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.ppt

FieldandWaveElectromagnetic电磁场与电磁波 第5讲 2 作业情况 1班 人合计 人情况 3 P 2 15 4 Review 1 GradientofaScalarField 2 DivergenceofaVectorField 3 DivergenceTheorem 5 Maintopic 1 CurlofaVectorField 2 Stokes sTheorem 3 TwoNullIdentities 4 Helmholtz sTheorem 6 7 8 1 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 9 称为矢量A对于方向en的环量强度 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 10 1 CurlofaVectorField Flowsource vortexsource vortexsink the net circulation SincecirculationasdefinedinEq isalineintegralofadotproduct itsvalueobviouslydependsontheorientationofthecontourCrelativetothevectorA Inordertodefineapointfunction whichisameasureofthestrengthofavortexsource wemustmakeCverysmallandorientitinsuchawaythatthecirculationismaximum 11 Inwords Eq statesthatthecurlofavectorfieldA denotedbycurlAor A isavectorwhosemagnitudeisthemaximumnetcirculationofAperunitareaastheareatendstozeroandwhosedirectionisthenormaldirectionoftheareawhentheareaisorientedtomakethenetcirculationmaximum Thecomponentof Ainanyotherdirectionauisau A whichcanbedeterminedfromthecirculationperunitareanormaltoauastheareaapproacheszero 12 2 旋度 概念在某点 旋度矢量的方向是使矢量A具有最大环量强度 环路所围面积的方向 的方向 其大小等于对该矢量方向的最大环量强度 记为 式中 en为旋度方向上的单位矢量 此式表明 矢量场的旋度大小可以认为是包围单位面积上的闭合曲线的最大环量 代表了 旋度 源的强度 绕任意方向的方向环量 旋度与该方向单位矢量的点积 投影 其方向为当面积的取向使得环量呈最大时 该面积的法线方向 右手定则 13 直角坐标系 球坐标系 柱坐标系 14 旋度运算规则 15 Example2 21 P57 58 16 Acurl freevectorfieldiscalledanirrotationaloraconservativefield Adivergencelessfieldiscalledasolenoidalfield 17 2 Stokes sTheorem Thesurfaceintegralofthecurlofavectorfieldoveranopensurfaceisequaltotheclosedlineintegralofthevectoralongthecontourboundingthesurface Stokes stheoremconvertsasurfaceintegralofthecurlofavectortoalineintegralofthevector andviceversa Italwaysimpliesanopensurfacewitharim Weremindourselvesherethatthedirectionsofdlandds an followtheright handrule 18 Example2 22 P60 19 20 3 TwoNullIdentities Thecurlofthegradientofanyscalarfieldisidenticallyzero theexistenceofVanditsfirstderivativeseverywhereisimpliedhere 3 1IDENTITYI AconversestatementofIdentityIcanbemadeasfollows ifavectorfieldiscurl free thenitcanbeexpressedasthegradientofascalarfield if then Anirrotational aconservative vectorfieldcanalwaysbeexpressedasthegradientofascalarfield 21 Thedivergenceofthecurlofanyvectorfieldisidenticallyzero 3 2IDENTITYII AconversestatementofIdentityIIisasfollows ifavectorfieldisdivergenceless thenitcanbeexpressedasthecurlofanothervectorfield Adivergencelessfieldisalsocalledasolenoidalfield Solenoidalfieldsarenotassociatedwithflowsourcesofsinks Thenetourwardfluxofasolenoidalfieldthroughanyclosedsurfaceiszero andthefluxlinescloseuponthemselves if then 22 4 Helmholtz sTheorem Inprevioussectionswementionedthatadivergencelessfieldissolenoidal andacurl freefieldisirrotational Wemayclassifyvectorfieldsinaccordancewiththeirbeingsolenoidaland orirrotational 1 solenoidalandirrotational Example Astaticelectricfieldinacharge freeregion 3 solenoidalandrotational Example Asteadymagneticfieldinacurrent carryingconductor 2 irrotationalbutnotsolenoidal Example Astaticelectricfieldinachargeregion 4 Neithersolenoidalnorirrotational Example Anelectricfieldinachargedmediumwithatime varyingmagneticfield 23 Themostgeneralvectorfieldthenhasbothanonzerodivergenceandanonzerocurl andcanbeconsideredasthesumofasolenoidalfieldandanirrotationalfield Helmholtz sTheorem Avectorfield vectorpointfunction isdeterminedtowithinanadditiveconstantifbothitsdivergenceanditscurlarespecifiedeverywhere Inanunboundedregionweassumethatboththedivergenceandthecurlofthevectorfieldvanishatinfinity Ifthevectorfieldisconfinedwithinaregionboundedbyasurface thenitisdeterminedifitsdivergenceandcurlthroughouttheregion aswellasthenormalcomponentofthevectorovertheboundingsurface aregiven Hereweassumethatthevectorfunctionissingle valuedandthatitsderivativesarefiniteandcontinuous 24 Thedivergenceofavectorisameasureofthestrengthoftheflowsourceandthatthecurlofavectorisameasureofthestrengthofthevortexsource Whenthestrengthsofboththeflowsourceandthevortexsourcearespecified weexceptthatthevectorfieldwillbedetermined Thus wecandecomposeageneralvectorfieldFintoanirrotational conservative partFiandasolenoidalpartFs BecauseofTwoNullIdentities 25 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 F r 矢量场的惟一性定理 26 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 亥姆霍兹定理 27 Example2 23 p65 Givenavectorfunction F ax 3y c1z ay c2x 2z az c3y z a Determinetheconstantc1 c2andc3ifFisirrotational b DeterminethescalarpotentialfunctionVwhosenegativegradientequalstoF 28 29 1 ProductsofVectors 2 OrthogonalCoordinateSystems CartesianCoordinates Positionvector ArbitraryVectorA summary 30 Dotproduct Crossproduct Differentiallength Differentialvolume Differentialsurface 31 3 GradientofaScalarField 4 DivergenceofaVectorField 5 DivergenceTheorem 32 6 CurlofaVectorField 7 Stokes sTheorem 8 TwoNullIdentities if then if then 33 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