.


:




:

































 

 

 

 


g cos 9




1

 

11.

¥ ( 41), , , ; . ¥ , , . (.

()

F . ¥, 10

] F dV. , ¥ , , , 1).

¥ , .

J) , F £2/8 , , , F (£2/8) .

6/=' = -^-E6D,

, = A- J ( 6D ) dV.

61=iI<D-6->?^+iI<6D-6')dv-iI(-)dv-

(11,2)

6* = !0 (0 :

J grad (D dV = J (d<g) df J div (d) dV.

, . divD = 0 div = 0, . . , , , 0 = const,

(D )df = 0^ (DG) df.

(, , , D (. . ).

&Dndf (f)($:ndf 4, .

, (11,2) ( =■= grad ).

==_ j_J(dE)6edK=jp6edv. (,3)

, , , = 0.

, , 6 , , (10,3), (10,4). , , . , -


, , .

@ ,

6<F = -6e-$Pdi/= ^, (11,4)

.

d¥ = <ydT <p>d. (11,5)

, , :

 

 

,

 

\ ] '

Ɨ- , " - , , (. V (11,4), (15,11)). , ?, , , 05 , .

D D = eE, 6<F, <F.

 

 

.

=i1<+)<D-)dyI (~).

, , E-f-0? = grad ( + 0) , .

F-f0(V, T) = -\^WdV. (11,7)

,

f-f0(l/, ) = -&&. (11,8)

(11,3), , ( D = eE) .

*/ = V,**, (11,9)

, ( 2). , , , . <xik ( 2) (11,6); ,

2< /

 

.

(11,7) , 1, . . = (1)/4 . , . .

= ? .

IF <F0= - J(S2dV, (11,10)

. - t^, = Vx6-,

f-f0 = -^g2. (11,11)

D (11,7) (11,8) . W


8 2

^=KF-i£)dv==S[F-^-im


dV, (11,12)


. ,

ed 2 1 /., >.

ST-------- +8-=-&-(-)(+^'

= V<P, = V<P0 . , (11,12) ( (11,7)) ( ) ( = 0, F = £2/8), .

 

, (11,7), , , (<?>.

. ,

¥ ~¥ = - J6-(D-e)E) dV.

 

12.

, , ( 15, 16) , . () .

, , , . 5, , . . .

(, ), , . . , =1. , . . ¥ ,

 

 

d¥ PdV- , (11,5)

d¥ = dT - dV - & .

:


= ¥ + PV.


(12,1)


( )

1 = ^ dT + VdP-tfbdV. (12,2)

. (. V (15,12)), ( V) ( ) . (11,8)

= 0-1/&&> (12,3)

. gb , ( 3~0 (11,8) V ).

V & (11,9), (12,3)

= 9(, )-*1%%, (12,4)

. , :

= 0(, )-^-& (12,5)

(. (11,11)).

V V0 ( ). , (12,5)

 

, ( , ).

Q, ( )J).

-1) , , = Q:1$p, % ,


0 , , . , (12,5)

(12,7)

Q .

 

3

1. , .

. (12,3) (8,10)

 

8 + 1 -


 
V
8

): - (J2 -1

(+1 ) (+1)2


 

) 1 )\ '


:


-)2
8

Q =

ne + 1 (+:

dV_ Jp
- , =
=

V I )~ .

, \,


-1

iZp_=JL2

? 8.

8

( 1). 1



( ) = 0


 

' 8


Q =


ù2

8


i ( 1)



■]■


1) >-, - (V V0)/V = 2/8>. = 1/3 . 4 5. 2) , , . . . -

2. ??<p ; 2).

. 1

 

 

: D = G. <ff <$,. = £/=(//, / . / V1^3. ^

Qf (JK^Ve. :

3. ,
.

. , ( ), . ').

(10,18)

 

 

.

 

4. ^D (. 2)
.

. ( ) .

3

_ TVE* ( _ TV& (

 

5. ,
h , ;
-
. n^ftc -
2. .

. , . Jp'. jF = (|2;2, 0 .

') (£2), . -

dS

, J SpdV, \ bpdV = 0 .


, = //, . . , 11, (8,2)

# "~ 2 Lo<P 2n2 26l + g2 '

_, 3 ]) -<" 21 + 2

13.

() , .

+(13,1)

D , e!k (, , ). D0 (13,1) , , . (. ), ,

Dt = &tkEk. (13,2)

zik :

£ik==eki- (13,3)

, (10,10) ,

*;

dEkdE; - dEk ~£ik

.

F ( (13,2))

= -^*' (13'4)

F

, eik . , , Eik (1), (2), (3). , > 1 (. 14).

eik .

, ; 1). 1 - . . eik.

, , , ; . , , .

, eik , . , ( 1, . . . , .

&ik , , . . , , ; , , , -

) . 98, 99.

, . .

D0 (13,1). , ; . , , ( ). , D , . D0 D , (13,1).

,

dF

 

^^-^-^- (13'6)

F = F + = F + = F + s^ (Di-Doi) (Dk-Dok).

(13,7)

, F F ,

(11,12), (13,7) (13,1). , * = 0, :

r= l(F(13.8)

, ( ) , .

, . -

J) , (. . , . 17). , , , .

, , , , ( ) ; d0.

, . , , . 32- , :

: ,,

: Cs, ,

: C2v,

: 4, Civ,

: Cs, C3v,

: 6, C6v. ' , , . 1 d0 - , Cs . d0 *).

*) , . ( ) , . , , . D0 .

, , . , . , , , , , , . , . , .

1. , .

. , 2 =D ( (8,1) ($ = 0, . . ). (13,1), 2£, + e,'^£fe = Doi. 1.

= Dpi. Di-Ej _ 3D0i

' 2-fe'>' 1 4 4(2 + <'>) "

<p = PV.

2. ).

. div D=4ne () ( ). D,' = e,-^Efe = = ,-£ dip/; , , ,-^,

1- 1 - 1 2 w

* = *' ~, = ' yeW\ z = z' Ӹ& (1)

= (),

<*><.><>

' = [e<x)eWe<z>]~'/2.

7

_1_ _1_

<*> ' eW ' <>

, ,

=

| eik xlxh

|| ,-£.

3. ( ), .

. (1) , ',

eikx'ixk = <*>*'2 + "/'2-f e<z'2 = 2.

(4,14) , :

7.



e(.v) / V eW 1 V 1 <>


) 2 6 .

4. , -

. ,

= @ + An,

, . : nD = n(S;

nflikEk = & + 1 = {.

 

11

, ( ),

=(1-)/. :

= :/.

5. ,
, ( ) (v.

. (8,2)

5_

A'~2+e(*>U*

( , 2), , , ,^. ( )

4 _ <*> 1., ^ = _,= __5,.

 

 

, -

6. .
£(> .

. (1) 2 . , , 1/<*>, 1/", 1/(>. , ( ) /}/"<*>, a/]AeW, /^ <*>, {\ *> ( (4,25)). (8,7),

1, <<'> '*> ('">_ dcp(g) ( ' ' ■ <*> ' ~~ ' ( ' '). ,

 


() () <)

() n(X) (g(X) _l)


14.

.

( ) (10,20)


 
 

.


 
 

(10,2) , ( ). ; ,


6^=-J^8>= 46e-I>-


(,1)


, ( ), . , , , ( ) . 2 .

- ( ). , , , , , ¥ . , , .

(14,1) , . . , . 2 - ¥ ¥ , 2. , , , , .

(14,1) , 1). , .

__- _j7____ 1 ' I Vnm \2(Wm W) \_[V _y.

n m n in

(14,2)

(. V (32,6)). E^ , ,

 

 

V' (14,2), , . ; .

, (14,2) , ¥ 11 , . ¥ ¥0 (11,7). , . , = (1)($/4,

*) , . , , .

 

 

, ¥¥ , > 1. 7 ,


, . . =(1)/4 .

{ > 1 eik . , .

15.

( ), , ( ) . .

f dV , dV, f .

, , - , , (. VII 2). . , dV, . , . aik,

(15,1)

V. a;k . ,

Gikdfk = e;knkdf

1- , df ( , ).

, , . , (oik = aki); , , .

(15,1) ,

, ,

 

 

, .

. -; . ) , ( , ) ( h), 1). ( ).

, () |; 1 . , ( ) , .

() oiknk. cr(-fenfe£,-. , ( - , \FdV ( ) hF. ,

eik$ink = 8(hF) = h8F + F6h. (15,3)

( ) ; , ' ( ), . 8F

tf-(#)r..+(#)..,*^+(1)..,*-)

*) , , . . , , - , , , , D .

= p8h/h. .

( - ) , . ( ) , ^

= () () = ? = ,

.

= |1, (15,5)

z .

6E = -in(El). (15,6)

(15,4) , bh = \z = \n,

oikhnk = (nD) (IE)-(in) U+ (in) F =

 

\ 4

:


0,- =


^ /,


,* + ^*. (15,7)


, , D . EiDk = EkD[ (15,7), , '). D = eE

F=F0(p,T)-e-g (15,8)

(. (10,17)); F0 . , 1 :

flFf) -'-■>(£),--'*

) , , , , , .


0 = 0(, ) , . (15,8) (15,7)

(, T)6lh-£[s-9 (|)r]'6,.fc + ^. (15,9)

1).

, , : aiknk = o'tkn'k, . ' ,

= 1. (15,10)

. , (15.7) (15.10) ,

EtDn = E'tDn.

Et £). , .

, , ( =1). , , atk (15,9),

-/,.(.+£(^) + (-£?) = -.+^(^-^).

Et E't, Dn = = D'n = E'n,

.(. )- = &(%)--£ + ). (15,11)

, .

de-dp

£2 8 [

4

*) . . 49.

. (15,2) (15,9),


div D = dDk/dxk = 0


F . > (*


, rot = 0. ,


f =
£2
--grade (15,12)

-gradP0(p, 7") + -^ grad

(. Helmholtz, 1881).

1, f div D/4n; divD = 4npCT,


">


(15,13)


, , , (. 3 16).

, 7, 1 . pde/dp = e 1 (15,12) :


f = -


VP0 + Vgrad£?- (15'14>


(15,12) , . , . , , grade

T-(jr).r+Wr

(15,12) :

f = -V/>.<P, )+{-

, , 0 pV£0 ( : pd£0 = dP0S0dT)


f =


8 \)


= VC.


(15,16)


£ (. (10,19)).

, f = 0 :

C= £.-S(g)r=const (15,17)

. . '1. , , (15,15) = const; , D = , . , f (15,15),

*'<P'r>-ir(l)r==Const' (158>

(15,17) , £0 0/.

, (15,12) (14,1), .

, , . : 1)

() () = ,

, , 2)

-(!)rpdivu'

: (. VII 1), div = pdivu. , :

6< = 60 J 8e^dl/ = \ P0divudV +

+ J£ [* + (|) pdivu] dV (15,19)

( ). (15,19) div

8<F = \uidV f, (15,12).


16.

 

( ), , (). , . , , ; , 1.

, ,


 
 

(, , ) . sik u!k.

(16,1)

0 , ( 2) , uik.

, , . F , (15,4)



dF


-ft-


 

(15,5),

 

8F uik ( dF/dulk),

 

4 . . , . .

, (15,7) 1):

 

 

(16,3) D . - ( -) , Dt=zikEk, F (13,4),

■ik=lk~^{aiEiEk+atE'6ik)-

(16,3) e,-fe = e06,-fe :

_((0) I 20"1 Eo + g2 F2fi. /1 4\

 

%] , .

) F , , , . , , : , - , . , Ujk ( (16,3) ; ). (16,3) dF/diiik, . 2) 17, . . .

2). , , . , . F, (16,2). , 8< 6<; .




 

, (15,6), , :

= -|(1)-[.].

= 1/2 rt ( , , = = [ - ]). (15,5),

6tp=^[vzi]=^[ni],

6 = -1 n (Ei) +1 [ [] = - i {n (1) +1 ()}. (16,2)

-5 D = nD) > + <) (")} = |f -^.

, (16,3) EtDk :

=f + £+h ^+<16-5)

, , , i /.

, (16,1)

tik = z\il + aiklmiilm, (16,6)

aiklm , i, k 1, ( i,k 1,). , .

( (16,4)), (16,6).

. , , , F , . , ( ) .


(f)aiknkdf, .

eiknk , aik (16,4), (15,9), . , . , (15,18). (15,9) ( ) , F - , , aik

/*=^(-6'*)' (16'7)

, - . :

JE(nE)-l£*njdf, (16,8)

K-4^^{[rE](nE)-i^[rnl|df. (16,9)

, , , (, , , ).

, ( ), , , , ; , . , , , . , , .

(11,3)

6 ==* $P6dl/,

- = (+) () = (uv)

? . u= const rot 05 = 0,

(u v) = (pv) () = (pv) ,

6< = l(Py)(SdV.

, 6<F = uF, *):

F= J(pv) 6-. (16,10)

, . , :

=S [+l [r-(pv)]dv. (16,)

, , (16,10)

f = ($pdvv)<S=(^v)e, (16,12)

, , , ¥ (11,8). (16,11) :

(16,13)

1. ( ), , , . .

. (16,8) ( =1), , . (8,2) £(') 3{ff./(2-|-e) ( ).

*) , , . , (?, , , , , .

E = DU = ^-e.

Er = Dri] = cos 0, = £</> = g sin 9,

0 - 05-

, 1)

 

16 ( + 2)-

2. -
.

4 5. 2). , 4 5.

1^ V_ '*> 1. 2

2J^~ 8 1+((*> 1) te

(. (8,9)), (16,1)

= 0 + 1





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==> ...

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