电磁场与电磁波第3讲坐标系变换矢量积分VIP免费

FieldandWaveElectromagnetic电磁场与电磁波第3讲作业情况1班：人合计：人情况:Review1.CartesianCoordinatesxzy1planexx1planeyy1planezzxazaya111,,PxyzoPositionvector:111ˆˆˆxyzopxayaza�111(,,)pxyzˆˆˆxxyyzzAAaAaAa�xxyyzzABABABAB�222xyyAAAAAA�ArbitraryVectorA:Dotproduct:Crossproduct:ˆˆˆˆˆˆ=xyzzyyzxxzzxyyxxyzxyzxyzABaABABaABABaABABaaaAAABBB�Differentiallength:ˆˆˆˆˆˆxxyyzzxyzdldladladladxadyadzaDifferentialvolume:dvdxdydzDifferentialsurfacechange:ˆˆˆˆˆˆˆsxxyyzzxyzdsdsaadsadsadsadydzadxdzadxdy2.CylindricalCoordinatesAP(r,,z)xyzor0r02z,,rzTherelationshipsbetweenthevariablesr,,zandthevariablesx,y,zarecosrxsinryzz22yxrxyarctanzzandyzxP00=0r=r0z=z0ˆraˆzaˆaO3.SphericalCoordinatesAφRxyzoP(R,,)0R002,,RTherelationshipsbetweenthevariablesR,,andthevariablesx,y,zaresincosxRsinsinyRcoszR222Rxyzzyx22arctanxyarctanandxzy=000R=R0=0ˆaˆRaˆaP0OR0MetricCoefficient（度规系数）坐标增量向长度增量变换的系数hhh123，，Maintopic1.TransformofCoordinateSystems2.IntegralsContainingVectorFunctions1.TransformofCoordinateSystemsABABBAcosThedotproductoftwovectors(1)islessthanorequaltotheproductoftheirmagnitudes;(2)canbeeitherapositiveoranegativequantity,dependingonwhethertheanglebetweenthemissmallerorlargerthan/2radians(90);(3)isequaltotheproductofthemagnitudeofonevectorandtheprojectionoftheothervectoruponthefirstone;and(4)iszerowhenthevectorsareperpendiculartoeachother.Question:wewanttoexpressinCartesiancoordinates.ˆˆˆrrzzAAaAaAa�ˆˆˆxxyyzzAAaAaAa�First:ˆˆˆˆˆ()ˆˆˆˆˆˆxxrrzzxrrxxzzxAAaAaAaAaaAaaAaaAaa�similarly:ˆˆˆˆˆˆˆˆˆˆˆˆˆˆyyrryyzzyzzrrzzzzzAAaAaaAaaAaaAAaAaaAaaAaa��So:coscos()2cos()cos2xryrzzAAAAAAAAyzxP00=0r=r0z=z0ˆraˆzaˆaOnow:ˆˆˆ[coscos()][cos()cos]22rxryzzAAAaAAaAa�Second:Moreover,Ar,A,andAzmaythemselvesbefunctionsofr,,andz.inthatcase,theytooshouldbeconvertedintofunctionsofx,y,andz,inthefinalanswer.Therelationshipsbetweenthevariablesr,,zandthevariablesx,y,zarecosrxsinryzz22yxrxyarctanzzandOver.coscos()2cos()cos2xryrzzAAAAAAAAItisconvenienttowritetherelationsbetweenthecomponentsofavectorinCartesianandCylindricalcoordinatesinamatrixform:cossin0sincos0001xryzzAAAAAA写成矩阵的形式圆柱-直角写成矩阵的形式直角-圆柱写成矩阵的形式球-直角写成矩阵的形式直角-球FromEXAMPLE2-11(p35-36)EXAMPLE2-9(p30-31)2.IntegralsContainingVectorFunctionsInelectromagneticsworkwehaveoccasiontoencounterintegralsthatcontainvectorfunctionssuchaslineintegral,surfaceintegral,andvolumeintegral.1)Lineintegral（work）Ingeneral,therearethreetypesoflineintegralsCCCdlFdlFdlˆˆˆxyzdldxadyadza＝ˆˆˆrzdldrardadzaˆˆˆsinRdldRaRdaRdaEXAMPLE2-14(p38-40)2)Surfaceintegral（flux）Ingeneral,therearethreetypesofsurfaceintegralsSSSdSFdSFdSˆˆˆrzdsarddzadrdzardrd2ˆˆˆsinsinRdsRddaRdRdaRdRdaˆˆˆxyzdsadxdyadxdzadydz1.Ifthesurfaceofintegration,S,isaclosedsurfaceenclosingavolume,thenthepositivedirectionforanisalwaysintheoutwarddirectionfromthevolume.2.IfSisanopensurface,thepositivedirectionforandependsonthedirectioninwhichtheperimeteroftheopensurfaceistraversed.Weapplytheright-handrule.EXAMPLE2-15(p41-42)3、volumeintegral（charge）Qdvdvdxdydzdvrdrddz2sindvRdRddSee:EXAMPLE2-12(p36)summary1.TransformofCoordinateSystems2.IntegralsContainingVectorFunctionshomeworkThankyou!Bye-bye!答疑安排时间：周一下午14：00~16：00地点：1401，1403P.2-15,2-22,2-24