电磁场与电磁波第5讲旋度和旋度定理零恒等式亥姆霍兹定理

1P.2-152Review1.GradientofaScalarField2.DivergenceofaVectorField3.DivergenceTheorem3Maintopic1.CurlofaVectorField2.Stokes’sTheorem3.TwoNullIdentities4.Helmholtz’sTheorem41).矢量场的环量矢量场A沿有向闭合曲线l

的线积分称为矢量场A沿该曲线的环量，以表示为>0,<0,=0如如何显示源的分布特性？？矢量场的旋度5SMlen称为矢量A对于方向en的环量强度正最大次之零负最大零注意：在每一点P处都有无穷多的方向环量，且大小可能不等。en1en261.CurlofaVectorFieldFlowsource;vortexsource;vortexsinkthe(net)circulationSincecirculationasdefinedinEq.isalineintegralofadotproduct,itsvalueobviouslydependsontheorientationofthecontourCrelativetothevectorA.Inordertodefineapointfunction,whichisameasureofthestrengthofavortexsource,wemustmakeCverysmallandorientitinsuchawaythatthecirculationismaximum.7Inwords,Eq.statesthatthecurlofavectorfieldA,denotedbycurlAorA,isavectorwhosemagnitudeisthemaximumnetcirculationofAperunitareaastheareatendstozeroandwhosedirectionisthenormaldirectionoftheareawhentheareaisorientedtomakethenetcirculationmaximum.ThecomponentofAinanyotherdirectionauisau·(A),whichcanbedeterminedfromthecirculationperunitareanormaltoau

astheareaapproacheszero.8直角坐标系球坐标系柱坐标系9旋度运算规则10Example2-21(P57-58)11Acurl-freevectorfieldiscalledanirrotationaloraconservativefield.Adivergencelessfieldiscalledasolenoidalfield.122.Stokes’sTheoremThesurfaceintegralofthecurlofavectorfieldoveranopensurfaceisequaltotheclosedlineintegralofthevectoralongthecontourboundingthesurface.Stokes’stheoremconvertsasurfaceintegralofthecurlofavectortoalineintegralofthevector,andviceversa.Italwaysimpliesanopensurfacewitharim.Weremindourselvesherethatthedirectionsofdlandds(an)followtheright-handrule.13Example2-22(P60)14153.TwoNullIdentitiesThecurlofthegradientofanyscalarfield

isidentically

zero.(theexistenceofVanditsfirstderivativeseverywhereisimpliedhere.)3.1IDENTITYIAconversestatementofIdentityIcanbemadeasfollows:ifavectorfieldiscurl-free,thenitcanbeexpressedasthegradientofascalarfield.ifthenAnirrotational(aconservative)vectorfieldcanalwaysbeexpressedasthegradientofascalarfield.16Thedivergenceofthecurlofanyvectorfield

isidentically

zero.3.2IDENTITYIIAconversestatementofIdentityIIisasfollows:ifavectorfieldisdivergenceless,thenitcanbeexpressedasthecurlofanothervectorfield.Adivergencelessfieldisalsocalledasolenoidalfield.Solenoidalfieldsarenotassociatedwithflowsourcesofsinks.Thenetourwardfluxofasolenoidalfieldthroughanyclosedsurfaceiszero,andthefluxlinescloseuponthemselves.ifthen174.Helmholtz’sTheoremInprevioussectionswementionedthatadivergencelessfieldissolenoidal,andacurl-freefieldisirrotational.Wemayclassifyvectorfieldsinaccordancewiththeirbeingsolenoidaland/orirrotational1).solenoidalandirrotationalExample:Astaticelectricfieldinacharge-freeregion3).solenoidalandrotationalExample:Asteadymagneticfieldinacurrent-carryingconductor2).irrotationalbutnotsolenoidalExample:Astaticelectricfieldinachargeregion4).NeithersolenoidalnorirrotationalExample:Anelectricfieldinachargedmediumwithatime-varyingmagneticfield.18Themostgeneralvectorfieldthenhasbothanonzerodivergenceandanonzerocurl,andcanbeconsideredasthesumofasolenoidalfieldandanirrotationalfield.

Helmholtz’sTheorem:Avectorfield(vectorpointfunction)isdeterminedtowithinanadditiveconstantifbothitsdivergenceanditscurlarespecifiedeverywhere.Inanunboundedregionweassumethatboththedivergenceandthecurlofthevectorfieldvanishatinfinity.Ifthevectorfieldisconfinedwithinaregionboundedbyasurface,thenitisdeterminedifitsdivergenceandcurlthroughouttheregion,aswellasthenormalcomponentofthevectorovertheboundingsurface,aregiven.Hereweassumethatthevectorfunctionissingle-valuedandthatitsderivativesarefiniteandcontinuous.19Thedivergenceofavectorisameasureofthestrengthoftheflowsourceandthatthecurlofavectorisameasureofthestrengthofthevortexsource.Whenthestrengthsofboththeflowsourceandthevortexsourcearespecified,weexceptthatthevectorfieldwillbedetermined.Thus,wecandecomposeageneralvectorfieldFintoanirrotational(conservative)partFi

andasolenoidalpartFs:BecauseofTwoNullIdentities:20

位于某一区域中的矢量场，当其散度、旋度以及边界上场量的切向分量或法向分量给定后，则该区域中的矢量场被惟一地确定。

已知散度和旋度代表产生矢量场的源，可见惟一性定理表明，矢量场被其源及边界条件共同决定的。VSF(r)矢量场的惟一性定理21

若矢量场

F(r)

在无限区域中处处是单值的，且其导数连续有界，源分布在有限区域V

中，则当矢量场的散度及旋度给定后，该矢量场

F(r)

可以表示为

式中V'zxyr

Or'

r–r'

F(r)亥姆霍兹定理22Example2-23(p65)Givenavectorfunction:F=ax(3y-c1z)+ay(c2x-2z)-az(c3y+z)

(a)Determinetheconstantc1,c2andc3ifFisirrotational;(b)DeterminethescalarpotentialfunctionVwhosenegativegradientequalstoF2324summary1.CurlofaVectorField2.Stokes’sTheorem253.TwoNullIdentities4.Helmholtz’sTheoremifthenifthenHelmholtz’sTheorem:Avectorfield(vectorpointfunction)isdeterminedtowithinanadditiveconstantifbothitsdivergenceanditscurlarespecifiedeverywhere.261.ProductsofVectors2.OrthogonalCoordinateSystemsReviewCartes