1、 1 k j i a 15 20 3 fxyz (,) f grad) ( div a f 1 fxyz x y z (,) ( ) 22212 2 fxyz x y z (,)222 3 fxyz x y z (,) l n ( ) 222 1 ) ( ) ( grad232 2 2k j i z y x z y x f ) 15 20 3 ( ) ( ) ( div232 2 2z y x z y x f a 2 ) ( 2 grad k j i z y x f ) 15 20 3 ( 2 ) ( div z y x f a 3 ) ( ) ( 2 grad1 2 2 2k j i z y
2、 x z y x f ) 15 20 3 ( ) ( 2 ) ( div1 2 2 2z y x z y x f a2 k j i a2 2 2z y x xyz2221 ) 0 , 0 , 0 ( 1 :2 2 2. . . P z y x z y xFFP dxdy z dzdx y dydz x2 2 2 8 4) 1 (103202 2 2 8 F F FF FFP Pdr r d dxdy y x dxdy zxy 82 2 FF FFP Pdzdx y dydz x 832 2 2 FFPdxdy z dzdx y dydz x 3 k j i r z y x | |r r 1 0
3、 ) ( div r r f fr () 2 0 ) ( divgrad r f fr () 1 rxr f r fxx r f2) ( ) () ) ( ( II , ) ( ) () ) ( (2ryr f r fyy r f IIrzr f r fzz r f2) ( ) () ) ( ( II ) ( ) ( 3 ) ( div r f r r f r f r 1 0 ) ( div r r f 0 ) ( ) ( 3 r f r r f 3) (rcr f c 2 ) () (r frxxr f II ) () (r frxxr f II ) () (r frxxr f II ) (
4、 ) ( ) (2232 2r frxr frx rr frxx ()$%&II ) ( ) ( ) (2232 2r fryr fry rr fryy ()$%&II ) ( ) ( ) (2232 2r frzr frz rr frzz ()$%&II 2divgrad ( ) ( ) ( ) frf rfr r 0 ) ( divgrad r f 0 ) ( ) ( 2 r f r r f 12()cfrcr 2 1,c c4. 2 2 ) ln(21grad r c r c c k j i r z y x 0 2r c ) , , (3 2 1c c c c ) ln(21r c r
5、c 2 2 u ) ( 2,) ( 2,) ( 2332211r c r c r c 2 II2 II2 II cczu ccyu ccxu r ccc r c r c2 2 221) ln(21grad 5. ()$%& xyarctan grad a 1 () (), xyz 2212201z 2 xy z224 , z ()$%& xyarctan grad a221(, 0 yxxy ) 22222rot0yzyxxyxyIII III0ijka=x z 1 22(,)( 2 ) ( 2 ) 1 , 0 Lx y zx y z z 22(,)( 2 ) ( 2 ) 1 , 0 xyz
6、x y z P Stokes zP dr o tdL P2 2FF Fas aS0 2 2cos , 2sin , 0 xyz 88 22dLLxdy ydxx y 2 FFas202 d 8 F 6. ) ( k j i r xyz M(,) 132 k j i n 2 2 rot ( ) ( ) ( ) xzy yxz zyxxyzxyz xyz xyzIII IIIijkri j k ) ( rot M 4 3 ) ( k j i r xyz M(,) 132 n 2PFP I Pr r dmM) (1lim ) ( rot M 31 2nn 7. k j i az y xa a a f
7、xyz (,)aaaxy,zf 1 0 ) div(rot a 2 0 ) (grad rot f 3 a a a : ) (rot rot ) grad(div 1 rotyyxx zzaaaa aayz zx xyII &) & II II &)$ $ $II II II%(%( %ai)(j k z y xa a a , , 30 ) div(rot ()$%&II IIII ()$%&II IIII ()$%&II IIII yaxaz xazay zayaxxyz xyza 2 rot (grad ) fyzfffxyzIIIIIIIIIIII0 xijk 3 kajaiaaz y
8、x II II II div div div) grad(divj i()$%&I II II I II ()$%&I II I II II z yayay xaz xay xaxazyx zyx2222 2222k()$%&II I II I II 2222zaz yaz xazyx k j i a()$%&II II ()$%&II II ()$%&II II yaxaxazazayaxyz xyzrot = ) (rot rot a i()$%&I II II II I IIz xazayay xaz x xy222222k j()$%&I II II II I II ()$%&I II
9、 II II I II z yayaxaz xay xaxazaz yayz z x xy yz22222 2 222222 a k j i a a : : : : z y xa a a ) (rot rot ) grad(div 8. q ) (430k j i E z y xrq 1 rxyz 222 10 E rot 03 34 4 3 3 ()$%&II ()$%&IIryzryzryz rzy 33xzzr xrII &) &)$ $II%( %(44330zx zxrr 33yxxr yrII &) &)$ $II%( %(44330 xy xyrr rot ,(,) xyz E0
10、 0 49. a k j i r z y x 1 0 ) ( D 2 Q r a 2 a r a 2 ) ( D D Q 3 a r a r r 2 2 2 Q 2 ) ) ( 1 z y xa a az y xz y xIIIIII D 2 Q ) ( r a0) () () ( I I I I I I zx a y ayz a x axy a z ay xx zz y 2 ()yzzxxyxyzaz ay ax az ay axIIIQD D IIIijkar2( )xyzaaa ijka 2 3 22 2()() ()( ) ) 2yx zayax azxyzII IQ2 2 2IIIr
11、ra ra 10. () ()( xy z d xyx z d yzx y d222222 ) z k j i a ) 2 ( ) 2 ( ) 2 (2 2 2xy z xz y yz x yazxaxayzazaxyaxyz xyzII IIII IIII II2 , 2 , 2 k j i a ) 2 ( ) 2 ( ) 2 (2 2 2xy z xz y yz x (,)222(0,0,0)(,) ( 2 ) ( 2 ) ( 2 )xyzUxyz x y zd x y x zd y z x yd zC F222 2 2 20001(2)( )23xyzx dx y dy z xy dz
12、x y z xyz C FFF 11. ) 0 (2 2 2 2 xy xy xy xy xj i a 0 x()$%& II ()$%& II2 2 2 2 22 22 22y xy xx y xxy x yy xy xy ) ( a(,)22(1,0)()()(,) (,)xyx y dx x y dyVxy Uxy Cxy F2222101arctan ln( )2xydx x y ydy C x y Cxxy x FF 12. k j i a xy z y x zx z y x yz z y x ) 2 ( ) 2 ( ) 2 ( 5 k j i az y xa a a xaz x y
13、 yzazaz y x xyaz xyzII IIII II) ( 2 , ) ( 22 2 yay x z zxaxyII II) ( 22 a (,) UUxyz (2 ) ( 2 ) ( 2 ) dU x y z yzdx x y z zxdy x y z xydz ) ( ) ( 2 2 2 2zxdy xdz zdx y ydz zdy x yzdx ) ( 2 2xydz xdy ydx z ) ( ) ( ) ( ) (2 2 2z y x xyz d xyz d z xy d yz x d (,) (,) ( ) Vxyz Uxyz x y zx y z C 13. 1 uy
14、xy 3232R 2 2 2) ( ) ( ln b y a x u ) , ( 2b a R 3 2 2 21z y xu ) 0 , 0 , 0 ( 3R 1 yyuyxux yyuxyxu6 , 6 , 3 3 , 622222 2 II II II II 02222 II IIyuxu uy xy 3232R 2 2 2 2 2) ( ) (,) ( ) ( b y a xb yyub y a xa xxu II II 2 2 22 2222 2 22 222 ) ( ) () ( ) (, ) ( ) () ( ) (b y a xb y a xyub y a xa x b yxu
15、II II 02222 II IIyuxu 2 2) ( ) ( ln b y a x u ) , ( 2b a R 3 2 2 2z y x r 3 21rxrxr xu II 523 4 3 223131rxr rxrxr xu II 6231 uyyr rrI Iy2223431133uy yyrr rrrI I5y 231 uzzr rrI Iz2223431133uz zzrr rrrI I5z 0 3352 2 23 222222 II II IIrz y xr zuyuxu 2 2 21z y xu ) 0 , 0 , 0 ( 3R14. u x y (,) 2R u2R C 0
16、 IIFCdsnu nuII C C 2R ) , cos( ) , cos( yyuxxu u II II II ) , cos( ) , cos( xyuyxu II II n C ) , ( y xF FII II IIC Cdxyudyxudsu green FF F()$%&II II IID Cdxdyyuxudsnu22220 ) , (0 0 0y x M 0) , ( ) , (20 0220 02 II IIyy x uxy x u u x y (,) 0 3 ) , (03 M O D 2222yuxuII II0 FF F()$%&II II IID Cdxdyyuxu
17、dsnu22220 u15. uu x y (,) vv x y (,) 2 2v u F p . 2 F 0:() Fp. 0 ) (2 1x xp x x ppvv uu pFFvv uupFxF II 7 12()pyy ppyyuu vvFpFp F u uyF IIvv ) ( ) ( ) 2 () (2 2 2 2 422xx xx x xpx xppvv uu v u pF vv uu F p pxF II ) ( ) ( ) 2 () (2 2 2 2 422yy yy y ypy yppvv uu v u pF vv uu F p pyF II : ) (pF ) ( ) (
18、 ) ( ) 2 (2 2 2 2 2 2 2 4y y x xpy y x xpv u v u pF vv uu vv uu F p p 0 .16. 1 | ) , , (2 2 2 z y x z y x B ) , , ( z y x F3 3R R ) 0 , 0 , 0 ( 6IBF 0 6 2 QBF 3R ) , , ( z y x g0 2 QFFFdxdydz gBF F F F 2 Q 2 Q 2 Q g g g ) ( Gauss 2 QFFFdxdydz gB 2 QFFFdxdydz gB ( dxdydz gFFF2 QB d g 2 FFIB dxdydz gF
19、FF2 Q B 0 ) 0 , 0 , 0 ( 6IBF 0 6 2 QBF 17. = D 1 | ) , (2 2 2 y x y x R ) , ( y x u D u D D D I Duyuxu4 II II2222 4 0 grad22 FF FFD Ddxdy u dxdy u 4 green Duuud xud yyxIIIIIFdxdyyuxuuyuxuDFF!#+,-II II II II ) ( ) ( ) (22222 2 D I ) , ( y x uDuuud xud yyxIIIIIF0 D 8 uyuxu4 II II2222 0 ) ( ) (2 2 2 !#
20、+,- II IIFFdxdy uyuxuD4 0 grad22 FF FFD Ddxdy u dxdy u 4 18. J P ) , , ( z y x u J J 0222222 II II IIzuyuxu 1 0 IIFFPdSnu n P J ) , , (0 0 0z y xFFP()$%&II dSnur ru z y x u1 ) , cos(41) , , (20 0 0n r ) , , (0 0 0z z y y x x r | |r r 1 n ) cos , cos , (cos Gauss FF FFP PII II II IIdSzuyuxudSnu) cos
21、cos cos ( 0 ) (222222 II II II FFFJdxdydzzuyuxu 2 r 2 ) cos( 2 II) grad ( unu ()$%&II FFPdSnur ru1 ) , cos(412n r FFP Rdxdy Qdzdx Pdydz 41 320) (ru r u x xPx 320) (ru r u y yQy 320) (ru r u z zRz ruurx xruxPxx II5203) (3 ruury yruyQ yy II5203) (3 9ruurz zruzRzz II5203) (3 0 II II IIzRyQxP ) , , (0 0
22、 0z y x 0 3 0SJ 0S 0S P Gauss 0S J02()1c o s ( , ) 11()44SuP Q Rud Sd xrr n xyz P JII I I &) $II I I%(FF FFFrn0 dydz 21c o s ( , ) 14uud Srr n PI &) $I%(FFrn021c o s ( , ) 14Suudrr n I &)$I%(FFrnS 3 r cos( ) 1 00 IIFFdSnuS 21c o s ( , ) 14uud Srr n PI &) $I%(FFrnFF 0) , , (412SdS z y x u3 0 3 FFP()$%&II dSnur ru z y x u1 ) , cos(41) , , (20 0 0n r 10
