УрМатФиз / УрМатФиз с теорией

Rn r = Anr +1 + B r 2+1 :

! " ! 2.4.4 r ! 0 ! , " B = 0

n =

0 1: % & , " ' r

! 0

2.4.11 * +

R r = A r2+1 :

2:4:12

1

u r ' =

u r ' = A r2+1 sin 2n + 1'

2:4:13

=0

=0

1

g ' = A a2+1 sin 2n + 1':

2:4:14

=0

2

nk

Z

sin 2n + 1' sin 2k + 1' d' =

:

0

4

2

Aka2k+1:

Z

g ' sin 2k + 1' d' =

2:4:15

0

4

A0a = 1 A1a3 = 3 An = 0 # n 6= 01:

2:4:16

u r ' = 3a 3r3 sin 3' a 1r sin ':

2:4:17

18.

u' '=0= u '= =2= 0

ur r=a= cos 3' + cos ':

19.

u' '=0= u' '= = 0

ur r=a= cos ' 2 cos 2':

20.

u '=0= u' '= =3= 0

u r=a= 3 sin 9'=2 2 sin 3'=2 :

23.

u' '=0= u' '= =4= 0 ur r=a= 2 cos 8'

4 cos 4':

24.

u '=0= u' '= =2= 0

ur r=a= sin ' + 2 sin 3':

25.

u '=0= u '= = 0

u r=a= 2 sin 3' sin ':

26.

u' '=0= u '= =4= 0

u r=a= 3 cos 2'

cos 6':

27.

u' '=0= u' '= =3= 0

ur r=a= 2 cos 9'

4 cos 3':

28.

u '=0= u' '= = 0

u r=a= 5 sin'=2

2 sin 3'=2:

29.

u '=0= u '= =4= 0

u r=a= 2 sin 4'

sin 8':

30.

u' '=0= u '= =3= 0

ur r=a= 3 cos 3'=2

cos 9'=2:

u '=0= 0

@u

@' '= =2= 0

2:4:19

u

r=a= 0

2:4:20

)! % !$! * r ! 0 2.4.4 , $

f r ' = r2 sin 5':

2:4:21

u r

=

An r sin 2n + 1'

2:4:22

n=0

,

r '. -

" #$ # %$ f r

:

1

f r

=

fn r sin 2n + 1'

2:4:23

n=0

fnr =

4

:4:24

f r ' sinn

+ 1' d'

n =

0 1

:

0

1

81

d 0

dAn

r 1

n + 12

9

X

r

An r + fn r = sin

n + 1' = 0:

r

dr @

dr

A

r

2

n=1

:

:4:25

! "

#$% , ', ( )*

#$% . + " , $,## % ) f

) ':

1

d

0rdAn r 1

n + 12

r = f

r n =

:

:4:26

A

n

n

0 1

r

dr @

dr A

r2

$

An a = 0 jAn r j M r ! 0:

:4:27

, ( 2 $ 2 '

0

f2 r = r2 fn r 0

n 6= :2

5 , $)

n 6=2 '

) 1 :

An r 0

n 6= :2

:4:28

1 d 0 dA2 r 1

52

2

9

2:4:29

r

@r

dr A

r2 A2 r =

r

dr

=

A2 a = 0

jA2 r j M

r ! 0:

2:4:30

A2 r = C2 r r5 + D2 r r

5

2:4:31

C2 r D2 r ) ' #' * # # (

8 C20 r r5 + D20 r r 5 = 0

: 5C20 r r4

5D20 r r 6 = r2:

#+ )

0

r 2

r 1

~

C2 r =

10 C2 r = 10 + C2

0

r8

r9

~

D2 r =

10

D2 r =

90

+ D2

:

2.4.29 :

~

5

~

5

r4

A2 r = C2r

+ D2r

+

9 :

0* "# 2.4.30 )

~

~

a 1

D2

= 0

C2 =

9

:

A2 r =

r4

a 1r5

:

2:4:32

9

u r ' = 1 r4

a 1r5 sin 5':

2:4:33

9

. #

4u = f x y D = f r ' : 0 r a 0 ' =2g

@u

u '=0= 0

@' '= =2= 0

u r=2= g '

& f r ' = r2 sin 5' g ' = 3 sin 3' sin ' ' ' '

2.4.17 , 2.4.33 , ) 2.4.1 2.4.2:

2.

u '=0= u '= =3= ur r=a= 0

f r ' = r sin 6':

3.

u'

'=0

= u

'= =2

= u

r=a

= 0

f r ' = r3 cos 3':

4.

u'

'=0

= u'

'=

= u

r=a

= 0

f r ' = r3 cos 3':

5.

u '=0= u' '= =3= ur r=a= 0

f r ' = r sin 9'=2:

6.

u

'=0

= u

'= =2

= u

r=a

= 0

f r ' = r2 sin 6':

7.

u'

'=0

= u

'=

= ur

r=a

= 0

f r ' = r2 cos 3'=2:

8.

u'

'=0

= u'

'= =4

= u

r=a

= 0

f r ' = r4 cos 8':

9.

u

'=0

= u'

'= =2

= u

r=a

= 0

f r ' = r3 sin ':

10.

u '=0= u '= = ur r=a= 0

f r ' = r sin 4':

11.

u' '=0= u '= =4

= ur r=a= 0

f r ' = r cos 2':

12.

u'

'=0

= u'

'= =3

= u

r=a

= 0

f r ' = r3 cos 3':

13.

u '=0= u' '= = ur r=a= 0

f r ' = r sin '=2 :

14.

u

'=0

= u

'= =4

= ur

r=a

= 0

f r ' = r3 sin 4':

15.

u'

'=0

= u

'= =3

= u

r=a

= 0

f r ' = r4 cos 3'=2 :

16.

u

'=0

= u'

'= =4

= ur

r=a

= 0

f r ' = r3 sin 4':

17.

u

'=0

= u

'= =3

= u

r=a

= 0

f r ' = r2 sin 3':

18.

u' '=0= u '= =2

= ur r=a= 0

f r ' = r cos ':

19.

u'

'=0

= u'

'=

= ur

r=a

= 0

f r ' = r3 cos 3':

20.

u

'=0

= u'

'= =3

= u

r=a

= 0

f r ' = r2 sin 3'=2 :

21.

u

'=0

= u

'= =2

= ur

r=a

= 0

f r ' = r2 sin 2':

22.

u'

'=0

= u

'=

= u

r=a

= 0

f r ' = r4 cos '=2 :

23.

u' '=0= u' '= =4= ur r=a= 0 f r ' = r cos 4':

24.

u

'=0

= u'

'= =2

= ur

r=a

= 0

f r ' = r3 sin 3':

25.

u

'=0

= u

'=

= u

r=a

= 0

f

r ' = r4 sin ':

26.

u' '=0= u '= =4= u r=a= 0

f r ' = r cos 6' :

27.

u'

'=0

= u'

'= =3

= ur

r=a

= 0

f r ' = r3 cos 6':

28.

u

'=0

= u'

'=

= u

r=a

= 0

f r ' = r2 sin 5'=2:

29.

u

'=0

= u

'= =4

= u

r=a

= 0

f r ' = r2 sin 8':

30.

u'

'=0

= u

'= =3

= ur

r=a

= 0

f r ' = r4 cos 9'=2:

1

@

@u

1

@2u

@2u

4u = r

r

! +

+

= 0

2:5:1

@r

@r

r2

@'2

@z2

@u

3 z

@r

r=a= g1 z = 2 sin 2h !

2:5:2

1r

@u

u z=0= g2 r = 3J0

!

z=h= g3 z 0

2:5:3

a

@z

! % ) %$%!

r ! 0

ju r j 1

2:5:4

4u = 1r @r@

2

r@u@r ! + @@zu2 = 0:

2:5:5

4u = 0 D = f r ' z : r a 0 ' 2 0 z hg

2:5:6

@u

@r r=a= 0

2:5:7

1r

!

@u

u z=0= g2 r = 3J0 a

@z

z=h= g3 z 0:

2:5:8

II.

4u = 0 D = f r ' z : r a 0 ' 2 0 z hg

2:5:9

@u

3 z

@r

r=a= g1 z

= 2 sin 2h !

2:5:10

u

=

@u

= 0:

2:5:11

) , * + z=0

@z z=h

* uI r z

2.5.6 | 2.5.8 * uII r z

2.5.9 | 2.5.11 ,

:

u r z = R r Z z :

2:5:12

2

2.5.12 2.5.6 :

1

R r

+

R r

Z z

0 0

r

r rR r Z z + Z z R z = 0

R r

= Z z

:

2:5:13