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