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2.

2.1. .

2.1.1. . ()

XI = (,2,),

*2 = 2(1,"2,"3), (1)

*3 = ("1."2,")-

, , 2, "

2, , \, 2, .

-. , , , - .

-.

1. :

X = , > = 5, Z = z ( [0,2], >0)

( xi, 2, , , z, , 2, , , z). z .

2. :

= , = rsinOsit^, z = rcos9 ( [0, ], [0,2], > 0).

2.1.2 . = (, , ) , F = F(i, 2, ) , F =

=

:

grady = = .,; (=1 ()

3

divF=(V,F)3 = Xa.^; i=i 2

dz 2 Fi F2 F3

rotF = [V, F] = ()

= divgrad = X =

.

, =

1 d / dv\ 1 2 ^

: V aJ + rW + az:

1 / 2 3v\ 1 / . 3v\ 1 32v

/ ~ (sin6 1 + ^55 r-.

(3)

V / /^sin G w

2 V / /^sinG V 0/ sin2 G

2.1.3 . v , . = (, , z), (), (, , z). (, , z) |, Q

= sJ(x-t,)2 + (y-r\)2 + (z-Qi.

, , , z ), , ,

2 2 2 , . . .

, (gjj) 5*

, :

(\

[rJp = Mcosa+ ^cosP + 9Ccosy'

cos a, cosP, cosy . -

III { + + f )dV = //(^osa + GcosP + ^cosy) V s

. P ud\v, Q u d2v, R u 83 V,

JJJ (grad , grad v) dV + JJJ uAvdV = JJ u-?dS. (4)

V V s

(4) v (4).

IIJ{uAV-VAu}dV = ll{ud?-V^)dS. (5)

V S

V, (5) v = 1 / . - , (5) v ,

5 v

. . = 4, V, . 2, dV, . = , A g V. , D (, ), L ( ). v, , :

d d l

ff{uAV-VAu)dS=j{ud?-Vd^dl, (8)

D L

? (!)) 4 ":= h/ 17 ? "'" 2nilAuln-ds' (9)

d

^

)2 )2

L, = + , = dq

.

2.1.4.

. , V, ,

= 0. 1
U.
S . . .

2.

5

3. SR R , . . , '.

uM=*bffuds-

Sr

4. ǰ : , , .

2.2. .

2.2.1. () . V , S. V (), V.

^p-dV (10)

v

, V . () - .

2.2.2. V () (10) x,y,z . ,

grad () = ֲ p(/>)grad(i) dV = - JJJ p(P) (-L) dV, (11) v v

r - -, r= rAP = (A: 1-F-(> Ҳ) j-F-( Q k, A A(x,y, z), P = P{%, 11, ). (1 /) = 0, A V,

() = JJJ() (-L) dV = 0, A$V.

v

, () , V, V.

, , V

u(A)^-rJJJp(P)dV = j, (12)

v <1> dz = JJJpdV .

, - - ( ) ( ), - , ( ) ( ), V. , () - 0 > -

(13)

.

2.2.3. . V R = const. , <, 9, ? = rsin9cos-, r>R,

= < [_(1]

, () '() > 0, "() r = R.

, , . = 4.

2.2.4. - . - .

1. gradw .

2. () 0 = sjx2 +y2 + z2 2! gradw| < .

, () , V,

v

.. . = (1/(4)).

2.3. .

. D , L. D () D.

() = JJp(P)ln-rdS (14)

d

.

() , ( ) (I/2) ( , z). 1(1/) .

. , D, () , . ,

gradA = () gradA In -dS=-ff () dS, (15)

d d

= ( - + ( - ]) j. ,

= II (/)(^-)<, D

, . , (. 2.2.2).

, , , , - , . .

() =(1/) + *(), u*(N)-+ ->-,

2! gradA *| < , . , In .

, (15) , D.

, D

= -2,

.

.

1. () grad . 2. ?>, L, D - .

() = /(4). 3.

= (P)dS

d

*() = () -/W 1(1/), \/2+2, *() 0 r2| gradA < , .

, () 1-3, D (), ..

() = II p(P)\n^dS.

d

2.3.3. . R: ?2 + 12 < R2 = const. KR

() = n/?2pln - = -,

= 2 KR. KR

, , , . (. 2.2.3).

2.4. .

2.4.1. . V , () V. , (. 2.9.1),

)

dV.

,-///!

v

, 5 (/>),

()

dS, (16)

S

5 , p(Z'). () S ; - .

(14) ( - ), 5 . () , 5 . , , .

2.4.2. . , .

, , - , S. , - ,

dz

<>

7'

. - - , \ {\ /+ \/_

(

4(). (17)

() = 2 { ( , , , . (17) :
(SW/PF^^M, <|8>
5 , -
= . , .
. ( )
J 42в = ^, R>R, U(R) = < R R
\ 4NRP=JF, R < R,
M 42 , .
, . , . . ,
'() - LIM '() -47.
-+ R-YR-0
. , L ( ) P(Z'). (14),
() = J P(P)LAJ-DL (20)
L
. L - , () . GRAD , Z = 0, , (1/2), Z, 2, L . , .
, L, (20) ,
. ,
GRAD = - J(P)^DL, = J{) (IN DL = 0.
L L
L , , . . .
, , . .
() = -+*(),

= J () DL * () 0 = \/2 + 2 | GRADA * \ <
< /2. [, ] = CONST. (20)

{,)= [ 1 [ IN{&-X)2+Y2}DT.
2.5. .
. , + , H (). , , . .
, , , 8 . H> 0 , , - EH V, ,
COS 8 (1/)
V, .
. . S, , V(P) . S , , . . . , S 5 , , . , .
(21) , .
. - S (1/2)H, H . (L//I)V(/>) , . , (1 /H)V(P), S. H S V
(22)
8 ( ).
, S, () , , Z . ,

, .
(22) : 21()| < , RI\GRADAU\ < , . , , /1, , /3. , 1_,() = 0.
, . V = CONST.

, 5 . COS8, .. - . , V 4NV ( ). , - , 2 (. . ).
, ,
U+(A)-U-(A)=ANV(A),

0()=1-{+() + .()},
. ,
+() = P(P)^^-DS + 2NP(A), (23)
S
,_() = II P(P)C-2^LDS-2NP(A). (24)
-
=, , +() , -() .
2.5.3. . , \ 2, , , , = V,
COS 6 1(1/) (A) = V=V-L/-. (25)
V. (25) , .
(25)
()= LV(P)^DL = LV(P)U\N)DL, (26)
L L L , a v(P) , L. (26) - , L v(/>). , , .

v = const (26) ,

F cos0

v dhp ,

l

, L . L. , L , , L,

/

cos9 >, ~

dl = ,

l

, L,

/

cos8 ,,

dl = 2nv,

l

, L, , 8 , . = \J2 + 2

MA)ll

vm |v(/')| L > 0 . (26)

. j . dl

| gradA | <2vmy -j < -j. L

(26) , L, , 2nv(A), v(A) . () , +() _() () ,

+() = () + rcv(A), _() = u0(A)-nv(A).

. , 2 ():

(sL-(sL

(. (17)), .

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