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VectorAnalysis&Field,()()VectorCalculus,Marsden&Tromba,FreemanIntroductiontoelectromagneticfileds(Secondedition),ClaytonR.Paul&SyedA.Nasar,Mc-GrawHillVectorAnalysis&FieldAbstract(FromVectorCalculus,Marsden&Tromba,Freeman)Wearenowpreparedtotietogetherthevectordifferentialcalculusandthevectorintegralcalculus.ThiswillbedonebymeansoftheimportanttheoremsofGreen,GaussandStokes.Weshallalsopointoutsomeofthephysicalapplicationsofthesetheoremstostudytheelectricityandmagnetism,hydrodynamics,heatconduction,anddifferentialequations.Green()DDYXXdxYdydxdyxy∂∂∂−=+∂∂∫∫∫()DDYXXdxYdydxdyxy∂∂∂+=−∂∂∫∫∫Ostrogradsky-GaussStokes1GreenJordanGreen221223Stokes31Stokes324Ostrogradsky-Gauss1/16VectorAnalysis&Field41Gauss425Hamilton66162636.4PoissonLaplace1.GreenGreenJordanlR2R2\l...Jordan.DDD.ll.ll+GreenVx(,)((,),(,))yXxyYxy=DC(1)DD∂()DDYXXdxYdydxdyxy∂∂∂+=−∂∂∫∫∫GreenDD.∂,xD:{(12,)|()(),}xyyxyyxaxb≤≤≤≤2/16VectorAnalysis&FieldAA’B’BX012'''''21(,())(,())[(,())(,())]baabABBAAABBAbaXdxXdxXdxXxyxdxXxyxdxXxyxXxyxdx=+=+=−−∫∫∫∫∫∫y1,y2.D21()21()(,())(,())byxbayxaDXXdxdydxdyXxyxXxyxdxyy∂∂==−∂∂∫∫∫∫∫,DDXXdxdxdyy∂∂=−∂∫∫∫GreenDDYYdydxdyx∂∂=∂∫∫∫yxD:{(21,)|()(),}xyxyxxycyd≤≤≤≤C’CD’DY012()12()[((),)((),)]dxydcxycDYYdxdydydxYxyyYxyydyxx∂∂==−∂∂∫∫∫∫∫3/16VectorAnalysis&Field12'''''12((),)((),)[((),)((),)]dccdCCDDCDCCDdcYdyYdyYdyXxyydxXxyydxXxyyXxyydx=+=+=−∫∫∫∫∫∫DDYYdydxdyx∂∂=∂∫∫∫DDXXdxdxdyy∂∂=−∂∫∫∫DDYYdydxdyx∂∂=∂∫∫∫Green()DDYXXdxYdydxdyxy∂∂∂+=−∂∂∫∫∫DD..D.Green()DDYXXdxYdydxdyxy∂∂∂−=+∂∂∫∫∫GaussStokes.YXxy∂∂−∂∂YXxy∂∂+∂∂.2.((,),(,))XxyYxy=VD(,)(,)(,)uXxyxyDvYxy=∈=X,YCD(tangent)(normal)nTn=×nTk.4/16VectorAnalysis&Field21CCC.Cds⋅∫VTVVΓds⋅∫VnCVΦ..C()()xxsyys==dxdydsds=+Tij(,)(,)dxdyXxyYxydsds⋅=+VT………………………………………….CCCdsXdxYdy⋅=+∫∫∫VT()(dxdydydxdsdsdsds=×=+×=−kijkij)nT………………………………………….CCdsXdyYdx⋅=−∫∫VnC.22Green()DDYXXdxYdydxdyxy∂∂∂+=−∂∂∫∫∫YXxy∂∂−∂∂rotdxdy∫VrotVGreenDDds∂⋅=∫∫VTGreen()DDYXXdxYdydxdyxy∂∂∂−=+∂∂∫∫∫YXxy∂∂+∂∂dxdydivVdivDDds∂⋅=∫∫∫VnVvT00KYXxy∂∂−∂∂.n.3Stokes5/16VectorAnalysis&FieldStokesGreen.31StokesStokesS∂S∂SS.S+S∂+S∂S.S+((,,),(,,),(,,))XxyzYxyzZxyz∂SC(1)()()()SSZYYXXZdydzdxdydzdxyzxyzxXdxYdyZdz++∂∂∂∂∂∂∂−+−+−∂∂∂∂∂∂=++∫∫∫vStokes(,,)SSXXdzdxdxdyXxyzdxzy++∂∂∂−=∂∂∫∫∫vS(,)zzxy=,SxOyDxy∂S+L+(,,)(,,(,))SSXxyzdxXxyzxydx++∂=∫∫SZ2π()cosdSdxdyγ=()cosdSdxdzβ=Green[0(,,(,))]xyDXxyzxydxdyy∂−∂∫∫(xyDXXzdxdyyzy∂∂∂−+∂∂∂∫∫)…………………(*)dS(coscos)SSXXXXdzdxdxdydSzyzyβγ++∂∂∂∂−=−∂∂∂∂∫∫∫∫6/16VectorAnalysis&Field22cos1zyXXzyβ∂−∂=∂∂++∂∂221s1XXcozyγ=∂∂++∂∂(coscos)(()coscos)()cos()SSSSXXdSzyXzXdSzyyXzXdSzyyXzXdxdyzyyβγγγγ++++∂∂−∂∂∂∂∂=−−∂∂∂∂∂∂=−+∂∂∂∂∂∂=−+∂∂∂∫∫∫∫∫∫∫∫Green(*)S(,)yyzx=(,)xxyz=(,,)SSYYdxdydydzYxyzdyxz++∂∂∂−=∂∂∫∫∫v(,,)SSZZdzdydxdzZxyzdzyx++∂∂∂−=∂∂∫∫∫vStokes..Stokes.(,,)(,,(,))()xyDSSXXzXxyzdxXxyzxydxdxdyyzy++∂∂∂∂==−+∂∂∂∫∫∫∫()cosdSdxdyγ=()cosdSdzdxβ=coscosdzdxdxdyβγ=z=z(x,y)n(,,1zzxy)∂∂−∂∂7/16VectorAnalysis&Field02222(cos,cos,cos),,11()()1()()zzyxzzzzyxyxαβγ∂∂−−∂∂==∂∂∂∂++++∂∂∂∂nn0//ncoscoscos1zzxyαβγ==∂∂−∂∂coscoszyβγ∂=−∂zdxdyy∂=−∂dzdx()xyxyDDxxzxxdxdydxdydxdzyzyyz∂∂∂∂∂−+=−+∂∂∂∂∂∫∫∫∫.2πγStokes.StokesSSdydzdzdxdxdyXdxYdyZdzxyzXYZ++∂∂∂∂=++∂∂∂∫∫∫StokescoscoscosSSdSXdxYdyZdzxyzXYZαβγ++∂∂∂∂=++∂∂∂∫∫∫S+xOyStokesGreen.32StokesGreen.Vrot(,,)ZYXZYXyzzxxy∂∂∂∂∂∂=−−−∂∂∂∂∂∂V8/16VectorAnalysis&FieldrotxyzXYZ∂∂∂=∂∂∂ijkV.SXdxYdyZdz+∂++∫∂S+StokesrotSSdd++∂⋅=⋅∫∫∫VrVs.4.Ostrogradsky-GaussGaussGauss.41Gauss(,,)((,,),(,,),(,,))xyzXxyzYxyzZxyz=V(,,ddxdyd)z=rGauss∂ΩYx(,,)Xxyz(,,)yz(,,)ZxyzC(1)()(coscoscos)XYZdvXdydzYdzdxZdxdyxyzXYZαβγ++Ω∂Ω∂Ω∂∂∂++=++∂∂∂=++∫∫∫∫∫∫∫wwdS∂Ω.+GaussGreenGreen.GaussXYZZdvZdxdyz+Ω∂Ω∂=∂∫∫∫∫∫w.9/16VectorAnalysis&Field∂ΩS+1S2S3SzS11:(,Szzxy=))22:(,zxy=3XYxyD21(,)21(,)[(,,(,))(,,(,))]xyxyzxyzxyDDZZdvdxdydzZxyzxyZxyzxydxdyzzΩ∂∂==−∂∂∫∫∫∫∫∫∫∫(*)1231221(,,(,)(,,(,)0[(,,(,)(,,(,)]xyxyxySSSDDDZdxdyZdxdyZdxdyZdxdyZxyzxydxdyZxyzxydxdyZxyzxyZxyzxydxdy+∂Ω=++=−++=−∫∫∫∫∫∫∫∫∫∫∫∫∫∫w(*).ZdvZdxdyz+Ω∂Ω∂=∂∫∫∫∫∫wYdvYdxdzy+Ω∂Ω∂=∂∫∫∫∫∫wXdvXdydzx+Ω∂Ω∂=∂∫∫∫∫∫wGauss()XYZdvXdydzYdzdxZdxdyxyz+Ω∂Ω∂∂∂++=++∂∂∂∫∫∫∫∫wGauss.420()XYZdsdxdydzxyz+Ω∂Ω∂∂∂Φ=⋅=++∂∂∂∫∫∫∫∫Vnw10/16VectorAnalysis&FieldXYZxyz∂∂∂++∂∂∂((,,),(,,),(,,))XxyzYxyzZxyz=V.divGaussVdivdSdv+Ω∂Ω⋅=∫∫∫∫∫VnVw.5.HamiltonHamilton∇xyz∂∂∂∇=++∂∂∂ijkNabla.u=u(x,y,z),graduuuuuxyz∂∂∂=++=∇∂∂∂ijk((,,),(,,),(,,))XxyzYxyzZxyz=V()()XYZXYZxyzxyz∂∂∂∂∂∂∇⋅=++++=++=∂∂∂∂∂∂VijkijkdivVrotxyzXYZ∂∂∂∇×==∂∂∂ijkVVGaussStokesdvdS+Ω∂Ω∇⋅=⋅∫∫∫∫∫VVnwSSdSds+∂∇×=⋅∫∫∫VVT6.....6.1.11/16VectorAnalysis&Fieldu=u(x,y,z)CuC{(,,)|(,,)}CMxyzuxyzC==.MC.u=u(x,y){(,)|(,)}CMxyuxyC==MC.graduuuuuxyz∂∂∂=++=∇∂∂∂ijkugradugraduu.6.2Stokes.CCd⋅=⋅∫∫VTVrvvd⋅Vr0d⋅Vr.lC∫v.M0(x0,y0,z0)SM(,,)XYZ=V0∂SnS+0Vn00Sds+∂Γ=⋅∫VnvM0S∆SdsSS∆⋅=∆∆∫Vnv∆ΓSM00limSSMMdsddSS∆∆→⋅Γ=∆∫VnvVM0(x0,y0,z0)n0.Stokes[()cos()cos()cos]limSSMXYXZYXdsyzzxxyddSSαβ∆∆→∂∂∂∂∂∂−+−+−∂∂∂∂∂∂Γ=∆∫∫γ12/16VectorAnalysis&Field00[()cos()cos()cos]MMdXYXZYXdSyzzxxyαβΓ∂∂∂∂∂∂=−+−+−∂∂∂∂∂∂γcoscoscosddtxyzxyzXYZXYZαβγΓ∂∂∂∂∂∂==∂∂∂∂∂∂ijkn⋅,,αβγrotVrotxyzXYZ∂∂∂=∂∂∂ijkVHamiltonrot=∇×VVrot|rot|cos,rotddSΓ=⋅=VnVnVnn..6.3GaussdSXdydzYdxdzZdxdy+Ω∂Ω⋅=++∫∫∫∫∫VnwV.∂Ω.M0(x0,y0,z0)M(,,)XYZ=V0∂Ω∂Ωn++00ds+∂ΩΦ=