Vector-analysis电磁场向量分析.ppt

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1、Chapter1VectorAnalysisGradient,Divergence,Rotation,Helmholtz’sTheory0.Scalarsandvectors,vectoralgebra,scalarproduct,Vectorproduct1.DirectionalDerivative&Gradient2.Flux&Divergence3.Circulation&Curl4.Solenoidal&IrrotationalFields5.Green’sTheorems6.UniquenessTheoremforVectorFields7.Helmholtz’sTheo

2、rem8.OrthogonalCurvilinearCoordinate1.DirectionalDerivative&GradientThedirectionalderivativeofascalaratapointindicatesthespatialrateofchangeofthescalaratthepointinacertaindirection.ThedirectionalderivativeofscalaratpointPinthedirectionoflisdefinedasPlThegradientisavector.Themagnitudeofthegrad

3、ientofascalarfieldatapointisthemaximumdirectionalderivativeatthepoint,anditsdirectionisthatinwhichthedirectionalderivativewillbemaximum.Inrectangularcoordinatesystem,thegradientofascalarfieldcanbeexpressedasWhere“grad”istheobservationoftheword“gradient”.Inrectangularcoordinatesystem,theoperato

4、risdenotedasThenthegradofscalarfieldcanbedenotedasItcanbeshownthatThesurfaceintegralofthevectorfieldAevaluatedoveradirectedsurfaceSiscalledthefluxthroughthedirectedsurfaceS,anditisdenotedbyscalar,i.e.2.Flux&DivergenceThefluxcouldbepositive,negative,orzero.Thedirectionofaclosedsurfaceisdefin

5、edastheoutwardnormalontheclosedsurface.Hence,ifthereisasourceinaclosedsurface,thefluxofthevectorsmustbepositive;conversely,ifthereisasink,thefluxofthevectorswillbenegative.Thesourceapositivesource;Thesinkanegativesource.Asourceintheclosedsurfaceproducesapositiveintegral,whileasinkgivesrisetoa

6、negativeone.FromphysicsweknowthatIfthereispositiveelectricchargeintheclosedsurface,thefluxwillbepositive.Iftheelectricchargeisnegative,thefluxwillbenegative.Inasource-freeregionwherethereisnocharge,thefluxthroughanyclosedsurfacebecomeszero.Thefluxofthevectorsthroughaclosedsurfacecanrevealthepro

7、pertiesofthesourcesandhowthepresenceofsourceswithintheclosedsurface.Thefluxonlygivesthetotalsourceinaclosedsurface,anditcannotdescribethedistributionofthesource.Forthisreason,thedivergenceisrequired.Where“div”istheobservationofthe