

! "

f a, b $

f (x) dx = F (x) + C,

F % f C %

!	& # $
!			!	!	!
	dg(x)			g (x) dx,	f (x) dg(x) f (x)g (x) dx.
	'
$
	(◦	!
	(◦	f (x) dx = f (x)
		f (x) dx = f (x)
	)◦	!
	)◦	f (x) dx = F (x) + C F %
		f (x) dx = F (x) + C F %
		f *
	◦◦	!
	◦◦		(x) dx = F (x) + C
	2.	F	(x) dx = F (x) + C

+◦ ,- .

/ #	f1(x) dx	f2(x) dx
α, β R #		!
!	!	!

(αf1(x) + βf2(x)) dx = α f1(x) dx + β f2(x) dx + C.

0 1◦

2◦
1 3◦		F (x) =
"	"
= α f1(x) dx + β	f2(x) dx

	§
αf1 + βf2 *
!	!

F (x) = α f1(x) dx + β f2(x) dx = αf1(x) + βf2(x).

' 2◦ 0 1◦ 2◦◦

, .

2 3

f 4 #

f (x) dx

5 F (x) = f (x)

# f

f (x) dx = F (x) + C 2◦

/ 4 " " 4 " "

/ 5 "

" " #

x1 dx = ln |x| + C

$ (−∞, 0) (0, +∞)

u, v
"		(x)v(x) dx
	u	(x)v(x) dx
!		!

u(x)v (x) dx = u(x)v(x) − u (x)v(x) dx + C.

F (x) + C

F = uv − u v dx = u v + uv − u v = uv ,

F uv

2◦

f a, b

ϕ: α, β → a, b					α, β
α, β		!
!		!

	(t) dt =			+ C.
f (ϕ(t))ϕ	(t) dt =		f (x) dx	+ C.	!"

x=ϕ(t)

%$ $ F ◦ϕ& F (x) =

=f (x) dx&

(F (ϕ(t))) = f (ϕ(t))ϕ (t),

' ' !

2◦

(	!"
!	!

	f (ϕ(t)) dϕ(t) =		+ C.
	f (ϕ(t)) dϕ(t) =	f (x) dx	+ C.

x=ϕ(t)

) *"$

" f (ϕ(t)) dϕ(t) ϕ(t) $ x&

+ f (x) dx& #$ t"

) + *

% & ϕ α, β &

% ξ, η ϕ( α, β )			#
ϕ−1 ,		!" & ξ, η
!	!

	f (x) dx =			+ C.
		f (ϕ(t))ϕ (t) dt

t=ϕ−1(x)

- *$

t dt

1 d(t

1 dx

+ a )

+ C1 =

2 t

+ a

2 x

x=t2+a2

ln |x|

+ C2 =

ln(t

+ a ) + C2.

x=t2+a2

* * %

z = x + iy,

x, y R& i . * & * *

/ x *

z x = Re z"& y .

zy = Im z"

*+ z1 = x1 + iy1& z2 = x2 + iy2

*$ & x1 = x2& y1 = y2 0

z = x + i0 % *

x 1 2 z = x

z = x + iy *

|z| = x2 + y2.

% z = x + iy

z¯ = x − iy.

z1 + z2 + *+ z1 = x1 + iy1& z2 = x2 + iy2 *

z = (x1 + x2) + i(y1 + y2).

z1z2 + *+ z1 = x1 + + iy1 z2 = x2 + iy2 *

z = (x1x2 − y1y2) + i(x1y2 + x2y1).

i · i = −1

! " #

$ %

&$ & # % 0 = 0 + i0 1 = 1 + i0 ! z C &

! z C z = 0 &

" $ $ $ #

$ " %

' " z2 − z1 $ $ z2 = x2 + iy2 z1 = x1 + iy1

z2 − z1 = x2 − x1 + i(y2 − y1).

( z z1 $ $ z1 z2 z2=0 z2

zz2 = z1 "

z = x + iy ) * $ ! #

zz2 = z1 z¯2 #

* z2 $ z = x + iy

z|z2|2 = z1z¯2 ! |z2| + " z2 |z2| > 0,

	z1z¯2		x1x2 + y1y2			−x1y2 + x2y1
z =	\|z2\|2	=			+ i		.
				x22 + y22		x22 + y22

- ! $ $

" .$ ! #

- ! ! $

" % ! "

) . z = x + iy #

(x, y) # .

0 $ #

% % %&$ 0 # * z z¯ %

" &

1 %& *$ #

z¯ = z, z1 + z2 = z¯1 + z¯1, z1 − z2 = z¯1 − z¯2, z1z2 = z¯1z¯2, zz¯ = |z|2.

2 % #

n N0 /

Pn(z) = Anzn + An−1zn−1 + . . . + A1z + A0, An = 0,

! z C Ak C k = 0 n

z0 Pn(z0) = 0

) " z0 C ! Pn(z) #

" (z − z0) " Pn(z)

Pn(z) = (z − z0)Qn−1(z) + r,

! Qn−1 + ! n − 1 r

Pn Pn(z)

z − z0

- "

3 ! Pn k N k n #

Pn(z) = (z − z0)kQn−k(z), Qn−k(z0) = 0,

! Qn−k + ! z0 % !

Pn k ) k = 1 z0 %

C ! Pn

	r
Pn(z) = An(z − z1)k1 (z − z2)k2 . . .(z − zr)kr ,
Pn(z) = An(z − z1)k1 (z − z2)k2 . . .(z − zr)kr ,	ki = n,
	i=1

z1 z2 zr " Pn

k1 k2 kr

# Pn # $

z0 = a + ib (b = 0)

k z¯0 = a − ib

% &

Pn(z) = (z − z0)kQn−k(z).

' $ !(

Pn(z) = (z − z0)kQn−k(z).

Pn(¯z) = (¯z − z¯0)kQn−k(¯z),

$ Qn−k

!( ) $

Qn−k

* z¯ z

Pn(z) = (z − z¯0)kQn−k(z).

z¯0 " Pn (

z¯0 # z0

+ z¯0 "

Pn k z¯0 = z0

Pn # k

z0 z¯0

* ' z0 = a + ib b = 0 (z − z0)(z − z¯0) = z2 + pz + q,

p, q R p2 − q < 0

(z − a − bi)(z − a + bi) = (z − a)2 + b2 = z2 − 2az + a2 + b2,

p2 − q = a2 − (a2 + b2) = −b2 < 0.

, -

Pn $

) ! !

	α1			αr 2	β1	2	βs
Pn(x) = An(x−a1) . . .(x−ar)				(x +p1x+q1) . . .(x +psx+qs),
			− qj < 0
r	s	2
i=1 ai + 2 j=1 βj = n		pj			.j = 1	s
i=1 ai + 2 j=1 βj = n		4			.j = 1	s
An R ai
Pn αi αi N
	x2 + pj x + qj = (x − zj )(x − z¯j ), Im zj = 0,
zj z¯j Pn Im zj = 0 βj
βj N



	P (x)
		! P
	Q(x)	! P
" Q			# $

$ !

P(x)

Q(x)

a α

− α ˜ ˜

Q(x) = (x a) Q(x), Q(a) = 0.

P (x)

x − a α

x − a

α−1

Q x

(

( )

)

(

)

A R

P (x)

α−1

(x − a)

Q(x)

" # A R

P (x)

−

P (x) − AQ(x)

x − a α

Q x

)

(

x − a

α ˜

(

)

(

)

Q x

( )

% & % %

A a % % %

% A =	P (a)	' (
	˜
	Q(a)
˜		˜

P (x) − AQ(x) = (x − a)P (x).

) $ x − a * +

P(x)

Q(x)

z0 = a + ib (b = 0) β

Q x2 + px + q = (x − z0)(x − z¯0)

2	β ˜	˜
Q(x) = (x	+ px + q) Q(x), Q(a + ib) = 0.

P (x)

Q(x)

M, N

M x + N

P (x)

β−1

q β

(

)

(

)

Q x

)

(

P (x)

β−1

(x + px + q)

Q(x)

" # * M, N R

P (x)

−

M x + N

P (x) −

(M x + N )Q(x)

, ,

Q x

(

β ˜

( )

)

(

)

Q x

( )

% & % % "

M N

, % x2 + px + q

a + ib % % % -

				˜
P (a + ib) − (M (a + ib) + N )Q(a + ib) = 0,
			P (a + ib)
	M (a + ib) + N =
						.
			˜
			Q(a + ib)
. * % % M N /
	P (a + ib)				P (a + ib)
M = Im		, N = Re					− M a .

	˜				˜
	bQ(a + ib)			Q(a + ib)

- $ ,

P Q

P(x)

!! Q(x)

α1	αr	2	β1	2	βs
Q(x) = (x −a1) . . . (x −ar)		(x + p1x + q1) . . . (x			+ psx + qs) ,

a1 ar

Q x2 + pj x + qj = (x − zj )(x − z¯j ) Im zj > 0 z1 zs

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