Сборник задач по высшей математике 2 том
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Puc. 53 |
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,n:BOtiHbIM HHTerpaJIOM Heo6xo,n:HMO nOCTaBHTb 3HaK MHHYC: |
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I-x |
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II= IlydZdX=- II(Z+X-l)dxdZ= IdX |
I(Z+X-l)dZ= |
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(S,n) |
Dzz |
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= IGZ2+XZ-Z)lo-X dx= 1[~(1-X)2+X(1-X)-(1-X)] dx= |
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=/(_lX2+X_1) dx= (_X3 +X2 _~) 11 =_1 |
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o
B) 1bo6pasHM nOBepXHOCTb S BMeCTe C Tpe6yeMoit B YCJIOBHH 3a,n:aqR
HOPMaJIbIO Ha pHC. 55.
113 reOMeTpHqeCKHX c006p8JKeHHit nOHflTHO, qTO e,n:HHHqHM HOPMaJIb n
(T. K. OHa - BHeWHfIfi HOPMaJIb) o6pa3yeT Tyrroti yrOJI C OCbIO Oz. TaIOKe flCHO, qTO OHa o6pasyeT OCTPblit yroJI C OCbIO Ox B Tex TOqKaX, r,n:e x ~ 0 H Tynoti - B Tex, r,n:e x < O. AHaJIOrHqHO, n o6pa3yeT OCTPblti (Tyrroit) yroJI C OCbIO Oy B TOqKaX, r,n:e BbIIlOJIHfleTCfI HepaBeHcTBo y > 0 (y < 0). ,n:JIfi BblqHCJIeHHfI nOTOKa BeKTopHoro rrOJIfi HarrHweM HHTerpaJI II po,n:a:
II = II Pdydz + Qdzdx + Rdxdy = II dydz + dzdx + zdxdy =
(S,n) (S,n)
= II dydz + II dzdx + If zdxdy.
(S,n) (S,n) (S,n)
BblqHCJIHM K8JK,n:blit H3 Tpex HHTerpaJIOB OT,n:eJIbHO. .ll:JIfi BblqHCJIeHHfI HHTe-
rpaJIa
II dydz
(S,n)
250
pa306beM nOBepXHOCTb 8 Ha .n;Be '1acTII:81 II 82 nJIOCKOCTblO Ozy (81 OTBe- 'meTTOft '1acTIInapa6oJIoII.n;a, r.n;e x ~ 0). Heo6xo.n;IIMOCTb pa3611eHlljI npo-
,UIIKTOBaHa, KaK Y)Ke OTMe'laJIOCbBbIme, TeM <paKTOPOM, 'ITOHOPMaJIb n Ha
81 o6pa3yeT OCTPbIft yroJI C OCblO Ox (T.e. COSet> 0), |
aHa 82 - Tynoft. |
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I1poeKII.lleft II 81 II |
82 Ha nJIOCKOCTb Ozy jlBJIjleTCjI o.n;Ha II Ta )Ke 06JIacTb |
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Dzy , nOKa3aHHM Ha p"C. 56. CJIe.n;oBaTeJIbHO, |
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IIdydz = |
II dydz + |
II dydz = IldydZ - |
Ildydz = o. |
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(8,0) |
(8,,0) |
(82,0) |
D. y |
D. y |
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--- --) |
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_.... ,' |
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:' z=l
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z=l
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Puc. 55 |
Puc. 56 |
3HaK MIIHyC nepe.n; BTOPbIM .n;BOftHbIM IIHTerpaJIOM nOCTaBJIeH nOCTOJIbKY, noCKOJIbKY Ha 82 HOPMaJIb 06pa3yeT Tynoft yroJI C OCblO OX (IIJIII, 'ITOTO )Ke caMoe, COSet < 0). 11:3 coo6proKeHllft C"MMeTp"" nOHjlTHO, 'ITOII
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II dzdx = O. |
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(8,0) |
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OCTaJIOCb BbI'IIICJIIITb |
II z'dxdy. |
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(8,0) |
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KaK OTMe'leHOBbIme, COS'Y< O. IIo9TOMY IIMeeM: |
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II zdxdy = - |
II(x 2 +y2)dxdy, |
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(8,0) |
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D 2y |
r.n;e Dxy - npOeKII.llj1 nOBepXHOCTII 8 Ha nJIOCKOCTb xOy (oHa 11306proKeHa Ha p"C. 57). ,UJIjI BbI'IIICJIeHlljInOCJIe.n;Hero IIHTerpaJIa nepeft.n;eM K nOJIjlpHbIM KOop.n;IIHaTaM:
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211" |
1 |
- II (x 2 + y2) dxdy = - |
II p2. p dpdcp = - |
I |
dcp I p3 dp = - ~. |
D 2y |
D~'I' |
0 |
0 |
251
Puc. |
57 |
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TaKHM o6pa30M, nOTOK BeKTopHoro nOJUI paBeH -i. |
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B'b/,"I,uc.ltum'b nomo'll: 6e'll:mOp'lt020 nOJ!Jl |
F(P, Q, R) "I,epe3 |
n06epX'ltOCm'b S 6 |
cmopo'lty, onpeiJeJ!JleMY70 'ltOpMa.lt'b70 n |
'II: n06epX'ltOCmu S, |
ec.ltu: |
5.3.2.F(2, -1, 1), S - KBa,n;paT: 0 ~ x ~ 1, 0 ~ y ~ 1, z = 1, HOPMa.JIb
n HanpaBJIeHa BBepx.
5.3.3.F(-y,x,z), S - '1acTbIJ;HJIHH.n;pa x 2 + y2 = 1, 3aKJIIO'IeHHMMe-
:lK,n,y nJIOCKOCTaMH z = 0 H Z = 1, n - BHemHaa HOPMa.JIb.
5.3.4.F(x, y, 0), S - '1acTbnJIOCKOCTH y + z = 1, pacnOJIO:lKeHHM B
nepBOM OKTaHTe Me:lK,n,y nJIOCKOCTaMH x = 0 H X = 1, n 06pa3yeT
OCTPbdi yroJI C OCblO Oy.
5.3.5.F(x, y, z), S - nOJIyc<pepa x 2 + y2 + z2 = R2, pacnOJIO:lKeHHM B
nOJIynpocTpaHCTBe z ;:: 0, n 06pa3yeT OCTPbrii yroJI C OCblO Oz.
5.3.6.F (y - z, z - x, x - y), S - '1acTbKOHyca z2 = x 2+ y2 , 3aKJIIO'IeHHM
Me:lK,n,y nJIOCKOCTaMH z = 0 H Z = 2, n o6pa3yeT Tynoii yroJI C OCblO
Oz.
5.3.7.F(1, 0, 0), S - nOBepXHOCTb nHpaMH.n;bI, OrpaHH'IeHHOiinJIOCKo-
CTaMH x + y + z = 1, x = 0, y = 0, z = o.
5.3.8. |
F(xy, yz, xz), S - '1aCTbc<pepbI x 2+y2+Z2 = R2, pacnOJIO:lKeHHruI |
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B nepBOM OKTaHTe, n |
- BHemHaa HOPMa.JIb K c<pepe. |
5.3.9. |
IIoJIb3yacb <p0pMYJIoii raycca-0cTporpa,n;cKoro BbI'IHCJIHTbnOTOK |
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BeKTopHoro F nOJIa '1epe33aMKHyTylO nOBepXHOCTb S B HanpaBJIe- |
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HHH BHemHeii HOPMa.JIH: |
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a) F = x 2 . i + y2 . j |
+ Z2 . k, S - nOBepXHOCTb Ky6a 0 ~ x ~ a, |
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o~ y ~ a, 0 ~ z ~ aj |
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6) F(x(z - y), y(x - |
z), z(y - x)), S - npOH3BOJIbHM 3aMKHYTrui |
nOBepXHOCTb.
Q a) BbI'IHCJIHM.n;HBepreHIJ;HIO nOJIa:
div F = (x2)~ + (y2)~ + (z2)~ = 2(x + y + z).
BOCnOJIb30BaBmHcb <p0pMYJIoii raycca-0cTporpa.n;cKoro, BbI'IHCJIHM nOTOK
252
6eKTOpHoro nOJI:H:
II= !!F.ndS= !!!divFdV=2 !!!(x+y+z)dxdydz=
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= 2 !dx !dy j{X + y + z) dz = 3a4. |
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IIpOMe:lKYTOqHble BblqIlCJIeHII:H, B CIIJIY IIX OqeBII)l.HOCTII, onyrn;eHbI.
6) IIycTb V - TeJIO, OrpaHllqeHHOe nOBepXHOCTblO S. Tor)l.a
II = !!!divFdV.
Ho |
v |
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divF = |
[x(z-y)l~+[y(x-z)l~+[z(y-x)l~ = |
(z-y)+(x-z)+(y-x) = O. |
CJIe)l.OBaTeJIbHO, II nOTOK paBeH O. |
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5.3.10. |
,il;oKa3aTb, qTO nOTOK nOCTO:HHHOro BeKTopHoro nOJI:H F = c qepe3 |
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JII06ylO 3aMKHyTylO nOBepxHocTb paBeH O.
5.3.11. ,il;oKa:lKIITe, nOJIb3Y:HCb cPopMYJIoii raycca-OCTpOrpa)l.CKoro, qTO nOTOK pa)l.lIyca-BeKTopa r qepe3 JII06ylO 3aMKHyTylO nOBepXHOCTb B HanpaBJIeHIIII BHeIIIHeii HOPMaJIII paBeH YTpoeHHoMY 06'beMYTeJIa, orpaHllqeHHOrO 9TOii nOBepXHOCTblO.
B 3ada"tax 5.3.12-5.3.14 6U"tUC.II.Um'b nomo'll: 6e'll:mop'ttoeo no.!I.R F "tepe3 3a- M'II:'ttymy70 n06epx'ttOCm'b S 6 'ttanpa6.11.e'ttUU 6'ttew,'tteti. 'ttOPMa.ll.U, ec.II.u:
5.3.12. |
F(x, z, y), S - |
nOJIHa:H |
nOBepXHOCTb |
n:IIJIIIH)l.pa x 2 + y2 = |
R2, |
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z=O,z=H. |
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5.3.13. |
F |
= |
xz . i |
+ y2 . j + X |
• k, S - nOJIHa:H nOBepxHocTb np"3MbI, |
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OrpaHllqeHHOii nJIOCKOCT:HMII X + y = 1, X |
= |
0, y = 0, z = |
0, |
z = 1. |
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5.3.14. |
F |
= |
(y2 - |
z) . i + xy . j |
- |
(y + x) . k, S |
- |
nOJIHa:H nOBepxHocTb |
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nllpaMII)l.bI, |
OrpaHllqeHHOii |
nJIOCKOCT:HMII |
X |
+ y + z = |
1, X |
= |
0, |
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y = 0, z = O. |
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5.3.15. |
MCnOJIb3Y:H 3a)l.aqy 5.3.11, HaiiTII nOTOK pa)l.lIyca-BeKTopa r |
qepe3 |
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nOJIHYIO nOBepXHOCTb nllpaMII)l.bI ABCD C BepIIIIIHaMII B TOqKax |
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A(-1,0,0), B(I, 1,0), C(I, -1,0), D(O, 2, 3). |
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5.3.16. |
HaiiTII nOTOK rpa)l.lIeHTa CKaJI:HpHOrO nOJI:H U = x 2 + y2 + z2 qepe3 |
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nOBepxHocTb YPOBH:H U = |
1 9Toro CKaJI:HPHOro nOJI:H B HanpaBJIe- |
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HIIII BHeIIIHeii HOPMaJIII. |
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5.3.17. |
HaiiTII nOTOK pOTopa BeKTopHoro nOJI:H F(yz, zx, xy) Qepe3 ccPepy |
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x 2 + y2 + Z2 = |
4 B HanpaBJIeHIIII BHeIIIHeii HOPMaJIII. |
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5.3.18. |
HaiiTII nOTOK BeKTopHoro nOJI:H F(x - |
1, Y + 3, z) Qepe3 60KOBYlO |
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nOBepXHOCTb KOHyca z2 = |
x 2 + y2, 3aKJIIOQeHHYlO Me:lK)l.y nJIOCK0- |
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1 B HanpaBJIeHIIII BHeIIIHeii HOPMaJIII.
~PacCMOTPIIM TeJIO V, rpaHlln:eii KOTOPOro CJIy:lKIIT KOHIIQeCKa:H nOBepx-
IiQcTb z2 = x 2 + y2 (S1) II nJIOCKOCTb z = 1 (S2) (CM. pllC. 58).
253
n |
z |
z=l |
y
x
Puc. 58
Ha IIOBepXHOCTH 8 = 81 U 82 , aBJUllOIlleitca 06'be)l.HHeHHeMIIoBepxHocTeit
81 1'182 , B03bMeM BHeWHlO1O HOPMaJIb n. llOTOK II qepe3 IIOBepXHOCTb 8 CKJIa- )l.bIBaeTCa 1'13IIOTOKOB III 1'1 II2 qepe3 IIoBepxHocTH 8 1 1'1 8 2 COOTBeTCTBeHHO. CJIe)l.OBaTeJIbHO, HHTepeCYIOIllHit Hac IIOTOK MO>KeT 6bITb Hait)l.eH KaK pa3HOCTb IIOTOKOB: III = II - II2. llOTOK II MO:>KeT 6bITb Hait)l.eH IIO cPopMYJIe raycca-OcTpOrpa)l.CKoro:
II = !!F. nd8 = !!!divFdV = 3 !!!dV.
8 v v
llOCJ1e)l.HHit HHTerpaJI IIpe)l.CTaBJIaeT co6oit 06'beMTeJIa V. TeJIo IIpe)l.CTaBJIa- eT co6oit KOHYC C BbICOTOit h = 1 1'1 Pa)l.HYCOM OCHOBaHHa R = 1. llo H3BeCTHOit 1'139JIeMeHTapHoit MaTeMaTHKH cPopMYJIe, ero 06'beMpaBeH ~7rR2 h = ~7r.
OTCIO)l.a II = 3 . ~7r = 'It. llOTOK II2 (Qepe3 IIJIOCKOCTb Z = 1) MO:>KeT 6bITb BbIQHCJIeH )l.OBOJIbHO IIpOCTO. BHewHeit e)l.HHHQHoit HOPMaJIblO K IIJIOCKOCTH aBJIaeTCa BeKTOp n(O, 0, 1). l109TOMY
II2 = !!F. nd8 = !!zd8.
82 82
llOCKOJIbKY Z = 1 Ha 8 2 , a 9JIeMeHT IIJIOIlla)l.H (d8) paBeH 9JIeMeHTY IIJIOIlla)l.H ee IIpoeKIIHH Ha IIJIOCKOCTb Oxy (dxdy), TO IIOCJIe)l.HHit HHTerpaJI CBO)l.HTCa K )l.BOitHOMY:
!! dxdy,
Dxy
r)l.e Dxy - KPyr C IIeHTpOM B HaQaJIe KOOp)l.HHaT 1'1pa)l.Hyca 1. OTOT HHTerpaJI BbIpa:>KaeT IIJIOIlla)l.b 9Toro Kpyra, KOTopaa paBHa 7r. CJIe)l.OBaTeJIbHO, HCK0MbIit IIOTOK Qepe3 KOHHQeCKYIO IIOBepXHOCTb paBeH III = II - II2 =
=0.
254
Haiimu nomo'll: 6e'll:mOp'ltoeo no.ll..f! F "tepe3 'lte3aM'II:'ltymy'lO n06epX'ltOCm'b S 6 'ltanpa6.n.e'ltUU 'ltOpMa.n.u n, ucno.n.'b3Y.R. rjjOPMY.n.y raycca-Ocmpoepaac'll:oeo:
5.3.19.F(I, 2, 3), S - 60KOBaJI nOBepXHOCTb KOHyca, OCblO KOTOPOro CJIy-
:lKHT OCb Oz, BepIIIHHa HaxO)l;HTCa B TOqKe M(h, 0, 0), a OCHOBa-
HHe - Kpyr pa;::J;Hyca R, JIe:lKaW;Hil: B nJIOCKOCTH Oxy.
5.3.20.F(x3, y3, 0), S - BepXHaa qaCTb c<I>epbI x 2 + y2 + Z2 = 1, pacno-
JIO:lKeHHaJI BbIme nJIOCKOCTH Oxy, n o6pa3yeT OCTPbIil: yrOJI C OCblO
Oz.
5.3.21. |
F(2x, -y, z), S - |
60KOBaJI nOBepxHocTb II,HJIHH)l;pa x 2 + y2 = R2, |
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pacnOJIO:lKeHHOrO Me:lK)l;y nJIOCKOCTaMH z = °H Z = H, n |
- |
BHem- |
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Haa HOPMaJIb. |
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5.3.22. |
F = zi +yk, S - |
qacTb nOBepXHOCTH napa60JIHqeCKoro II,HJIHH)l;pa |
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z = 1 - x 2 , OTCeqeHHaJI nJIOCKOCTaMH y = 0, y = 1, |
z = 0, n - |
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HOPMaJIb, o6pa3YlOW;aJI OCTPbIil: yroJI C OCblO Oz. |
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5.3.23. |
F = (y - l)i + j - |
yk, S - qacTb II,HJIHH)l;pa x 2 + y2 = 1, pacnOJIO- |
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:lKeHHaJI Me:lK)l;y nJIOCKOCTaMH z = °H X + y + z = 5, n |
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BHemHaa |
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HOPMaJIb. |
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5.3.24. |
Hail:TH nOTOK rpa;::J;HeHTa CKaJIapHoro nOJIa U = x 2 |
+ yz |
qepe3 |
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qaCTb c<I>epbI x 2 + y2 + Z2 = R2, Y ~ °B HanpaBJIeHHH e)l;HHHqHoil: |
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HOPMaJIH, o6pa3ylOw;eil: OCTPbIil: yrOJI C OCblO Oy.
B'b/,"tuc.n.um'b nomo'll: 6e'll:mOp'ltoeo no.ll..f! F "tepe3 n06epX'ltOCm'b S 6 cmopo'lty, onpeae.ll..f!eMY'lO eau'ltu"t'ltoii 'ltOpMa.n.'b'lO n 'II: n06epX'ltOCmu S:
5.3.25.F = xi - zj + y2k, S - npaMoyrOJIbHHK: °:::;; x :::;; 2, °:::;; y :::;; 1,
HOPMaJIb n HanpaBJIeHa BBepx. 5.3.26. F = x 2i - 2xyj + zk, S - c<I>epa:
n - BHemHaa HOPMaJIb.
(x -1)2 + (y - 2)2 + (z - 3)2 = 9,
5.3.27. F = (1 - yz)i + (1 + xz)j + 2(x + y)k, S - qaCTb napa60JIOH)l;a z = x 2 + y2, 3aKJIlOqeHHaJI Me:lK)l;y nJIOCKOCTaMH z = 0, z = 1, n -
HOPMaJIb, o6pa3YlOW;aJI Tynoil: yroJI C OCblO Oz.
5.3.28.F = zi + (1 - z)j + xyk, S - qacTb nJIOCKOCTH x + y = 1, orpa-
HHqeHHaJI nJIOCKOCTaMH z = 0, z = 1, x = 0, y = 0, n - HOPMaJIb,
o6pa3YlOW;aJI OCTPbIil: yroJI C OCblO Ox.
5.3.29.F(O,O,z), S - qacTb KOHyca Z2 = x 2 + y2, 3aKJIlOqeHHaa Me:lK)l;y
nJIOCKOCTaMH z = 0, z = 1, n - HOPMaJIb, o6pa3YlOW;aJI Tynoil:
yroJI C OCblO Oz.
5.3.30.F(x2, y2, Z2), S - 60KOBaJI nOBepxHocTb II,HJIHH)l;pa, 3aKJIlOqeHHaJI
Me:lK)l;y nJIOCKOCTaMH z = 0, z = 2, n - BHemHaa HOPMaJIb.
5.3.31. Hail:TH nOTOK pa;::J;Hyca-BeKTopa r qepe3 6oKOBYlO nOBepxHocTb
nHpaMH)l;bI, BepmHHa KOTOPOil: HaxO)l;HTCa B TOqKe A(4, 5, 3), a
255
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OCHOBaHHeM CJIy>KHT qeTbIpexyroJIbHHK C BepmHHaMH B(O, 0, 0); |
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C(1, 1,0), D(3, -1,0), E(2, -2,0). |
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5.3.32. |
HaiiTH rrOTOK BeKTopHoro rroml F(yz,x+2yz, Z2_Z) qepe3rrOBepx- |
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HOCTb rrap8.JIJIeJIerrHrre,ll;a, rrOCTpoeHHoro Ha BeKTopax OA, OB II |
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OC, r,ll;e 0(0,0,0), A(1,-2,1), B(3,2,1), C(1,0,-1). |
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KOHTponbHble Bonpocbl III 60nee CnO)l(Hbie 3aAaHIIIH |
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5.3.33. |
TIoKa3aTb, qTO rrOTOK rpa,ll;HeHTa CK8.JIjlpHOrO rrOJIjI U, jlBJIjllOIJJ;ero- |
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CjI rapMOHHqeCKOii <pYHKII.Heii (T. e. y,ll;OBJIeTBOpjllOIJJ;eii ypaBHeHHIO |
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t::.U = 0) qepe3 JII06ylO 3aMKHyTylO rrOBepXHOCTb paBeH 0. |
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5.3.34. |
TIoKa3aTb, qTO rrOTOK grad (c . r), r,ll;e r - |
Pa,ll;HYC-BeKTOp, a |
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c - <pHKcHpoBaHHbIii BeKTOp, qepe3 rrpOH3BOJIbHYlO 3aMKHyTylO |
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rrOBepXHOCTb paBeH 0. |
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5.3.35. |
HaiiTH rrOTOK rrOJIjI c x r qepe3 rrOBepXHOCTb c<pepbI x 2+y2+Z2=R2 |
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B HarrpaBJIeHHH BHemHeii HOPM8.JIH. |
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5.3.36. |
OTpe30K KPHBOii z = Vfj, JIe>KaIJJ;Hii B rrJIOCKOCTH Ozy Me>K,ll;y TOq- |
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KaMH 0(0,0,0) H A(O, 1, 1), BpaIJJ;MCb BOKpyr OCH Oz 06pa3yeT |
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rroBepxHocTb S. HaiiTH rrOTOK BeKTopHoro rrOJIjI F(y, x, z - 1) qe- |
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pe3 rrOBepXHOCTb S B HarrpaBJIeHHH BHemHeii HOPM8.JIH. |
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5.3.37. |
HaiiTH rrOTOK BeKTopHoro rrOJIjI F(x3 , y3, Z3) |
qepe3 c<pepy: |
a)x 2 + y2 + Z2 = R2;
6)x 2 - X + y2 + z2 =°B HarrpaBJIeHHH BHemHeii HOPM8.JIH.
5.3.38. |
HaiiTH rrOTOK BeKTopHoro rrOJIjI F ( x |
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B HarrpaBJIeHHH BHemHeii HOPM8.JIH. |
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5.3.39. |
TIOJIb3YjlCb ¢opMYJIoii |
raycca-OcTpOrpa,ll;CKOrO |
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r,ll;e Ve - map C II.eHTpOM B TOqKe M, a Se - OrpaHHqHBalOIJJ;M erO c¢epa, 1Yo 1- 06beM 9Toro mapa. I1crroJIb3Yjl 9TOT ¢aKT rrOKa3aTb, qTO ,ll;HBepreHII.HjI He 3aBHCHT OT BbI60pa rrpjlMoyrOJIbHOii CHCTeMbI KOOp,ll;HHaT.
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256
noml U B HanpaBJIemlH BHewHeit HOPMaJIH K nOBepXHOCTH S. ,[LoKa3aTb, 'ITO
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5.3.41. IIycTb U, V - )l.BIDK)I.bI HenpepbIBHO )l.H<p<pepeHIJ;HpyeMble CKaJIapHble nOJIa B npOCTpaHCTBeHHO O)l.HOCBa3Hoit 06JIaCTH O. IIycTb TeJIO V BMeCTe co cBoeit rpaHHIJ;eit S JIe)KHT B 0, a ~~ H ~~ -
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nOBepXHOCTH S. ,[LoKa3aTb, 'ITO: |
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§ 4. U.LJlPKYIUIU.LJI~ BEKTOPHOrO nOl1~ |
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IIycTb F = Pi + Qj + Rk - |
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BeKTopHoe nOJIe, 3a,n;aHHOe B HeKoTopoil: 06JIaCTH |
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n C lR3 , H <PYHKIIHH P(x, y, z), Q(x, y, z), R(x, y, z) - HenpephlBHO ,n;H<p<pepeHIIHpyeMhI B 06JIacTH n. IIycTb L - rJIa,n;KaH KpHBaH, pacnOJIOlKeHHaH B 06JIaCTH n.
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u; = fF. dr = fPdx+ Qdy+ Rdz.
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B CJIyqae, Kor,n;a BeKTopHoe nOJIe F(P, Q) - nJIOCKOe, ero IIHpKyJIHIIHH B,n;OJIb 3aMKHYTOil: KPHBOil: L 3a,n;aeTCH HHTerpaJIOM:
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TeopeMa 5.2 (CToKe). nYCTb S - rnaAKal'lopllleHTlllpyeMal'lnOBepXHOCTb, a L -
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KplllBal'l, l'IBmllOLllal'lCl'IrpaHlillIeili nOBepXHOCTIll S. |
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ilia ee CTOpOH. nYCTb BeKTopHoe none F(P, Q, R) - |
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PyeMo Ha S III L. TorAa |
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nplll'ieMHanpaBneHllle 06XOAa |
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BeKTopa n |
OHO npOIIICXOAIilT npOTIllB 'iacoBoiliCTpenKIll. |
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JIeBhIiI: HHTerpaJI B <popMYJIe (4.2) rrpe)l:CTaBJIHeT co6oil: IJ;HPKYJIHIJ;HIO BeKTOpHoro rrOJIH F B)l:OJIb KOHTypa L, a rrpaBbliI: - rrOTOK pOTopa ,noro rrOJIH qepea rroBepXHOCTb S. IIo9ToMY <P0PMYJIY CToKca Y)l:06HO 3arrHChIBaTb B BeKTopHoil: <p0pMe:
IF.dr= jjrotF.ndS= jj(rotF)ndS,
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T. e. rrOTOK pOTopa BeKTopHoro rrOJIH F qepe3 opHeHTHpOBaHHYIO rrOBepXHOCTb S paBeH IJ;HPKYJIHIJ;HH rrOJIH F B)l:OJIb KOHTypa L 9TOil: rrOBepXHOCTH (rrpOXO)l:HMOrO B rroJIOlKHTeJIbHOM HarrpaBJIeHHH). MCrrOJIb3YH orrepaTOp raMHJIbTOHa, <P0PMYJIY CTOKca MOlKHO 3arrHcaTb B BH)l:e:
IF. dr = jjCV' x F)·ndS.
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B CJIyqae, KOr)l:a BeKTopHoe rrOJIe F(P, Q) - rrJIOCKOe, <popMYJIa CTOKca rrpHHHMaeT BH)l: <P0PMYJIbI rpHHa:
IPdx+Qdy= jj (~~ - ~~) dxdy.
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<t>OPMYJIY CTOKca qaCTO rrpHMeHHIOT !l:JlH BblqHCJIeHHH IJ;HPKYJIHIJ;HH BeKTopHoro rrOJIH. O)l:HaKO CJIe)l:yeT rrOMHHTb, qTO )l:JIH Toro, qTo6bI MOlKHO 6bIJIO rrpHMeHHTb <P0PMYJIY CTOKca K KOHTYPY Len, Heo6xo)l:HMO, qTo6bI HaIIIJIaCb rrOBepXHOCTb S,
IJ;eJIHKOM JIelKaru;ax B n, rpaHHIJ;eil: KOTOPOil: 6hIJI 6bI KOHTYP L.
06JIacTb n, 06JIa)l:aIOru;aH TaKHM CBOil:CTBOM, Ha3bIBaeTCH n06epxnocmno oanOC6.R3noiJ. 06JIacTbIO. BOJIee TOqHO, 06JIaCTb n c R3 Ha3bIBaeTCH n06epxnocmno OanOC6.R3noiJ., eCJIH !l:JlH JII06oro 3aMKHYToro KOHTypa Len Hail:)l:eTCH rrOBepXHOCTb
Sen, rpaHHIJ;eil: KOToporo HBJIHeTCH KOHTYP L.
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5.4.1. |
HaihH pa60TY nJIOCKoro BeKTopHoro nOJIa |
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qacTb napa60JIbI y = x 2, KOHIreBbIMH TOqKaMH |
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KOTOpoii CJIY)I{aT TOqKH A(O,O) H B(2,4); |
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6) F(y,x), |
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apKa n:HKJIOH.n;bI |
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sint, |
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= 1 - cost, |
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o:::;; t :::;; 211". |
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o a) BblqHCJIHM pa60TY nOJIa, npHMeHaa <P0PMYJIY (4.1): |
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A = / x 2 dx + yx dy |
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(T. K. nOJIe nJIOCKOe, |
TO R = 0). nOCKOJIbKY B.n;OJIb KPHBOii |
L nepeMeHHble |
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CBa3aHbI paBeHCTBOM Y = |
x 2, TO dy = 2x dx, |
H KPHBOJIHHeiiHblii HHTerpaJI |
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CBo.n;HTca K onpe.n;eJIeHHoMY HHTerpaJIY: |
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A = |
/x 2 dx + x 2 . X . 2x dx = f(x 2 + 2X4) dx = (lx3 + ~x5) |
1 = |
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211" |
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A = |
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/[(1- cost)(1- cost) + (t - sint) sintjdt = |
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211" |
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211" |
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/[1-2 cost+cos2 t+tsint-sin2 tj dt = |
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5.4.2.HaiiTH pa60TY BeKTopHoro nOJIa F(x, y, z) B.n;OJIb JIHHHH L, aBJIaIOru:eiica nepeCeqeHHeM napa60JIHqeCKoro n:HJIHH.n;pa z = y2 C nJIOC-
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KOCTbIO Z + X = |
1 OT TOqKH A(O, 1, 1) )1:0 TOqKH B(I, 0, 0). |
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o 3a.n;a.n;HM JIHHHIO L napaMeTpHqeCKH: nOJIO)l{HB y = t, nOJIyqHM Z = |
t2 , |
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a x = |
1 - Z = 1 - t2. Tor.n;a dx = |
-2tdt, |
dy = dt, dz = |
2tdt. TOqKe |
A |
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COOTBeTcTByeT 3HaqeHHe napaMeTpa t = 1, a TOqKe B - |
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o. |
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TaKHM 06Pa30M, |
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o |
t2) . (- |
2t) dt + t dt + t2 . 2t dt = |
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A = |
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P dx + Qdy + R dz = PI - |
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= j(-2t+2t3 +t+23)dt= /(4t 3 -t)dt= (t4_ t;) 1:= |
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=O-(I-~)=-~. • |
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HaiJ.mu pa60my n.llOC'Il:020 |
6e'll:mOp'H.020 no.!!Sl F = Pi + Qj 600.ll'b 'Il:pU60iJ. L: |
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5.4.3. |
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F = -ti+ lj, L - qacTb OKPY)I{HOCTH x 2 +y2 = |
R2, JIe)l{aru:aa B |
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I qeTBepTH H np06eraeMaa npOTHB qacoBoii CTpeJIKH.
259