Сборник задач по высшей математике 2 том
.pdf5.4.4. |
F = |
-xi+yj, L 3a,!l.aHa lIapaMeTpHqeCKH ypaBHeHH~MH x = .jcost, |
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Y = |
.jsint, °:::;; t :::;; ~. |
5.4.5.F = 3yi + xj, L - JIOMaH~ ABC, r,!l.e A(O, 0), B(I, 1), C(2,4).
HaiJ.mu pa60my npocmpaHcmeeHHOiW ee'ICmOpHOeo nOJIJI F(P, Q, R) eiJo.!l:b 'lCpueoiJ. L:
5.4.6.F(x, 2y, -z), L - OTpe30K AB lIP~MOii, 3a.n.aBaeMoii ypaBHeHH~MH
x - I y |
z+1 |
1, -2). |
-2- = I |
= ~,r,!l.e A(I,O, -1), B(3, |
5.4.7.F(x, y, 1), L - lIepBbIii BHTOK BHHTOBOii JIHHHH, 3a.n.aHHoii lIa-
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paMeTpHqeCKH |
ypaBHeHH~MH x = R cos t, Y = |
R sin t, Z |
= |
at, |
5.4.8. |
°:::;; t :::;; 27r. |
qacTb OKP)')KHOCTHx = cos t, Y |
= sin t, z |
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F(z, 1, 2y), L - |
= |
1, |
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tl = 0, t2 = ~. |
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5.4.9.HaiiTH pa60TY rpa.n.HeHTa CKaJI~PHOI'O lIOJI~ U = xyz B,!l.OJIb OT-
pe3Ka lIP~MOii AB, r,!l.e A(I, 2, 3), B(3, -2,3).
5.4.10. HaiiTH pa60TY pOTopa BeKTopHoro lIOJI~ F(y, x) B,!l.OJIb KOHHqeCKoii ClIHPaJIH x = tcost, Y = tsint, z = t, °:::;;Z,t :::;; 27r.
5.4.11.HaiiTH lI.HpKyJI~lI.HIO lIJIOCKOI'OBeKTopHoro lIOJI~ F(P, Q) lIO 3aMKHyTOii KPHBOii L B lIOJIOlKHTeJIbHOM HalIpaBJIeHHH:
a)F( -y, x), L - OKPY:>KHOCTb, 3a.n.aBaeM~ ypaBHeHHeM
x2 + (y + 1)2 = R2;
6) F(2y, x), L - |
KOHTYP TpeyrOJIbHHKa ABC, r,!l.e A(O, 0), B(I, 0), |
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C(I,I). |
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a a) 3alIHweM lIapaMeTpHqeCKHe ypaBHeHH~ OKPY:>KHOCTH: x |
= R cos t, |
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Y = Rsint - |
1, °:::;; t :::;; |
27r. HaXO,!l.HM dx = |
-Rsintdt, dy = |
Rcostdt. |
Tor,!l.a lI.HpKyJI~lI.H~ lIOJI~ F |
B,!l.OJIb KPHBOii L 6Y,!l.eT paBHa: |
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21T |
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U;= !F.dr= !-ydx+xdy= ![(Rsint-l)Rsint+R2 cos2 t]dt= |
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L |
0 |
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21T |
21T |
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! (R2 - Rsint) dt = |
(R2t + Rcost)lo |
= 27rR2. |
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6) IIepe'bl.iJ. cnoco6.
KOHTYP L eCTb 06'be,!l.HHeHHeOTpe3KOB AB, BC H CA. n09TOMY lI.HpKyJI~lI.H~ lIOJI~ F B,!l.OJIb KPHBOii L 6Y,!l.eT paBHa:
U;= !F.dr= !F.dr+ !F.dr+ !F.dr.
L |
AB |
BC |
CA |
°H, |
BblqHCJIHM Ka:>K,!l.bIii |
H3 HHTerpaJIOB. B,!l.OJIb OTpe3Ka AB HMeeM y |
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CTaJIO 6bITb, dy = 0. |
CJIe,!l.OBaTeJIbHO, |
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!F·dr= ! 2ydx+xdy=0.
AB |
AB |
260
B.n;OJIb OTpe3Ka BC HMeeM x = 1 H dx = O. n09TOMY
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1 |
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JF . dr = J2y dx + x dy = Jdy = 1. |
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BC |
BC |
0 |
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HaKOHeu;, B.n;OJIb OTpe3Ka CA HMeeM y = X |
H dy = dx. CJIe.n;OBaTeJIbHO, |
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JF . dr = |
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0 |
20 |
-l |
J2y dx + x dy = J3x dx = 3~ |
11 = |
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CA |
CA |
1 |
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TaKHM 06pa30M, U;HPKYJIgU;Hg nOJIg F B.n;OJIb KOHTypa L 6y.n;eT paBHa: U =
3 |
1 |
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=0+1- 2 |
=-2' |
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Bmopoi1. cnoco6. |
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BbI'IHCJIHMU;HPKYJIgU;HIO, npHMeHHB <P0PMYJIY rpHHa: |
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U= fPdx+Qdy= JJ(~~ - ~P) dxdy, |
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L |
D |
Y |
r.n;e 06JIaCTbIO D gBJIgeTCg TpeyrOJIbHHK ABC. B |
HaweM CJIY'IaeP = 2y, |
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8Q |
8P |
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Q = x. CJIe.n;oBaTeJIbHO, 8x |
1, a 8y = 2. Tor.n;a U;HPKYJIgU;Hg nOJIg F |
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B.n;OJIb L paBHa
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Hai1.mu 'qUptcy.II.JI'qU1O nJtoctcoeo 6etcmOpHoeo no.II.JI F(p, Q) 600Jt'b tcPU60i1. L (Hanpa6JteHue o6xooa - nOJtrotCumeJt'bHoe):
5.4.12.F (y2 , 2xy), L - npOH3BOJIbHbIii 3aMKHYTbIii KOHTYp.
5.4.13.F(y, -x), L - OKPY)KHOCTb x 2 + y2 = R2.
5.4.14.F(y, -x), L - OKPY)KHOCTb (x - 1)2 + (y + 1)2 = R2.
5.4.15.F(2x - xy2, -2xy), L - JIOMaHM ABA, r.n;e A(O, 0), B(I, 1), KpH-
BM AB - KYCOK napa60JIbI y = x 2 , a BA - oTpe30K npgMoii.
5.4.16.F(xy, 1), L - rpaHHu;a KBa.n;paTa 0 ~ x ~ 1, 0 ~ y ~ 1.
5.4.17. BbI'IHCJIHTb U;HPKYJIgU;HIO npOCTpaHCTBeHHoro BeKTopHoro nOJIg
F = -xi + xj + yk B.n;OJIb 9JIJIHnCa L, nOJIY'IalOIIJ;erOcgnepece- '1eHHeMU;HJIHH.n;pa x 2 + y2 = 1 C nJIOCKOCTblO x + y + z = 1 (npH B3rJIg.n;e C nOJIO)KHTeJIbHoro HanpaBJIeHHg OCH 0 Z o6xo.n; KOHTypa
L COBepWaeTCg npOTHB '1acoBoiiCTpeJIKH).
Q IIep6'b1,i1. cnoco6.
3anHweM napaMeTpH'IeCKHeypaBHeHHg 9JIJIHnca: x = cos t, Y = sin t, Z ==
= 1 - cos t - sin t. npH H3MeHeHHH napaMeTpa t OT 0 .n;o 271' nOJIY'IaeMTpe6y-
261
II: = f F . dr = f P dx + Q dy + R dz = f -x dx + x dy + y dz =
L |
L |
L |
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21T |
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21T |
211". |
1[(- cos t) . (- sin t) + cos t . cos t + sin t(sin t - cos t)] dt = |
1dt = |
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0 |
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Bmopoi1. cnoco6.
BbI'lHCJIHMII,HpKymuIHIO, npHMeHHB <P0PMYJIY CTOKca, npH'leMB Ka'leCTBe nOBepXHOCTH B, OrpaHH'lHBaeMOiiKPHBOii L, BbI6epeM 'laCTbnJIOCKOCTH x + + y + z = 1, JIe:>KaIIIeii BHYTPH II,HJIHH.n;pa x 2 + y2 = 1. E.n;HHH'lHYIOHOPMaJIb K nJIOCKOCTH BbI6epeM TaK, 'lT06bI,rJIg):VI C ee KOHII,a, HanpaBJIeHHe 06xo.n;a KOHTypa L npoxo.n;HJIO npOTHB 'lacoBoiiCTpeJIKH. TaKoii e.n;HHH'lHOiiHopMa-
JIbIO 6y.n;eT BeKTop n (~, ~, ~). ITo <p0pMYJIe CToKca HMeeM:
II: = f F . dr = 11 rot F . n dB =
L |
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= II[(R~ - |
Q~) coso + (P; - R~) cos{3 + (Q~ - P;) cos,"),] dB = |
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s |
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= If [(1 - |
0) . -.L + (0 - |
0) . -.L + (1 - |
0) . -.L] |
dB = If~dB. |
s |
v'3 |
v'3 |
v'3 |
s v'3 |
Bw'lHCJIeHHenOCJIe.n;Hero HHTerpaJIa CBe.n;eM K BbI'lHCJIeHHIO.n;BoiiHoro HHTerpa.TIa no 06JIacTH Dxy , gBJIgIOIIIeiicg npOeKII,Heii nOBepXHOCTH B Ha nJIOC-
KOCTb Oxy. |
oTOii 06JIacTbIO 6y.n;eT Kpyr x 2 + y2 |
~ 1. |
ITOCKOJIbKY dB = |
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dxdy |
f<) |
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= -'--=I v 3dxdy, TO OKOH'laTeJIbHOnOJIY'laeM: |
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cos'")' |
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II: = 11Ja dB = 112 dxdy = 211 dxdy = 2B(Dxy) = 211". |
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D zy |
D zy |
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Hai1.mu '4uptcYJlJl'4U1O |
eetcmOpH020 |
nOJlJl F(P, Q, R) |
eaO.!I:b |
3aMtcHymo2o |
tcOH- |
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mypa L: |
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5.4.18.F(x2, y2, Z2), L - OKPY:>KHOCTb, napaMeTpH'leCKHeypaBHeHHg KO-
Topoii: x = cos t, Y = sin t, z = 1, HanpaBJIeHHe 06xo.n;a - B CTDPOHY YBeJIH'leHHgnapaMeTpa t.
5.4.19.F(y + z,z + x,x + y), L - OKPY:>KHOCTb, nOJIY'laIOIIIMCgnepece-
'leHHeMc<pepbI x 2 + y2 + z2 = R2 H nJIOCKOCTH x + y + z = 0,
HanpaBJIeHHe 06xo.n;a- npOTHB 'lacoBoiiCTpeJIKH, eCJIH cMoTpeTb
C KOHII,a OCH Oz.
5.4.20.F(x - y, y - z, z - x), L - KOHTYP TpeyrOJIbHHKa ABC, A(l, 0, 0),
B(l, 1,0), C(l, 0,1).
262
5.4.21.F ( - x 2 : y2 ' x 2 : y2 ,0) ,
a) L - OKP}')KHOCTb: X = cos t, Y = sin t, z = hj HanpaBJIeHHe 06xo.n;a - B CTOPOHY YBeJIHqeHHH napaMeTpaj
6) L - OKP}')KHOCTb:x = cos t+2, y = sin t+2, z = hj HanpaBJIeHHe
06xo.n;a - B CTOPOHY YBeJIHqeHHH napaMeTpa.
5.4.22.F(x - 2y - z,x - z,y + x), L - KOHTYP TpeyrOJIbHHKa ABC, r.n;e
A(I, 0, 0), B(O, 1,0), C(O, 0,1).
5.4.23.HaiiTH nOTOK pOTopa BeKTopHoro nOJIH F(y, z, x) qepe3 nOBepx-
HOCTb napa60JIOH.n;a z = 4 - x2 - y2, pacnOJImKeHHYIO BbIwe nJIOCKOCTH Oxy B HanpaBJIeHHH HOPMaJIH, Y KOTOpoii cos'Y> O.
o lIpH peweHHH .n;aHHoii 3a,n:aqH HMeeT CMbICJI BOCnOJIb30BaTbCH Teope-
Moii CToKca, nOCKOJIbKY BbIqHCJIeHHe nOTOKa, npOBe.n;eHHoe Henocpe.n;cTBeHHO, CJIOJKHee, HeJKeJIH BbIqHCJIeHHe IT;HPKYJIHIT;HH no rpaHHIT;e nOBepxHocTH. rpaHHIT;eii napa60JIOH.n;a HBJIHeTCH OKPYJKHOCTb L, JIe}KaIn;aH B nJIOCKOCTH xOy, napaMeTpHqeCKHe ypaBHeHHH KOTOpoii: x = 2 cos t, Y = 2 sin t, z = O.
TaKHM 06pa30M,
II = !!rot F.n dB = f F.dr = f y dx + z dy + x dz =
S |
L |
L |
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k |
k |
~(I-cos2t)dt = -2 (t - ~sin2t) |
k |
= ! 2sint(-2sint)dt = -4! |
10 = -41r. |
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0 |
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3a.Me"taHue. ECJIH B aHaJIOrHqHbIX 3a,n:aqaX BbIqHCJIeHHe IT;HPKYJIHIT;HH 3aTpy.n;HHTeJIbHO, TO MO}KHO BHOBb BOCnOJIb30BaTbCH TeopeMoii CTOKca, pacCMOTpeB .n;pyrylO, 60JIee npOCTYIO nOBepXHOCTb, KOTOpylO OrpaHHqHBaeT .n;aHHbIii KOHTyp. B HaweM CJIyqae - 9TO Mor 6bITb Kpyr B nJIOCKOCTH Oxy:
x2 +y2:::; 4. |
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5.4.24. |
HaiiTH nOTOK pOTopa BeKTopHoro nOJIH F(XZ2,y2z,z + y) qepe3 |
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qacTb nOBepXHOCTH KOHyca (z - 1)2 = x 2 + y2, pacnOJIO}KeHHYIO |
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Me}K.n:y nJIOCKOCTHMH z = 0 H Z = 1 B HanpaBJIeHHH BHeWHeii HOp- |
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MaJIH. |
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5.4.25. |
HaiiTH nOTOK pOTopa BeKTopHoro nOJIH F(z, x, y) qepe3 qacTb cclJe- |
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pbI x 2 + y2 + |
z2 = 1 (z ~ 0) B HanpaBJIeHHH BHeWHeii HOPMaJIH. |
5.4.26. |
TeJIo BpaIn;aeTCH C nOCTOHHHOii yrJIoBoii CKOPOCTblO W BOKpyr OCH |
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Oz. BbIqHCJIHTb IT;HpKyJIHIT;HIO nOJIH JIHHeiiHbIX CKopOCTeii B.n;OJIb |
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OKPY}KHOCTH e.n;HHHqHOrO pa,n:Hyca, IT;eHTp KOTOpoii JIe}KHT Ha OCH |
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BpaIn;eHHH, a |
nJIOCKOCTb, B KOTOpoii JIe}KHT OKPY}KHOCTb, nepneH- |
.n;HKYJIHpHa OCH Oz, B HanpaBJIeHHH OCH BpaIn;eHHH.
5.4.27. HaiiTH IT;HpKyJIHIT;HIO BeKTopHoro nOJIH F = yi+2xj B.n;OJIb JIOMaHOii
ABC, r.n;e A(I, 1), B(4, 2), C(0,4).
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5.4.28. |
HaftTH Il,HPKYmlIl,HIO BeKTopHoro 1I0ml F (-y, x) B.n;OJIb Kap.IJ:HOH.n;bI. |
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x = 2 cos t - cos 2t, Y = 2 sin t - sin 2t B CTOPOHY YBeJIH'IeHH8; |
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lIapaMeTpa. |
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5.4.29. |
IIoKa3aTb, 'ITOIl,HPKYJI8Il,H8 pa.n;Hyca-BeKTopa r B.n;OJIb JII060ro 3a- |
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MKHYToro KOHTypa paBHa 0. |
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5.4.30. |
HaftTH Il,HPKYJI8Il,HIO BeKTopHoro 1I0JI8 F (-y, x, 1) |
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a) B.n;OJIb OKPY)KHOCTH (x - 3)2 + y2 = 1, z = 1; |
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6) B.n;OJIb OKPY)KHOCTH x 2 + z2 = 1, y = 0. |
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5.4.31. |
IIoKa3aTb, 'iTOIl,HPKYJI8Il,H8 1I0CT08HHoro BeKTopHoro 1I0JI8 F = C |
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B.n;OJIb JII060ft rJIa.n;Koft 3aMKHYToft JIHHHH L paBHa 0. |
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5.4.32. |
HaftTH Il,HPKYJI8Il,HIO BeKTopHoro 1I0JI8 F(z, 2,1) B.n;OJIb JIOMaHoft |
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ABC, r.n;e A(I, 3, 2), B(O, 0,1), C( -1, -1, 1). |
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5.4.33. |
HaftTH Il,HPKYJI8I.I,HIO BeKTopHoro 1I0JI8 |
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F = (z + 2x - 3y)i + (x + y - 2z)j + yk |
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B.n;OJIb KOHTypa TpeyrOJIbHHKa |
ABC, r.n;e A(2, 0, 0), B(O, 3, 0), |
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C(O, 0,1). |
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5.4.34. |
HaftTH Il,HpKyJI8Il,HIO rpa.n;HeHTa |
CKaJmpHOrO 1I0JI8 U = x 3 y2 Z |
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B.n;OJIb 9JIJIHllca: x 2 + y2 = 1, x + z = 3. |
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KOHTponbHble Bonpocbl M 60nee CnO>KHble 3aAClHMSI |
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5.4.35. IIpHBecTH IIpHMepbI 06JIaCTH n: |
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a) 8BJI8IOIIl,eftc8 IIpocTpaHcTBeHHo 0.n;HOCB83HOft, HO He 1I0Bepx- |
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HOCTHO 0.n;HOCB83Hoft; |
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6) 8BJI8IOIIl,eftc8 1I0BepXHOCTHO 0.n;HOCB83HOft, HO He IIpOCTpaHCT- |
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BeHHO 0.n;HOCB83HOft; |
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B) He 8BJI8IOIIl,eftc8 HH 1I0BepXHOCTHO, HH IIpOCTpaHCTBeHHO O.n;HO- |
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CB83HOft. |
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5.4.36. |
BepHo JIH, 'ITOeCJIH B 06JIacTH n POTOP BeKTopHoro 1I0JI8 F paBeH |
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0, TO Il,HpKyJI8Il,H8 9TOro BeKTopHoro 1I0JI8 F 110 JII060MY 3aMKHy- |
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TOMY KOHTYPY L, pacIlOJIO)KeHHOMY B n paBHa O? |
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5.4.37. |
BepHo JIH, 'iTO1I0TOKH pOTopa BeKTopHoro 1I0JI8 F 'Iepe3.n;Be pa3- |
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HbIe 1I0BepXHOCTH Sl H S2, HMeIOIIl,He O.n;HY H TY )Ke rpaHHIl,y L |
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COBlla.n;aIOT? |
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§ 5. nOTEHLI,lI1AJ1bHbIE 1I1 COJ1EHOll1AAJ1bHbIE nOJ1~
~BeKTopHoe IIOJIe F Ha3bIBaeTCH nomeH'4UaJ!'bH'btM, eCJIH OHO HBJIHeTCH rpa)l;H-
eHTOM HeKOToporo CKaJIHpHOrO IIOJIH U, T. e. F = grad U = 'ilU. |
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B cJIy'lae,eCJIH IIOJIe F(P, Q, R) IIOTeHI.I,HaJIbHO, BbIIIOJIHHIOTCH paBeHCTBa |
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p_ aU |
Q _ aU |
R- aU |
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- ax' |
- ay' |
- az' |
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'ITOpaBHOCHJIbHO TOMY, 'ITOBhlpalKeHHe P dx + Q dy + R dz = dU HBmleTCH rrOJI-
HbIM .n;H<p<pepeHU;HaJIOM HeKoTopoit <PYHKU;HH U(X, y, Z). 2ha <PYHKU;HH Ha3bIBaeTCH
nome"'4UaJtOM BeKTOpHOrO rrOJIH F.
TeopeMa 5.33 • nYCTb 061laCTb |
n nOBepxHocTHO |
OAHOCBl'I3Ha III |
cPyHKlIlII1II P, Q, |
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R - HenpepblBHo AlllcPcPepeHlIlllpyeMbl |
B n. TorAa |
BeKTopHoe |
nOlle F(P, Q, R) |
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nOTeHlIlllallbHO TorAa III TOllbKO TorAa, KorAa BblnOIlHl'IlOTCl'IpaBeHcTBa: |
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IIpHBe.n;eHHaH TeopeMa <paKTH'IeCKHyTBeplK.n;aeT, 'ITOBeKTopHoe rrOJIe F rrOTeH- U;HaJIbHO Tor.n;a H TOJIbKO Tor.n;a, Kor.n;a rot F = 0, T. e. rrOJIe HBJIHeTCH 6e3BUXpeB1JtM.
YCJIOBHe rot F = 0 HBJIHeTCH TaKlKe Heo6xo.n;HMbIM H .n;OCTaTO'lHbIMYCJIOBHeM Toro, 'ITOKPHBOJIHHeitHhlit HHTerpaJI
! Pdx+Qdy+Rdz
AB
He 3aBHCHT OT <pOPMhl KPHBOit, coe.n;HHHlOIu;eit TO'lKHA H B B 06JIaCTH n (B rrpe.n;- rrOJIOlKeHHH, eCTeCTBeHHO, 'ITOn- rrOBepXHOCTHO O.n;HOCBH3HaH), a TaKlKe Toro, 'ITO U;HPKYJIHU;HH rrOJIH F rro JI1060MY 3aMKHYTOMY KOHTYPY paBHa HYJIlO, T. e.
IF ·dr = O.
L
ECJIH rrOJIe F rrOTeHU;HaJIbHO, TO ero rrOTeHU;HaJI U MOlKeT 6bITb Hait.n;eH rrYTeM peIIIeHHH CHCTeMbI YPaBHeHHit C '1aCTHbIMHrrpOH3Bo.n;HbIMH:
U~ = P, U~ = Q, U~ = R.
TaKlKe MOlKHO HaitTH rrOTeHU;HaJI U Herrocpe.n;CTBeHHhlM HHTerpHpOBaHHeM rro HeKOTOPOMY rrYTH MoM:
U= ! Pdx+Qdy+Rdz.
MoM
IIPH aTOM, B CHJIY He3aBHCHMOCTH aToro HHTerpaJIa OT <pOPMbI rrYTH, rrYTb MoM BbI6HpalOT B BH.n;e JIOMaHoit MoMlM2M, B.n;OJIb KalK.n;oro H3 3BeHbeB KOTOPOit H3-
MeHHeTCH JIHIIIb o.n;Ha Koop.n;HHaTa, a OCTaJIbHhle OCTalOTCH rrOCTOHHHhlMH. B aTOM cJIY'Iae.n;Ba H3 Tpex .n;H<p<pepeHU;HaJIOB B KPHBOJIHHeitHoM HHTerpaJIe 06pallJ;alOTCH B HOJIb, H rrOTeHU;HaJI BbI'IHCJIHeTCHB BH.n;e CYMMbI:
U = ! P(x, Yo, zo) dx + |
! Q(x, y, zo) dy + |
! R(x, y, z) dz, |
MoMl |
M1M2 |
M2M |
3TepeMa 0 Heo6xop;HMOM H p;OCTaTO'IHOMYCJIOBHH nOTeHlIHaJIbHOCTH BeKTopHoro nOJIH.
265
r,ll;e KWK,ll;blit H3 HHTerpaJIOB - CYTb 06hIQHhIit Onpe,ll;eJIeHHhIit HHTerpaJI no COOTBeTcTByrow;eit nepeMeHHoit, a OCTaJIbHhIe nepeMeHHble (HH,ll;eKCHpoBaHHhIe H HeHH- ,ll;eKCHpOBaHHhIe) HrparoT pOJIb KOHCTaHT.
ECJIH nOTeHIJ;HaJI BeKTopHoro nOJIH F H3BeCTeH, TO
JPdx + Qdy + Rdz = JdU = U(B) - U(A).
AB AB
~ BeKTopHoe nOJIe F Ha3hIBaeTCH co.!!eHouOMbH'btM, eCJIH OHO HBJIHeTCH POTOPOM
HeKOToporo BeKTopHoro nOJIH A, T. e. F = rot A = V'x |
A. IIoJIe A Ha3hIBaeTCH |
aelCmOpH'btM nomeHtjua.!!OM nOJIH F. |
~ |
TeopeMa 5.44 • nYCTb o611acTb n npocTpaHcTBeHHo OAHOCB!'!3HaIII KoopAIIIHaTbl P, Q, R BeKTopHoro nOll!'! HenpepblBHo AlllcI>cI>epeH~lIIpyeMbl B n. TorAa BeKTopHoe
nOlle F(P, Q, R) COlleHOIIIAallbHO B TOM III TOllbKO B TOM CllY'lae,KorAa
div F = oP + oQ + oR = 0 ax oy oz
B Ka)f(AOA TO'lKe0611aCTIII n.
ECJIH BeKTopHoe nOJIe COJIeHOH,ll;aJIbHO, TO ero nOTOK Qepe3 JIro6yro 3aMKHyTyro nOBepXHOCTb paaeH HyJIro.
5.5.1. llOKa3aTb, qTQ nOJIe F(2x+yz)i+xzj + (xy+2z)k nOTeHU;Ha.JIbHO
H HaiiTH era nOTeHU;Ha.JI.
Q lloKroKeM, qTO rot F = O. |
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rot F = (R~ - Q~)i + (P~ - R~)j + (Q~ - |
P;)k = |
= (x - |
x)i + (y - y)j + (z - z)k = O. |
CJIe,n;OBaTeJIbHO, nOJIe F nOTeHU;Ha.JIbHO. Haii,n;eM nOTeHU;Ha.JI U (x, y, z) noml
F ,II,BYMjI pa3HbIMH cnoc06aMH.
I cnoco6. CocTaBHM CHCTeMY ypaBHeHHii C qacTHbIMH npOH3Bo,n;HhIMH:
{U~U~ == xz,2x + yz, U; =xy+2z.
IIHTerpHpYjl nepBoe ypaBHeHHe no x, nOJIyqaeM:
U = J(2x + yz) dx = x 2 + xyz + <p(y, z)
(3,n;ecb pOJIb KOHCTaHTbI HHTerpHpOBaHHjI HrpaeT JIl06M <PYHKU;HjI <p(y, z),
H60 ee qacTHM npOH3Bo,n;HM no x paBHa HyJIlo). ,lJ;a.JIee, ,n;H<p<pepeHu;HpYjl
4TeopeMa 0 Heo6xo)J;HMOM H )J;OCTaTOQHOM YCJIOBHH COJIeHOH)J;8JIbHOCTH nOJIH.
266
ITOJIyqeHHYIO <PYHKIJ:HIO U ITO ITepeMeHHOii y H HCnOJIb3Y5I BTOpOe paBeHCTBO CHCTeMbI, nOJIyqaeM ypaBHeHHe XZ + <p~(y, Z) = xz, oTKy.n:a <p~(y, Z) = O. MH-
TerpHpy51 nOJIyqeHHOe YPaBHeHHe no nepeMeHHoii y, nOJIyqHM <p(y, z) = 1/J
IIo.n:cTaBJI5I5I Haii.n:eHHoe 3HaqeHHe <PYHKIJ:HH <p(y, z) B <PYHKIJ:HIO U, npHXO- ,UHM K paBeHCTBY: U = x2 + xy z + <p(y, z) = x 2 + xyz + 1/J (z). HaKOHeIJ:, ,UH<p<pepeHu;HPY5I <PYHKIJ:HIO U no nepeMeHHoii z H HCnOJIb3Y5I nOCJIe.n:Hee pa-
BeHCTBO CHCTeMbI, nOJIyqaeM: xy + 1/J'(z) = xy + 2z, oTKy,Ua 1/J'(z) = 2z, T. e. |
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1/J(z) = Z2 |
+ C. TaKHM 06pa30M, npHXO,UHM K OKOHqaTeJIbHOMY BH.n:y nOTeH- |
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rrHaJIa: U = x2 + xyz + z2 + C. |
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II cnoco6. BblqHCJIHM nOTeHIJ:HaJI Henocpe.n:CTBeHHbIM HHTerpHpOBaHHeM. |
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IPHKCHPY51 |
TOqKY Mo(xo, Yo, zo), |
pacCMOTPHM npOH3BOJIbHYIO TOqKY |
M(x, y, z). Tor.n:a |
JPdx + Qdy + Rdz. |
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U(x,y,z) == U(M) = |
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MoM |
JIHHHIO HHTerpHpOBaHH5I (B CHJIY He3aBHCHMOCTH TaKOrO HHTerpaJIa OT <popMbI nYTH) BbI6epeM B BH.n:e JIOMaHOii MoMlM2M, r.n:e OTpe30K MoMl na-
paJIJIeJIeH OCH OX, OTpe30K MlM2 - OCH Oy, a OTpe30K M2M - OCH Oz.
B.n:OJIb MoMl HMeeM y = Yo HZ = zo, a, CJIe,UOBaTeJIbHO, dy = dz = 0, B,UOJIb
MlM2 Y:lKe x - |
nOCT05lHHO H z = zo, oTKy,Ua dx = dz = 0, a |
B,UOJIb M2M |
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06e nepeMeHHble, x H y - nOCT05lHHbI, a, 3HaqHT, dx = dy = O. Tor,Ua |
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x |
Y |
z |
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U(M) = J(2x + yozo) dx + JxZo dy + |
J(xy + 2z) dz = |
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Xo |
Yo |
Zo |
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=(x 2 + Yozox) IX +xyzol Y |
+(xyz + z2)IZ = |
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Xo |
Yo |
Zo |
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= (x2 + Yozox - |
x6 - Yozoxo) + (xyzo - |
xyozo) + (xyz + z2 - |
xyzo - |
z5) = |
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= x2 + xyz + z2 - (x6 + xoYozo + z5) = x 2 + xyz + z2 + C. |
• |
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51a.l/,JI1omCJI .l!U c.I!eaY'lOw,ue ae'ICmOpH'bte no.II.R nomeH'qua.l!'bH'bt.MU?
5.5.2. |
F = r. |
5.5.3. |
F = |
xi + yxj + zyk. |
5.5.4. |
F(x2, _y2,XZ). |
5.5.5. |
F = |
xyi - zj + xk. |
5.5.6.F = y2(1 - z)i + 2xy(1 - z)j - (xy2 - 3z2)k.
IIo'ICa3am'b, "tmo c.I!eaY'lOw,ue ae'ICmOpH'bte no.II.R nomeH'qua.l!'bH'bt, u Hai1.mu ux nomeH'qua.l!'bt:
5.5.7. F = x2i + y2j + z 2k. |
5.5.8. F = yzi + xzj + yxk. |
5.5.9.F(z - 2x,z - 2y,x + y).
5.5.10.F(y2 z3, 2xyz3 + z2, 3xy2 z2 + 2yz + 1).
5.5.11.IIoKa3aTb, qTO nJIOCKOe nOJIe
F(2 xy3 + 2xy sin(x2 y), 3x2y2 + x 2 sin(x2 y))
nOTeHIJ:HaJIbHO, H HaiiTH ero nOTeHIJ:HaJI.
267
5.5.12. |
He HCnOJIb3yg TeopeMY 5.3 nOKa3aTb, qTO POTOP nOTeHIJ;HaJIbHOro |
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nOJIg paBeH HYJIIO. |
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5.5.13. |
Henocpe,n:cTBeHHbIM BbIqHCJIeHHeM nOKa3aTb, qTO u:HPKYJIgIJ;Hg |
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rJIa,II,Koro nOTeHIJ;HaJIbHOro nOJIg F B,n:OJIb JII060fi 3aMKHYTofi KpH- |
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Bofi L paBHa O. |
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5.5.14. |
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HafiTH IJ;HPKYJIgIJ;HIO BeKTopHoro nOJIg F( yz2, xz2, 2xyz) B,n:OJIb 9JI- |
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JIHnca: x 2 + y2 = 4, x + 2y + 3z = 6. |
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5.5.15. |
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HafiTH IJ;HPKYJIgIJ;HIO BeKTopHoro nOJIg F(3x2y2z,2x3yz,x3y2) |
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B,n:OJIb KOHTypa ABC, r,n:e A(1, 0, 0), B(O, 1,0), C(O, 0,1). |
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5.5.16. |
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BbIqHCJIHTb pa60TY CHJIOBOrO nOJIg |
F(yz, xz, yx) B,n:OJIb |
o,n:Horo |
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BHTKa BHHTOBOfi JIHHHH x = cos t, Y = |
sin t, z = t (0 ::::; t ::::; |
211"). |
Q IIpH t |
= °nOJIyqHM HaqaJIbHYIO TOqKY KPHBOfi Mi (1,0,0), npH t |
= 211" - |
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KOHeqHYIO TOqKY M2(1, 0, 211"). TaK KaK BeKTopHoe nOJIe nOTeHIJ;HaJIbHO (CM. 3a,II,aqy 5.5.8), TO pa60Ta CHJIOBOro nOJIg He 3aBHCHT OT clJOPMbI nYTH. II09T0My BbI6epeM B KaqeCTBe nYTH MiM2 npgMOJIHHefiHbIfi OTpe30K. B,n:OJIb Hero
x = 1, y = 0, dx = |
dy = 0, H, CJIe,n:OBaTeJIbHO, pa60Ta |
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A = |
! |
yz dx + xz dy + yx dz = !211" |
0 dz = o. |
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M,M2 |
0 |
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.r:wyrHM cnoco6oM pa60TY MO}KHO 6bIJIO 6bI HafiTH KaK pa3HOCTb nOTeHIJ;H-
aJIOB B TOqKaX M2 H Mi. ,lJ;JIg 9TOro Haxo,n:HM CHaqaJIa nOTeHIJ;HaJI U = |
xyz |
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(CM. 3a,n:aqy 5.5.8). Tor,n:a A = U(M2) - U(Mt} = |
1·0·211" - 1·0·0= O. |
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5.5.17. |
BblqHCJIHTb pa60TY BeKTopHoro nOJIg F(z3 - |
y3,-3xy2,3xz2) OT |
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TOqKH A(1, 1, 1) ,n:o TOqKH B(2, 0,1). |
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2xz2,x - 2y,2x2z) |
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5.5.18. |
BbIqHCJIHTb pa60TY BeKTopHoro nOJIg |
F(y + |
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B,n:OJIb nOJIyoKpY}KHOCTH 60JIblIIOro pa,II,Hyca cclJepbI (1 + X)2 + y2 + z2 = 1
OT TOqKH A(-1, 1,0),n:o TOqKH B(-1, -1,0).
5.5.19.F(xy, -y - x, z - zy). 5.5.20. F(x2yz, 2xyz, -z2(xy + x)).
5.5.21.F = (y2 + z2)i - (xy + z3)j + (y2 + zx)k.
5.5.22.F = (x 2yz - x 3)i + yx3j + (x 2z - y)k.
5.5.23. |
.HBJIgeTCg JIH npOCTpaHCTBeHHoe BeKTopHoe nOJIe r x |
c |
(r,n:e c - |
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nOCTOgHHbIfi HeHYJIeBofi BeKTop): |
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a) nOTeHIJ;HaJIbHbIM; |
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6) COJIeHoH,n:aJIbHbIM? |
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5.5.24. |
.HBJIgeTCg JIH npocTpaHcTBeHHoe BeKTopHoe nOJIe F = |
} |
. r: |
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a) nOTeHIJ;HaJIbHbIM; |
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6) COJIeHoH,n:aJIbHbIM? |
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5.5.25. |
BbIqHCJIHTb nOTOK BeKTopHoro nOJIg F(xy2 - Z, -xy + |
z, zx - zy2) |
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qepe3 nOBepxHocTb 9JIJIHnCOH,n:a |
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222 |
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!L+~+~=1 |
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a2 |
b2 |
c2 |
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B HanpaBJIeHHH BHelliHefi HOPMaJIH.
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fI6.!1.H.10mc.R JtU cJteay'lOw,ue nOAA nomeH'quaJtbH'btMU?
5.5.26.F = (yz2 - 1)i + xz2j + 2xyzk.
5.5.27.F = (x 2 + y - z)i + (xy - xz)j + x 2zk.
5.5.28.F = cos(2y + 3z)i - 2ysin(2y + 3z)j + 3zsin(2y + 3z)k.
5.5.29.IIoKa3aTb, 'ITOBeKTopHoe nOJIe
F(yz(2x + y + z), xz(x + 2y + z), xy(x + y + 2z))
nOTeHIJ;HaJIbHO, H Hail:TH ero nOTeHIJ;HaJI.
5.5.30. IIoKMaTb, 'ITOBeKTopHoe nOJIe
F = (6xy - 2x)i + (3x2 - 2z)j + (1 - 2y)k
nOTeHIJ;HaJIbHO, H Hail:TH ero nOTeHIJ;HaJI.
5.5.31.F = (x 2 - yz + 2)i - 2xyj + (yx 3 - 1)k.
5.5.32.F(xy - yz + xz)i + (yz - xz + xy)j + (xz - xy + yz)k.
5.5.33.F(x2y,y2 z - y2x,xy - yz2).
B'bt"l,UcJtumb pa60my 6e'K:mOpHOeo nOAA F om mo"l,'K:U A ao mo"l,'K:U B:
5.5.34.F(3x2,2y,1), A(1,2,-1), B(0,1,1).
5.5.35.F(y2 + 2xz,z2 + 2xy,x2 + 2yz), A(O,O,O), B(1, -1, 1).
5.5.36.F = r . r, r.IJ:e r - Pa;:J;Hyc-BeKTop TO'lKH, a r = Irl, A(O,O,O),
B(6,2,3).
5.5.37.F = 4r2 . r, r.IJ:e r - Pa;:J;HYC-BeKTOp TO'lKHH r = Irl, A(0,3,4),
B(3,4,0).
5.5.38.IIoKa3aTb, 'ITOeCJIH BeKTopHbIe nOJI}! F 1 , F2 nOTeHIJ;HaJIbHbI H C -
'1HCJIO,TO Fl + F2 H C· Fl - TaK)Ke nOTeHIJ;HaJIbHbIe BeKTopHbIe nOJI}!.
5.5.39. IIoKa3aTb, 'ITOeCJIH BeKTopHbIe nOJI}! Fl H F2 COJIeHOH.IJ:aJIbHbI, TO
Cl . Fl + C2 . F2 - COJIeHOH.IJ:aJIbHOe BeKTopHoe nOJIe (Cl H C2 -
HeKOTopbIe KOHCTaHTbI).
KOHTponbHble BonpOCbl M 60nee CnO)l(Hbie saAaHMR
5.5.40. IIpHBecTH npHMep BeKTopHoro nOJI}!:
a) nOTeHIJ;HaJIbHOro H COJIeHOH.IJ:aJIbHOrO;
6) nOTeHIJ;HaJIbHOrO, HO He COJIeHOH.IJ:aJIbHOrO;
B)He nOTeHIJ;HaJIbHOrO, HO COJIeHOH.IJ:aJIbHOrO;
r)He nOTeHIJ;HaJIbHOrO H He COJIeHOH.IJ:aJIbHoro.
5.5.41.IIoKa3aTb, 'ITOnOTeHIJ;HaJI U nOTeHIJ;HaJIbHOro H COJIeHOH.IJ:aJIbHOro nOJI}! F Y.IJ:OBJIeTBOp}!eT ypaBHeHHIO JIanJIaca: b..U = 0.
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