电磁场与电磁波英文版

Chapter1VectorAnalysisGradient梯度,Divergence散度,Rotation,Helmholtz’sTheory1.DirectionalDerivative方向导数&Gradient2.Flux通量&Divergence3.Circulation环量&Curl旋度4.Solenoidal无散&Irrotational无旋Fields5.Green’sTheorems6.Uniqueness唯一性TheoremforVectorFields7.Helmholtz’sTheorem8.Orthogonal正交Curvilinear曲线Coordinate坐标1.DirectionalDerivative&GradientThedirectionalderivativeofascalaratapointindicatesthespatialrateofchangeofthescalaratthepointinacertaindirection.lPPllPΔ)()(lim0ΔThedirectionalderivativeofscalaratpointPinthedirectionoflisdefinedasPlPllΔPThegradientisavector.Themagnitude幅度ofthegradientofascalarfieldatapointisthemaximumdirectionalderivativeatthepoint,anditsdirectionisthatinwhichthedirectionalderivativewillbemaximum.zyxzyeeexgradzyxzyxeeegradInrectangularcoordinatesystem直角坐标系,thegradientofascalarfieldcanbeexpressedasWhere“grad”istheobservationoftheword“gradient”.Inrectangularcoordinatesystem,theoperator算符isdenotedasThenthegradofscalarfieldcanbedenotedasThesurfaceintegral面积分ofthevectorfieldAevaluatedoveradirectedsurfaceSiscalledthefluxthroughthedirectedsurfaceS,anditisdenotedbyscalar,i.e.2.Flux&DivergenceSdSAThefluxcouldbepositive,negative,orzero.Thedirectionofaclosedsurfaceisdefinedastheoutwardnormalontheclosedsurface.Hence,ifthereisasourceinaclosedsurface,thefluxofthevectorsmustbepositive;conversely,ifthereisasink,thefluxofthevectorswillbenegative.Thesourceapositivesource;Thesinkanegativesource.Asourceintheclosedsurfaceproducesapositiveintegral,whileasinkgivesrisetoanegativeone.FromphysicsweknowthatSq0dSEIfthereispositiveelectricchargeintheclosedsurface,thefluxwillbepositive.Iftheelectricchargeisnegative,thefluxwillbenegative.Inasource-freeregionwherethereisnocharge,thefluxthroughanyclosedsurfacebecomeszero.Theflux通量ofthevectorsthroughaclosedsurfacecanrevealthepropertiesofthesourcesandhowthesourcesexistedwithintheclosedsurface.Thefluxonlygivesthetotalsourceinaclosedsurface,anditcannotdescribethedistribution分布ofthesource.Forthisreason,thedivergenceisrequired.VSVΔdlimdiv0ΔSAAWhere“div”istheobservationoftheword“divergence,andVisthevolumeclosedbytheclosedsurface.Itshowsthatthedivergenceofavectorfieldisascalarfield,anditcanbeconsideredasthefluxthroughthesurfaceperunitvolume.Inrectangularcoordinates,thedivergencecanbeexpressedaszAyAxAzyxAdivWeintroducetheratio比率ofthefluxofthevectorfieldAatthepointthroughaclosedsurfacetothevolumeenclosedbythatsurface,andthelimit极限ofthisratio,asthesurfaceareaismadetobecomevanishinglysmallatthepoint,iscalledthedivergenceofthevectorfieldatthatpoint,denotedbydivA,givenbyUsingtheoperator,thedivergencecanbewrittenasAAdivSVVdddivSAADivergenceTheoremSVVddSAAorFromthepointofviewofmathematics,thedivergencetheoremstatesthatthesurfaceintegral面积分ofavectorfunctionoveraclosedsurfacecanbetransformedintoavolumeintegral体积分involvingthedivergenceofthevectoroverthevolumeenclosedbythesamesurface.Fromthepointoftheviewoffields,itgivestherelationshipbetweenthefieldsinaregiona区域ndthefieldsontheboundary边界oftheregion.ThelineintegralofavectorfieldAevaluatedalongaclosedcurveiscalledthecirculationofthevectorfieldAaroundthecurve,anditisdenotedby,i.e.3.Circulation环量&Curl旋度ldlAIfthedirectionofthevectorfieldAisthesameasthatofthelineelementdleverywherealongthecurve,thenthecirculation0.Iftheyareinoppositedirection,then0.Hence,thecirculationcanprovideadescriptionoftherotationalpropertyofavectorfield.Fromphysics,weknowthatthecirculationofthemagneticfluxdensityBaroundaclosedcurvelisequaltotheproductoftheconductioncurrentIenclosedbytheclosedcurveandthepermeability磁导率infreespace,i.e.wheretheflowingdirectionofthecurrentIandthedirectionofthedirectedcurveladheretotherighthandrule.Thecirculationisthereforeanindicationoftheintensityofasource.Il0dlBHowever,thecirculationonlystandsforthetotalsource,anditisunabletodescribethedistributionofthesource.Hence,therotationisrequired.SlSΔdlimcurlmax0ΔnlAeAWhereentheunitvectoratthedirectionaboutwhichthecirculationofthevectorAwillbemaximum,andSisthesurfaceclosedbytheclosedlinel.Themagnitudeofthecurlvectorisconsideredasthemaximumcirculationaroundtheclosedcurvewithunitarea.Curlisavector.IfthecurlofthevectorfieldAisdenotedby.ThedirectionisthattowhichthecirculationofthevectorAwillbemaximum,whilethemagnitudeofthecurlvectorisequaltothemaximumcirculationintensityaboutitsdirection,i.e.AcurlInrectangularcoordinates,thecurlcanbeexpressedbythematrixaszyxzyxAAAzyxeeeAcurlorbyusingtheoperatorasAAcurlStokes’TheoremlSdd)curl(lASAlSdd)(lASAorAsurfaceintegralcanbetransformedintoalineintegralbyusingStokes’theorem,andviseversa.Thegradient,thedivergence,orthecurlisdifferentialoperator.Theydescribethechangeofthefieldaboutapoint,andmaybedifferentatdifferentpoints.lSdd)(lASAFromthepointoftheviewofthefield,Stokes’theoremestablishestherelationshipbetweenthefieldintheregionandthefieldattheboundaryoftheregion.Theydescribethedifferentialpropertiesofthevectorfield.Thecontinuityofafunctionisanecessar