HaiJ.mu ee7l:mOpHble JlUHUU nJloc1I:ux ee1l:mOpHblX nOJleiJ.:
5.1.18. F = x ·i+y .j. 5.1.19. F = y. i+x .j.
5.1.20.F = (y, -x).
HaiJ.mu ee1l:mOpHble JlUHUU npOCmpaHCmeeHHblX ee1l:mOpHblX nOJleiJ.:
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5.1.21. |
F = a . i + b . j |
+ c· k, r,n,e a, b, c - KOHCTaHTbI. |
5.1.22. |
F = (y + z) . i |
+ (x + z) .j + (x + y) . k. |
5.1.23. |
F( -y, x, a), r,n,e a - |
KOHCTaHTa. |
5.1.24. |
,ll;oKa3aTb, qTO grad (u· v) = U·grad v + v· gradu. |
a ITo OIIpe,n,e.nemIJO rpa,n,HeHTa CKaJUlpHoro IIOJUI HMeeM: |
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grad ('1.1' |
v) = ((uv)~, (uv)~, (uv)~). |
IIcIIOJIb3YH IIpaBHJIO ,n,H<p<pepeHlJ,HpOBaHHH "pOH3Be,n,eHHH <PYHKIJ;Hit, a TaIOKe IIpaBHJIa CJIOlKeHHH BeKTopOB H YMHOlKeHHH HX Ha qHCJIO B Koop,n,HHaTHoit <popMe, IIOJIyqaeM:
grad (u· v) = (u~v + uv~,u~v + uv~,u~v + uv~) =
= |
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uxv, uyv, uzv |
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uVx' uvy, uVz |
= v· ux, uY' Uz |
U· vx, vy, vz = |
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5.1.25. |
,ll;oKIDKHTe CBoitcTBa rpa,D,HeHTa CKaJIHPHOro IIOJIH: |
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a) grad (U + C) = grad U, r,n,e C - KOHCTaHTaj |
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6) grad (C· U) = C· grad U,r,n,e C - |
KOHCTaHTaj |
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B) grad (*) = 12 (v· gradu - U· grad v). |
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5.1.26. |
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v |
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= cp' (u) . grad u. |
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,ll;oKIDKHTe paBeHCTBO grad (cp(u)) |
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5.1.27. |
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HaitTH grad (u· cp(u)). |
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5.1.28. ,ll;oKa3aTb, qTO gradf(u,v) = au . gradu + av . gradv. |
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B cJleaY?O!4UX 3aaa"l.aX r |
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= x . i + y . j |
+ z . k |
u r |
= Irl. |
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5.1.29. |
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BbIqHCJIHTb grad r. |
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a Hait,n,eM .D.JIHHY |
BeKTopa r: r = Irl = J |
x 2 + y2 + Z2. |
,ll;aJIee |
Haxo,n,HM |
qacTHbIe "POH3Bo,n,HbIe <PYHKIJ;HH r: |
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- J x2 + y2 + z2 |
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Jx2 + y2 |
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CJIe,n,OBaTeJIbHO, rpa,n,HeHTOM CKaJIHpHOrO IIOJIH r |
6y,n,eT BeKTopHoe IIOJIe |
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gradr = (~,~,~) = ¥(x,y, z) = ~. |
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5.1.30. |
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,ll;oKMaTb, qTO gradr2 = 2r. |
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5.1.31. |
HathH gradf(r). |
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5.1.32. |
HaitTH grad(c· r), r)l,e c - |
<pHKcHpoBaHHbIit BeKTOp. |
5.1.33. |
HaitTH npOH3BO)l,HYIO nOJUI U = r B HanpaBJIeHHH BeKTopa r. |
a KaK |
H3BeCTHO, npOH3BO)l,HaJI no |
HanpaBJIeHHIO e)l,HHHqHOrO BeKTopa t |
<PYHKIJ;HH U =U(x, y, z) MO>KeT 6bITb Hait)l,eHa no <popMYJIe: ~~ =grad U ·t.
HaxO)l,HM rpa)l,HeHT CKamlpHoro nOJIH U: grad U = grad r = ~ (CM. 3a)l,a-
qy 5.1.29). E)l,HHHqHbIM BeKTopOM, HMeIOIIJ;HM TO :>Ke HanpaBJIeHHe, qTO H
BeKTOp r, 6Y)l,eT BeKTOp t = 1;1 |
= ~. Tor)l,a |
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= 1. |
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5.1.34. HaitTH npOH3BO)l,HYIO nOJIH U = } B HanpaBJIeHHH rpa)l,HeHTa CKaJIHpHOrO nOJIH v = r.
5.1.35. HaitTH nOBepXHOCTH YPOBHH CKaJIHpHOrO nOJIH
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+ 2"' |
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5.1.36. |
HaitTH nOBepXHOCTH ypOBHH CKaJIHPHOI'OnOJIH U = eX2 +v2 . |
5.1.37. |
IIo~a3aTb, qTO )l,JIH CKaJIHpHOrO nOJIH U = x 2 +y2 +Z2 BeKTopHbIe |
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nOJIH grad U H grad Igrad UI KOJIJIHHeapHbI. |
5.1.38. |
HaitTH yrOJI Me:>K)l.y rpa)l,HeHTaMH CKaJIHPHbIX nOJIeit U = xyz H |
V = yz + zx + xy B TOqKe Mo(l, -1,2).
5.1.39.B KaKHX TOqKax npOCTpaHCTBa rpa)l,HeHTbI CKaJIHPHbIX nOJIeit U =
=x 2 + y2 + Z2 H V = x 2 _ y2 + Z2 nepneH)l,HKYJIHPHbI?
5.1.40. HaitTH nOBepXHOCTH ypOBHH CKaJIHpHOrO nOJIH 1grad UI, r)l,e CKa-
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JIHpHOe nOJIe U 3a)l,aHO paBeHCTBOM U = xy + yz + ZX . |
5.1.41. |
.II:oKa3aTb, qTO JIHHHH YPOBHH nJIOCKHX CKaJIHPHbIX nOJIeit U = xy |
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H V = x 2 - y2 nepneH)l,HKYJIHPHbI B Ka:>K)l,Oit TOqKe nJIOCKOCTH, |
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KpOMe HaqaJIa KOOp)l,HHaT. |
5.1.42. |
HaitTH BeKTopHbIe JIHHHH nOJIH F = yz . i + xz . j + xy . k. |
5.1.43. |
HaitTH BeKTopHbIe JIHHHH nOJIH F = r, r)l,e r = x . i + y . j + z . k. |
5.1.44. |
HaitTH r x grad r, r)l,e r = Irl, a r = x . i + y . j + z . k. |
KOHTponbHble BonpOCbl III 60nee CnO)l(Hbie saACIHIIIR
5.1.45. MorYT JIH pa3HbIe CKaJIHpHbIe nOJIH 06JIa)l,aTb O)l,HHM H TeM :>Ke Ha60pOM nOBepxHocTeit YPOBHH?
5.1.46. BepHo JIH, qTO eCJIH nOBepxHocTH ypOBHH Y CKaJIHPHbIX nOJIeit U
H V O)l,HHaKOBbI, TO 9TH nOJIH Y)l,OBJIeTBOpHIOT yCJIOBHIO
U - V = const?
5.1.47. MorYT JIH pa3Hble rrOBepXHOCTH ypOBHg CKaJIgpHOrO rrOJIg U rrepe-: ceKaTbCg?
5.1.48. MO)KeT JIH Y pa3HbIX BeKTopHbIX rrOJIeit 6bITb O,!l;HH H TOT)Ke Ha60p BeKTOpHbIX JIHHHit?
5.1.49. TIpHBecTH rrpHMep ,!l;BYX rrpOCTpaHCTBeHHbIX CKaJIgpHbIX rroJIeit, Y KOTOPbIX rrOBepXHOCTH ypOBHg OPTOI'OHaJIbHbIB K~,!l;Oit TO'IKe rrpocTpaHcTBa.
5.1.50. B,!l;OJIb KaKHX JIHHHit rpa,!l;HeHT CKaJIgpHOrO rrOJIg U = xy +yz + zx
COXpaHgeT CBoe HarrpaBJIeHHe?
5.1.51. HaitTH CHJIOBble JIHHHH BeKTopHoro rrOJIg
F{nz - ly, lx - mz, my - nx).
5.1.52. HaitTH rrpOH3BO,!l;HYIO CKaJIgpHOrO rrOJIg U B HarrpaBJIeHHH rpa,!l;H- eHTa CKaJIgpHOrO rrOJIg V.
5.1.53. KaKoBa CBg3b Me)K,!l;y rrOBepXHOCTgMH ypOBHg CKaJIgpHOrO rrOJIg U
H BeKTopHbIMH JIHHHgMH grad U.
5.1.54.BepHo JIH, 'ITOeCJIH JIHHHg, ypaBHeHHe KOTOPOit x 2+y2 = 1, gBJIg-
eTCg JIHHHeit YPOBHg HeKOToporo CKaJIgpHOrO rrOJIg U, TO JIHHHg x 2 + y2 = 2 TO)Ke gBJIgeTCg JIHHHeit ypOBHg Toro )Ke CKaJIgpHOrO
rroJIg?
§2. AViBEPrEHLI,VlH VI POTOP BEKTOPHOrO nOJlH. OnEPATOP rAMVlJlbTOHA
AIilBepreHu,IIIH III POTOP
~,Lf'U6epeeH:4'Ueii (pacxoiJ'U.MOCm'b1O) BeKTopHoro nOJUI F(P, Q, R) Ha3b1BaeTCH
CK3J1HpHOe nOJIe, onpe.n;eJIHeMOe paBeHCTBOM
divF = ~~ + ~~ + ~~ = P~ + Q~+ R~.
~Pomopo.M, BeKTopHoro nOJIH F(P, Q, R) Ha3bIBaeTCH BeKTopHoe nOJIe, onpe.n;e-
JIHeMOe CJIe.n;yIOIIIHM 06pa30M:
)JjrH y.n;o6CTBa 3anOMHHaHHH pOTopa npHHHTa <pOPM3J1bHaH 3arrHCb:
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{Jz ' |
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r.n;e «YMHOlKeHHe» CHMBOJIOB .n;H<p<pepeHIJ;HpOBaHHH Ha O.n;HY H3 <PYHKIJ;HtI: nOHHMaeTCH KaK B3HTHe COOTBeTCTBYIOIIIetl: '1aCTHotl:npOH3Bo.n;Hotl: 3TOtl: <PYHKIJ;HH.
CPU3U'I,eC~ui:t CM'bICJI pomopa2: eCJIH BeKT0P-<PYHK:U;HH v HBJIHeTCH nOJIeM CKOPoCTet!: TBep,!I;OrO TeJIa, Bparn;aIOrn;erOCH BOKpyr HenO,!l;BHlKHOt!: TO'lKH,TO C TO'lHOCTbIO AO '1HCJIOBOrO MHOlKHTeJIH POTOP BeKTOpHOrO nOJIH V npe,!l;CTaBJIHeT co6ot!: MrHoBeHHYIO yrJIOBYIO CKOPOCTb W 3TOrO Bparn;eHHH: W = ~rotv.
POTOP BeKTOpHOrO nOJIH Ha3bIBaIOT HHOr,!l;a 6UXpeM BeKTOpHOrO nOJIH.
OnepaTOp raMMnbTOHa
~Onepamop raMUJI'bmO'Ha HJIH OnepaTOp V ('Ha6J1a) Onpe,!l;eJIHeTCH <P0PMYJIOt!:
TIpHMeHeHHe 3TOrO OnepaTOpa K CKaJIHPHblM H BeKTOpHblM nOJIHM C <P0PMaJIb-
HOt!: TO'lKH3peHHH COOTBeTcTByeT Onepa:U;HH «YMHOlKeHHH» Ha BeKTOp C KOOp,!l;HHa-
a a a
TaMH ax' oy' oz:
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VxF= (;x';Y';z) x (P,Q,R) = ax |
oy |
oz |
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HeTPY,!I;HO 3aMeTHTb, 'ITO CTOHrn;He B npaBbIX '1aCTHX nOCJIe,!l;HHX Tpex paBeHCTB BbIpalKeHHH CYTb rpa,!l;HeHT, ,!I;HBepreH:U;HH H POTOP nOJIet!::
VU=gradU, V·F=divF, VxF=rotF.
OnepaTOp llannaca
~Onepamop JIanJiaca (o603Ha'iaeMblt!:V2 = V . V HJIH ~) Onpe,!l;eJIHeTCH <Pop-
MYJIOt!:
V2 = ~+ ~+~.
ox2 oy2 OZ2
TIpHMeHeHHe 3Toro onepaTopa K CKaJIHPHbIM H BeKTopHbIM nOJIHM Onpe,!l;eJIHeTCH paBeHCTBaMH:
~U == V 2U = (~ + ~ + ~) U = 02U |
+ 02U + 02U, |
ox2 oy2 OZ2 |
ox2 |
oy2 OZ2 |
~F == v2F = V 2 (Pi + Qj + Rk) = (V2P)i + (V2Q)j + (V2R)k.
~ CKaJIHpHOe nOJIe U(x, y, z) Ha3bIBaeTCH 2apMO'HU'I,eC~UM, eCJIH ~U == O. $
20 <pH3H'IeCKOMCMbICJIe ,n;HBepreHIJ;HH 6y,n;eT CKa3aHO B CJIeA)'lOrn;eMnaparpaq,e.
5.2.1. BbI'IHCJIHTb)l,HBepreHlJ;HIO H pOTOp BeKTOpHOI'Onoml
F = xy2 . i - yz· j + z2 . k.
Q ITo onpe)l,eJIeHHIO, div F = P~ + Q~ + R~. B HameM CJIY'IaeP = xy2, Q = -yz, R = Z2. OTCIO)l,a HaxO)l,HM P~ = y2, Q~ = -z, R~ = 2z. CJIe)l,OBa-
TeJIbHO, divF = y2 - |
Z + 2z = y2 + Z. |
BbI'IHCJIHMPOTOP nOJIg F: |
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ty |
tz = (R~ - Q~)i+ (P~ - R~)j + (Q~ - P;)k = |
rotF = ax |
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= (0 + y)i + (0 - O)j + (0 - 2xy)k = (y, 0, -2xy). •
BW"{UC./I,Umb ouaepae'H'qu'IO U pomop ae1CmOp'HOao nOM F:
5.2.2.F = c, I')l,ec - nOCTOgHHbIit BeKTOp.
5.2.3.F = r, I')l,er = x . i + y . j + z . k.
5.2.4.F(yz, xz, xy).
5.2.5.F G(y2 + Z2), ~(z2 + x2), ~(X2 + y2)).
5.2.6.F = x2y . i + y2 Z . j + z2 X . k.
5.2.7.BbI'IHCJIHTb)l,HBepI'eHlJ;HIO H POTOP I'pa)l,HeHTaCKaJIgpHOI'OnOJIg
U = U(x,y,z).
5.2.8.BhI'IHCJIHThdiv f'I')l,er(x, y, z) H r = Irl.
("\.r |
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zkH>< |
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+ r'. |
an)l,eM |
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x(Jx2 + y2 + Z2)~ |
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5.2.9. BbI'IHCJIHTbdiv(r . r). |
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5.2.10. BhI'IHCJIHTbdiv(f(r) . r). |
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5.2.11. |
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,n:oKa3aTb paBeHCTBO div(f· F) = gradf· F + f· divF. |
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Q ITYCTb F |
= P . i +Q .j + R . k. TOI')l,af .F =f |
.P . i + f |
.Q .j + f .R· k H |
div(f . F) = (f P)~ + (fQ)~ + (f R)~ = (f~ . P + f .P~) + (f~ . Q + f· Q~)+ + (f~ . R + f . R~) = (f~ . P + f~ .Q + f~ .R) + f . (P; + Q~ + R~).
B nepBoti cK06Ke CTOHT CKaJUlpHOe npOH3Be,!l;eHHe rpa,n,ueHTa CKa.J1HpHOrO no-
J151 I Ha BeKTOp F, a BO BTOPOti - ,!l;HBepreHIJ;HH BeKTopHoro nOJIH F. TaKHM |
06pa30M, div(f . F) = grad I· F + I .div F. |
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5.2.12. |
,I1;oKa3aTb CBoticTBa JIHHetiHocTH ,!l;HBepreHIJ;HH: |
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a) div(F l +F2) = divF1 +divF2; |
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6) div(c • F) = c·div F, r,!l;e c - |
KOHCTaHTa. |
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5.2.13. |
,I1;oKa3aTb paBeHCTBO: div(U ·c) = gradU ·c,r,!l;e c |
- nOCToHHHblti |
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5.2.14. |
BbI'IHCJIHTbdiv(U . grad V). |
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5.2.15. |
,I1;oKa3aTb paBeHcTBo div(F l x |
F 2) = F2 . rot Fl - |
Fl' rot F 2 • |
5.2.16. ,I1;oKa3aTb cBoticTBa JIHHetiHocTH pOTopa:
a) rot (Fl + F 2) = rot Fl + rot F 2;
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6) rot (c • F) = c . rot F, r,!l;e c - |
npOH3BOJIbHaH nOCT05lHHaH. |
5.2.17. |
,I1;oKa3aTb, 'ITOrot (f. F) = I· rot F + grad I |
x F. |
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o I1YCTb F = P . i |
+ Q . j |
+ R . k. Tor,!l;a I . F = I . P . i |
+ I . Q . j + I . R . k. |
Rati,!l;eM rot (f . F): |
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rot (f . F) = ax |
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(fP)~] + k[(fQ)~ - (fP)~] = |
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I;Q - IQ~]-j[/~R + IR~ - |
I;P - |
IP~]+ |
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IP;] = J[i(R~ - Q~) - j(R~ - |
P;) + k(Q~ - |
P;)]+ |
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+ [i(f~R - I;Q) - |
j(f~R - |
I;P) + k(f~Q - |
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I~P)] = |
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5.2.18. |
,I1;oKa3aTb, 'ITOdiv(rot F) = O. |
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5.2.19. |
BbI'IHCJIHTbPOTOP BeKTopHoro nOJIH F = r·c,r,!l;e c - |
nOCT05lHHblti |
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BeKTOp, a |
r - |
MO,!l;yJIb pa,!l;Hyca-BeKTopa r. |
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5.2.20. |
,I1;JIH BeKTopHoro nOJIH F(xy, yz, zx) BbI'IHCJIHTbrot (rot F). |
5.2.21. |
RatiTH grad (div F), eCJIH F( xyz2,xy - z,zx2). |
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5.2.22. |
3anHcaTb C nOMOIIJ;bIO onepaTopa Ha6JIa V BeKTopHble nOJIH: |
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a) grad (div F); |
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6) rot (grad U). |
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o a) ,I1;HBepreHIJ;HH BeKTopHoro nOJIH F C nOMOIIJ;bIO onepaTopa V 3anHCbI-
BaeTCH TaK: V . F. rpa,n,ueHT CKa.J1HpHOrO nOJIH U '1epe3onepaTOp V Bblpa- )KaeTCH CJIe,!l;yIOIIJ;HM 06pa30M: VU. CJIe,!l;OBaTeJIbHO,
grad (div F) = V(div F) = V(V . F).
6) POTOP BeKTopHoro nOJIH F C HCnOJIb30BaHHeM onepaTopa V 3anHCbI-
BaeTCH CJIe,!l;yIOIIJ;HM 06pa30M: V x F. CJIe,!l;OBaTeJIbHO, |
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rot (grad U) = V x (grad U) = V x (VU). |
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5.2.23. 3anlfcaTb C nOMOID;bIO onepaTOpa raMlfJIbTOHa CJIe.l1YIOID;lfe BbIpa-
)KeHlfa: |
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a) div(grad U); |
6) div(rot F); |
B) rot (rot F); |
r) rot (grad (divF)). |
5.2.24. ,il;oKa3aTb CJIe.l1YIOID;lfe paBeHCTBa: |
a) V' x (V'U) = 0; |
6) V' . (V' x F) = 0; |
B)V' . (V'U) = V' 2U == t::.U.
5.2.25.,il;oKa3aTb cBotkTBa JIlfHeitHocTIf onepaTopa raMlfJIbToHa:
a)V'(CIUI + C2U2) = CI . V'UI + C2 . V'U2;
6)V'. (CIF I + C2F 2) = CI(V'· Fd + C2(V'· F 2)j
B)V' x (CIFI +C2F 2) = CI(V' x F I) +C2(V' x F 2), r,ll;e CI , C2 -
npOlf3BOJIbHbIe nOCToaHHble, |
UI , U2 - CKaJIapHbIe nOJIa, |
a F I , |
F2 - BeKTopHbIe nOJIa. |
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5.2.26. ,il;oKa3aTb paBeHCTBa (U, V - |
CKaJIapHbIe nOJIa, F, FI If |
F2 - |
BeKTopHbIe nOJIa): |
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a) V'(U· V) = UV'V + VV'Uj
6) V'(U . F) = U(V' . F) + (V'U) . Fj
B) V'(FI x F 2) = F2 . (V' x F I) + Fl· (V' x F 2); r) V' x (UF) = U(V' x F) + (V'U) x F.
B'bt"l,Uc.!/,umb auaepee'H'4u'HJ u pomop ae'K:mop'Hoeo nOJIJI F:
5.2.27.F = y . i + z . j + x . k.
5.2.28.F = xy . i + yz . j + zx . k.
5.2.29. F = (x3 + y2 + z) . i + (y3 + Z2 + x) . j + (Z3 + x2 + y) . k.
F zy. xz· xy k
5.2.30.=X' l +y'J+z' .
5.2.31.RaitTIf yrOJI Me)K.l1Y pOTopaMIf BeKTopHbIX nOJIeit FI(X2y,y2z,z2x)
If F2(z,x,y) B |
TO'lKeMo(l, 1, 1). |
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5.2.32. RaitTIf ,ll;JIlfHy |
pOTopa BeKTopHoro nOJIa |
F(x - z2, yz, x 2 + y2) B |
TO'lKeM(l, 2, -1).
5.2.33.B KaKlfX TO'lKaxnpocTpaHcTBa POTOP BeKTopHoro nOJIa
F(y2 + Z2,Z2 + x2,X2 + y2)
nepneH,ll;lfKYJIapeH OCIf Ox?
5.2.34.B KaKlfX TO'lKaxnpocTpaHcTBa POTOP BeKTopHoro nOJIa
F(x2y,y2z,Z2X)
nepneH,ll;lfKyJIapeH nJIOCKOCTIf x + y + z = 2?
5.2.35.B KaKlfX TO'lKaXnpocTpaHcTBa POTOPbI BeKTopHbIX nOJIeit
FI(xy,yz,zx) If F2(z,x,y)
KOJIJIlfHeapHbI?
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B c.n,eoy'lOw,ux 3aOa"taX r = x· i + y. j + z· k u r |
= Irl: |
5.2.36. |
BblqHCJIHTb div(c x r), r,!l;e c - |
nOCToflHHblit BeKTOp. |
5.2.37. |
BbIqHCJIHTb divb(r·a), r,!l;e a H b - |
nOCTOflHHble BeKTOpbI. |
5.2.38. |
BbIqHCJIHTb divr(r·a), r,!l;e a - |
nOCToflHHbIit BeKTOp. |
5.2.39. |
BblqHCJIHTb rot (c x r), r,!l;e c - |
nOCToflHHbIit BeKTOp. |
5.2.40. |
BblqHCJIHTb rotr(r·a), r,!l;e a - |
nOCToflHHbIit BeKTOp. |
5.2.41. |
BblqHCJIHTb rot b(r·a), r,!l;e a H b - |
nOCTOflHHble BeKTOpbI. |
5.2.42.,il;oKa3aTb CBoitcTBa onepaTopa JIanJIaca:
a) 6(Cl Ul + C2 U2 ) = Cl 6Ul + C2 6U2 ;
6) bo(Ul . U2 ) = Ul . 6U2 + 2(V'Ul ) . (V'U2 ) + U2 • boUl.
KOHTponbHble Bonpocbl M 60nee CnO)l(Hbie saW-HMR
B c,//,eoy'lOw,ux 3aoa"tax r = x . i + y . j + z . k u r = Irl:
5.2.43.BblqHCJIHTb ,!l;HBepreHIIHIO H POTOP BeKTopHoro nOJIfi F = f(r) . c, r,!l;e c - nOCToflHHblit BeKTOp.
5.2.44. ,il;oKa3aTb, 'ITOeCJIH div(f(r) . r) = 0, TO f(r) = ~. r
5.2.45.BblqHCJIHTb div(grad f(r». KaKoBa ,!l;OJDKHa 6bITb <PYHKIIHfI f(r),
qTo6bI div(gradf(r» |
= 0. |
5.2.46. BbIqHCJIHTb rot [c x |
f(r) . r], r,!l;e c - nOCToflHHblit BeKTOp. |
5.2.47.,il;oKa3aTb paBeHCTBO: grad (div F) = V'2F + V' x (V' x F).
5.2.48.,il;oKa3aTb paBeHCTBO: rot (rot F) = V'(V' . F) - V'2F.
5.2.49. BeKTopHoe nOJIe F 3a,!l;aHO paBeHCTBOM: F = c·lnr, r,!l;e r - MO,!l;yJIb pa,!l;Hyca-BeKTopa TOqKH. HaitTH V'F, V' x F H V'2F.
§ 3. nOTOK BEKTOPHOrO no/ul
IIycTb B 06JIacTH n C IR3 3a,n;aHO HeKOTopoe BeKTopHoe nOJIe F = Pi+Qj+Rk, rAe P(x, y, z}, Q(x, y, z}, R(x, y, z} - HenpepblBHO AHq,q,epeHIIHpyeMble B 06JIacTH n q,YHKIIHH. IIycTb Sen - rJIa,n;KaH opHeHTHpyeMaH nOBepxHoCTb, Ha KOTOPOit BbI6paHa onpe,!l;eJIeHHaH CTopOHa, 3aAaaaeMaH e,!I;HHH'IHOit HOPMaJIbIO n(cosa,cosi3, cos')'}K STOit nOBepXHOCTH.
~ IIomo'ICoM 6e'ICmOp'H.020 no.l!J! F 'tepe3 n06epX'H.OCm'b S 6 'H.anpa6.11e'H.UU eaU'H.U't'H.o11. 'H.OPMa.IIU n Ha3bIBaIOT nOBepXHocTHblit HHTerpaJI rrepBoro pOAa:
II = jjF. ndS = jj(Pcosa + Qcosi3 + Rcos')'}dS. |
(3.1) |
s |
s |
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ECJIH 0603Ha'lHTb'1epe3Fn npoeKIIHIO BeKTopa F Ha HanpaBJIeHHe BeKTopa n, TO, Y'lHTbIBaH,'ITOHMeeT MeCTO paaeHCTBO F . n = IFI . Inl COS t.p = IFI . COS t.p = Fn
(rAe t.p - yrOJI Me:lKAY BeKTopaMH F H n), q,OpMyJIy MH BbI'IHCJIeHHHnOTOKa MO:lKHO 3arrHcaTb B q,opMe, KOTopaH He 3aBHCHT OT Bbl60pa CHCTeMbI KoopAHHaT:
TIoBepxHocTHbIiI: HHTerparr rrepBOrO pO,!l;a B <p0pMYJIe (3.1) CBH3aH C rrOBepXHOCTHhIM HHTerparrOM BToporo pO,!l;a paBeHCTBOM:
II= !!(P cos cr + Qcos(3 + Rcos")')dS = !!Pdydz+Qdzdx+Rdxdxy, (3.2)
S |
(S,n) |
KOTopoe ,!I;aeT ellIe O,!l;HH crroco6 BhI'IHC.JIeHHHrrOTOKa. |
t1>u3u'teC'lCuit CM'btC.II |
nomo'ICa: eC.JIH BeKToP-<PYHKD;HH F eCTb rrOJIe CKopocTeil: |
TeKYllIeil: lKH,!I;KOCTH, TO |
rrOTOK II 3Toro BeKTopHoro rrOJIH '1epe3 rrOBepXHOCTb S |
paBeH 0611IeMY KOJIH'IeCTBYlKH,!I;KOCTH, rrpOTeKaIOllIeil: '1epe3S 3a e,!l;HHHD;y BpeMeHH.
IlcrroJIb3YH rrOHHTHe rrOTOKa, MOlKHO rrOHHTb ljju3U'teC'lCuit CM'btC.II nomo'ICa (CM. TaKlKe 3a,!1;a'lY5.3.39). )J;HBepreHD;HH BeKTopHoro rrOJIH F B TO'lKeM eCTb rrpe,!l;e.Jl OTHomeHHH rrOTOKa rrOJIH '1epe3c<pepy S ,!I;OCTaTO'lHOMarroro pa,!l;Hyca, OKpYlKaIOIu:yIO TO'lKYM, K 06'beMYV mapa, OrpaHH'IeHHOrO3TOil: c<pepoil:, rrpH CTpeMJIeHHH
,!I;HaMeTpa d mapa K HYJIIO: div F = !~~!!Fn . dS
S
(J)0pMyna raycca-OCTpOrpaACKOrO
TeopeMa 5.1 (OcTporpaAcKllliii). nYCTb S - 3aMKHYTal'l rllaAKal'l Opl'leHHlpyeMal'lnOBepXHOCTb, l'IBlll'llOl.I.Ial'lCl'IrpaHI'I~eA Tella V 1'1 n(coscr, cos (3, cos")')- eAI'I- HI'IYHal'lBHeWHl'Il'IHopMallb K S. nyCTb BeKTopHoe nOlle F(P, Q, R) - HenpepblBHO
Al'lcI>cI>epeH~l'IpyeMo Ha S 1'1 B V. TorAa
!!(P cos cr + Qcos(3 + Rcos")')dS = !!!(~~ + ~~ + ~~) dV. (3.3)
S v
BhIpalKeHHe, CTOHllIee rro,!l; 3HaKOM HHTerparra B rrpaBoil: '1aCTHpaBeHCTBa (3.3), rrpe,!l;CTaBJIHeT co6oil: ,!I;HBepreHD;HIO BeKTopHoro rrOJIH F, HHTerparr, CTOHIlIHiI: C.JIeBa, rrpe,!l;CTaBJIHeT co6oil: rrOTOK BeKTopHoro rrOJIH F '1epe3rrOBepXHOCTb S B HarrpaBJIeHHH BHemHeil: HopMarrH. TI03TOMY <p0pMYJIa (3.3) MOlKeT 6hITb rreperrHcaHa B BH,!I;e:
!!F.ndS =I!!divFdV.
EC.JIH HCrrOJIb30BaTb orrepaTOp raMH.JIbTOHa, TO <popMYJIa raycca-OCTpOrpa,!l;CKOrO
(3.3) MOlKeT 6hITb 3arrHcaHa B C.JIe,!l;yIOllIeil: <p0pMe:
!!F.ndS= !!!V.FdV.
S |
v |
<l>0PMYJIY raycca-ocTpOrpa,!l;CKoro |
'1aCTO rrpHMeHHIOT ,!I;.JIH BhI'IHC.JIeHHH rrOTOK3 |
BeKTopHoro rrOJIH '1epe33aMKHYTYIO rrOBepXHOCTb S. O,!l;HaKO C.JIe,!l;yeT HMeTb B BH-
,!I;y, 'ITO,!I;.JIH rrpHMeHeHHH 3Toil: <P0PMYJIhI Heo6xo,!l;HMO, '1To6hIBeKTopHoe rrOJIe 6hIJIO HerrpephIBHo ,!I;H<p<pepeHD;HPyeMhIM BHyTpH rroBepxHocTH S. 2ho YC.JIOBHe Bcer,!l;3
x + y -
!!F.ndS = !!OdS = O.
m(l, 2, 3).
6y.n;eT BhIIIOJIHeHO, eCJIH 06JIacTb 11, B KOTOPOtl: paccMaTpHBaeTCH IIOBepXHOCTb S,
npOCmpaHCm6e'H.'H.O OiJ'H.OC6S13HCJ.H:
~ 06JIacTb 11 C ]R3 Ha3hIBaeTCH npOCmpa'H.Cm6e'H.'H.O OiJ'H.OC6S13'H.011., eCJIH H3 TOrO,
'ITO3aMKHYTaH IIOBepXHOCTb S JIelKHT B 11, CJIe.n;yeT, 'ITOTeJIO V, rpaHHu;etl: KOTo-
pom HBJIHeTCH IIOBepXHOCTb S, TOlKe JIelKHT B 11. $
5.3.1. BbI'IHCJUITbnOTOK BeKTOpHOrO nOJHI F(P, Q, R) '1epe3nOBepXHOCTb
S B CTOpOHy, Onpe,!l;eJIaeMYIO BeKTOpOM e,!l;HHH'IHOitHOPMa.JlH n K
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nOBepXHOCTH S, eCJIH: |
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a) F(4, -5,2), a S - '1acTbnJIOCKOCTH x + 2y + 3z = 6, pacnOJIO- |
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)l{eHHM B OKTaHTe x ~ 0, y ~ 0, z ~ 0, n |
06pa3yeT OCTPhIit yrOJI C |
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OCblO OZj |
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x + y - z = 0, pacnoJIo- |
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6) F(O, y, 0), S - |
'1acTbnJIOCKOCTH 1 - |
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)l{eHHM B OKTaHTe x ~ 0, y ~ 0, z ~ 0, a n 06pa3yeT OCTPblit yroJI |
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C OCblO Oz; |
'1acTbnapa60JIOH,!l;a z = x 2 + y2, y,!l;OBJIeTBOpa- |
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B) F(I, 1, z), S - |
a |
IOID;M YCJIOBHIO z ~ 1, an - BHeIIIHaa HOPMa.Jlb K napa60JIOH,!l;y. |
a) XOPOIIIO H3BeCTHO, 'ITOHOPMa.JlbHbIM BeKTopOM K nJIOCKOCTH aBJIaeT- |
ca BeKTOp, KOOp,!l;HHaTbI KOToporo CYTb K09cPqmIJ:HeHThI npH HeH3BeCTHhIX B
ypaBHeHHH nJIOCKOCTH. B HaIIIeM CJIy'lae9TO BeKTOp TIOCKOJIbKy m· F = 1· 4 + 2· (-5) + 3·2 = 0, TO HOPMa.Jlb m K nJIOCKOCTH, (a, 3Ha'lHT, H e,!l;HHH'IHMHOPMa.Jlb n K 9Toit nJIOCKOCTH) nepneH,!l;HKyJIapHa BeKTopHOMY rrOJIIO. Ho TOr,!l;a
8 |
8 |
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6) BbI'IHCJIHMnOTOK BeKTopHoro nOJIa C nOMOID;blO nOBepXHOCTHoro HH- |
Terpa.Jla II pO,!l;a (cP0pMYJIa (3.2» |
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II = !!Pdydz + Qdzdx + Rdxdxy = |
!!ydzdx |
(8,n) |
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(8,n) |
(B HaIIIeM CJIy'laeP = R = 0, Q = y). ,I1;JIa BbI'IHCJIeHHanOCJIe,!l;HerO HHTerpa.Jla H306pa3HM Ha '1epTe)l{enOBepXHOCTb S (pHC. 53) H ee npOeKIJ:HIO Dxz
Ha nJIOCKOCTb Oxz (pHC. 54)
HOPMa.Jlb n K nJIOCKOCTH 1 - z = 0, o6pa3YIOID;aa OCTPblit yroJI C OCblO 0 z, 06pa3yeT Tynoit yroJI C OCblO Oy (9TO O'leBH,!l;HOH3 '1epTe)KajO,!l;HaKO HeCJIO)l{HO nOKa3aTb, 'ITOH~HylO CTOPOHY nOBepXHOCTH S 3a,!l;aeT e,!l;HHH'IHM
HOPMa.Jlb n (.]a, -.]a, .]a); 3,!l;eCb COS'Y> 0, a |
cos (3 |
< 0, CJIe,!l;OBaTeJIbHO, |
n o6pa3yeT OCTPblit yrOJI C OCblO 0 Z H Tynoit - |
C OCblO Oy). TI09TOMY rrpH |
CBe,!l;eHHH nOBepXHOCTHoro HHTerpa.Jla K ,!l;BOitHOMY no |
06JIaCTH Dxz nepe,!l; |
