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Национальный исследовательский университет «МЭИ»

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УрМатФиз / УрМатФиз с теорией

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<<< < Предыдущая 1 2 3 45 / 365 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 > Следующая >>>

2.2.6 2.2.1 :
1	d 0	r	dR r 1		' +	d2	R r	= 0

r	dr @			dr A		d'2	r2
r	dr @			dr A		d'2	r2
				R r + R r			'
					r	=	'	:	2:2:7
					r
				R r			'
					r2

2.2.7 ! " r, | '. %

& " " ' D D , & !

. (' ) &		:
	R r + R r			'
	r	=				= :

	R r			'
	r2
( * ) O+,
	R r + 1R r				R r = 0		2:2:8

	r		r2
' + ' = 0:							2:2:9
2.2.6 " ) 2.2.5 )
0 = 2		0 = 2 :					2:2:10

0 " ) 2.2.9 , 2.2.10 " ' 1 ) 2- |4 " " ' 5 ) 1 ' 5 - ! 1. 6 7 ) 2.2.9 , 2.2.10 . 1 . . 8 '-) " *9 ' ! *

*9 1
		0 = 0		0 ' = 1<			2:2:11
n = n 2							2:2:12
n = n 2	n ' = An cos n' + Bn sin n' n =				1 1	:	2:2:12
% =		+, 2.2.8 = 0 = 0 = n = n 2:
			R0 r +	1R0 r = 0			2:2:13
				r

n r + 1rRn r

= n:

r2 Rn r = 0 n = 1 1:

. 4

R0 r = C0 + D0 ln r

Rn r = Cnrn + Dnr n n = 1 1:

:2:14

:2:15

! " # ! $% D = fr ag, " ' '%-


# ' ! r ! 0	! D0 = 0 Dn = 0: )
! " #	+ ,!
R0 r = C0 Rn r = Cnrn n =			:	:2:16
		1 1

! " # $% D = fr ag, " ' % -

' ! r ! 1		! D0 = 0 n = 0 n = 1 1: )
" #				+ ,!
R0 r = C0		Rn r = Dnr n n =						:	:2:17
R0 r = C0		Rn r = Dnr n n =					1 1	:	:2:17
/!$, + # !
un r ' = Rn r0n ' n =						:
un r ' = Rn r0n ' n =					0 1	:
1									+
$!2 ++ 3 # ! 3 un r ':
	a0	1	n
ur ' = n=0 un r ' =	2 + n=1 r			an cos n' + bn sin n'					:2:18

' % , #!' 4!'! 5$6 ' 2 +'7 ' 7 '# ' 55 6 ' !2 ' + + r '. 8 4!'+ $'455 6 ! a0 = 2C0 an = AnCn bn = BnCn " ! . 9 + 3, ' !

% # ' '

g ' = a0 +	1
g ' = a0 +	an an cos n' + bn sin n' :	:2:19
2	n=1

' # ' 7 ! ! ' ' " '7 " ! ' 5$6 g' ;2 ' ! %' '+ ! # $' ! + 5$6 fcos n' sin n'g n = 0 1: ! , 3' +:


			2
an =	1	Z g ' cos n' d'	bn =		1		Z g ' sin n' d'	2:2:20
	a				an
		0	0
			n =			:
				0 1

u r ',2.2.18 , ! " an bn " # $ 2.2.20 .

% ! " an bn & , $'# 2.2.20 . ( $ 2.2.182.2.2 $

	cos 2' = a0	1
4 sin2 ' = 2 1	cos 2' = a0	+ X an an cos n' + bn sin n' :
	2	n=1

+ ! ! " $ ", - , , $, $

a0 = 4 a2 = 2a 2 an = 0 n 6= 0n 6= 2bn = 0 n = 1 1: 2:2:21

( 2.2.21 2.2.18 $ 2.2.1 , 2.2.2 , 2.2.3 :

u r ' = 2 1 a 2r2 cos 2' :

2.2.1 , 2.2.2 , 2.2.4 ' 0' " ", ", un r ':

u r ' = X un r ' =	a0 + X r n an cos n' + bn sin n'	2:2:22
1	1
n=0	2 n=1

$ $ , ! 0 " & & " $0 $ $" r '. % ! ! " a0 = 2C0, an = AnDn, bn = BnDn ".

1 ,, $ 2.2.22 2.2.2 :

a0 1

g ' = + X a n an cos n' + bn sin n' :

2 n=1

2 ! 3 0 , ! "

	a	n 2			a	n 2
an =			Z g ' cos n' d'	bn =			Z g ' sin n' d' n =		:	2:2:23
								0 1
			0				0

u r ',2.2.22 , an bn 2.2.23 .

" an bn # , $2.2.23 . % 2.2.222.2.2

4 sin	2	' = 2 1	cos 2' =	a0	+ n=1 a	n	an cos n' + bn sin n' :
				2

* + - + + ,

a0 = 4 a2 = 2a2 an = 0 n 6= 0 n 6= 2 bn = 0 n = 1 1: 2:2:24

% 2.2.24 2.2.22 2.2.1 , 2.2.2 , 2.2.4

	u r ' = 2 1 a2r	2 cos 2' :
.	2.2.1 \| 2.2.3
	u r ' = 2 1 a 2r2 cos 2'/		2:2:25
	2.2.1 , 2.2.2 , 2.2.4
	u r ' = 2 1 a2r	2 cos 2' :	2:2:26
2.2.1.		3

4u = 0 D = fr ag De = fr ag.

1.	u r=a= sin ' + 2 cos ' D = fr ag:
2.	ur		=	2 cos 2' sin ' De = fr ag:
			r=a
3.	u	r	r=a=	2 cos 2' 3 sin 3' D = fr ag:
4.	ur u			= 3 cos ' + sin 2' De = fr ag:
				r=a

5.	ur + 3u			= sin ' + cos 2'				D = fr ag:
				r=a
6.	u		= 3 sin2 '		De = fr ag:
		r=a
7.	u		= 2 sin2 2'		D = fr ag:
		r=a
8.	ur		= 2 sin ' 3 cos 2'				De = fr ag:
			r=a
9.	u	r	r=a= 3 cos 2' 2 sin '				D = fr ag:
9.	u		r=a= 3 cos 2' 2 sin '				D = fr ag:
10.	ur 3u			= cos ' sin '				De = fr ag:
				r=a
11.	ur + 2u			= cos ' sin '				D = fr ag:
				r=a
12.	u		= 4 sin ' + cos 2'			De = fr ag:
		r=a
13.	u r=a= 3 cos 2' + sin '					D = fr ag:
14.	u		= sin2 ' De = fr ag:
		r=a
15.	u		= 2 cos2 '		D = fr ag:
		r=a
16.	ur u			= 3 cos ' sin '				De = fr ag:
			r=a
17.	ur + u			= sin 2' + cos '				D = fr ag:
			r=a
18.	u		= cos 3' + 2 sin '			De = fr ag:
		r=a
19.	u	r	r=a= 2 sin 2' 3 cos '				D = fr ag:
19.	u		r=a= 2 sin 2' 3 cos '				D = fr ag:
20.	ur		= 2 cos ' + sin 2'				De = fr ag:
			r=a
21.	ur r=a= 3 cos ' + 2 sin 2'						D = fr ag:
22.	ur 3u			= sin ' + cos 2'				De = fr ag:
				r=a
23.	ur + 2u			= sin 2' + cos '				D = fr ag:
				r=a

24.	u	= 2 cos2 ' De = fr ag:
		r=a
25.	u r=a= 3 sin 3' cos '				D = fr ag:
26.	ur r=a= 3 sin 2' cos '				De = fr ag:
27.	ur r=a= sin ' + cos ' D = fr ag:
28.	ur u r=a= sin ' cos 2'					De = fr ag:
29.	ur + u = cos 2' + sin '					D = fr ag:
		r=a
30.	u	= 2 sin2 2' De = fr ag:
		r=a
2.2.2. ! ""#
						2
		4u = 1r	@	r@u@r ! +			1	@'@ u2 = f r '	2:2:27
		4u = 1r	@r	r@u@r ! +			r2	@'@ u2 = f r '	2:2:27
" # # # % & % " # %
					u r=a= 0 :				2:2:28
1 & D = fr ag " (# % #&									#" -
* r ! 0 2.2.3
			f r ' = r2 cos 2'+						2:2:29

2 & D = fr ag " (# % & #" *

r ! 1 2.2.4
f r ' = r 2 cos 2':	2:2:30
.

.	%#, # " #,
*# "#("	% - . % #* # / * " #( " D
D " & % " # % 2.2.28 , # # , 1 " - -

. 2 "" .	3 , ( % " #,
*# "#("	% - . % #4 " #* # / * " , # # -

" # # # * % #4 ', % # *# "#(" % - . %

| 2.2.9 , 2.2.10 f n ' g, n = 0 1. 2.2.11 2.2.12 .

2.2.27 , 2.2.28

" #

$ An r

n r

0 1

u r

= A0 r +

An r cos n' + Bn r sin n'

2:2:31

n=1

) ) " #, " * *

) r '. -

f x y * * # , :

f r

= f0c r +

fnc r cos n' + fns r sin n'

2:2:32

n=1

" $

f0c r =

f r

fnc r =

f r

cos n' d'

2:2:33

fns r = 1

f r

sin n' d':

. 2.2.31 2.2.32 2.2.27 )

dA0 r 1

+ f0 r = +

dAn r

+ X

2 An r + fn r = cos n'+

2:2:34

n=1

+ 1

0rdBn r 1

n2 Bn r + fns r 9 sin n' = 0:

dr @

n=1

2.2.34 ) * # ,

+, , $

f g +:

dA0 r

1 =


1		d 0			dAn r 1			n2
r				@r			A r2 An r =
		dr				dr
1		d 0			dBn r 1			n2
			@r				A r2 Bn r =
r	dr					dr

fn r n =	1 1		2:2:35
c

fns r n = 1 1:

2.2.31 2.2.28! An r Bn r:

An a = 0 Bn a = 0 n = 0 1: 2:2:36

% ! & ' D ( - ' * ! 2.2.3

jAn r j M jBn r j M n =

0 1

r ! 0:

2:2:37

% , ! & ' D ( '- * 2.2.4

jAn r j M jBn r j M n = 0 1 r ! 1: 2:2:38

. & /01 2.2.35 , 2.2.36 , 2.2.37 2.2.38 3 & ( * ! An r Bn r,

2.2.31 , ! & .

%, ! ! & 4 fnc r fns r n = 0 1! , ' 3 2.2.33 .

5 , 2.2.27 , 2.2.28 , 2.2.29 . %- ( 2.2.32 2.2.29

r cos 2' = f0c r + X fnc r cos n' + fns r sin n':

n=1

8 4 -! ! ,

f2c r = r2 fnc r 0 n 6= 2 fns r 0 n = 1 1:

9 & , 01 2.2.35 , 01A2 r : 8 * , & 2.2.35 , 2.2.36 3, , ! & A2 r:

An r 0 n 6= 2 Bn r 0 n =

1 1

2:2:39

A2 r:

8 1	d	0rdA2 r 1			4	A2 r =	r2
					2			2:2:40
r dr @			dr	A	r
			jA2 r j M				r ! 0:	2:2:41
A2 a = 0
:				! 2.2.40 #

		A r = Cr2 + Dr 2:
% . 4 = 2 &4.4 .
					! 2.2.40 ' #( #

) '# *+ ,--) # '# % % A2 r

. *#( 1 2.2.40

A2 r = C2 r r2 + D2 r r 2

2 r D2 r + #' ' '# *

C20 r r2 + D20 r r 2 = 0

0		0	3	=		2	:
C2 r2r 2D2 r r				=		r	:
#' +
0	r			r2			~
C2 r =	4	C2 r =			8 + C2
0	r5			r6		~
D2 r =	4 D2 r =			24	+ D2:

& '# 2.2.43 2.2.42 ,

~	2	~	2	r4
A2 r = C2r		+ D2r		12	:

2:2:42

2:2:43

! 2.2.40

2:2:44

3 ' % 2.2.41 +
~	= 0	~	=	a2
D2	= 0	C2	=	12:

3# , % 2.2.40 , 2.2.41 #'

A2 r =	a2r2	r4	2:2:45
	12	:

2.2.39 , 2.2.45 2.2.31 -

	2.2.27 , 2.2.28 , 2.2.29 :
	u r ' =	1	a2r2	r4 cos 2':	2:2:46
	u r ' =		a2r2	r4 cos 2':	2:2:46
		12
& ' . () * 2.2.32					2.2.30

r2	cos 2' = f0c r +		fnc r cos n' + fns r sin n':
		n=1

- .// 0 ) ) 1 /- 0 2 2 ,


f2c r = r 2	fnc r 0 n 6= 2 fns r 0 n =			:
			1 1
3 ,	45 2.2.35 ), 45
2 A2 r : - ' , )		2.2.35 , 2.2.36 , 2.2.38
6 ) 2,		2 A2 r:

An r 0

n 6= 2

Bn r 0

n =

1 1

7 6

2 A2 r:

d 0

dA2 r 1

A2 r = r

A2 a = 0 jA2 r j M

r ! 1:

2:2:47

2:2:48

2:2:49

2.2.48

! "# $%% & ' & A2 r () " 1 2.2.48

A2 r = C2 r r2 + D2 r r 2

2:2:50

2 r D2 r # ! ! ' "

8 C20 r r2 + D20 r r 2 = 0

: C20 r2r 2D20 r r 3 = r2:

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