Lecture 5 Fundamental of Field Theory

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1、Lecture5FundamentalofFieldtheory(cont)Correction2.2.3ProductsofThreeVectors1)ScalarTripleProductvvvvvvvvvA(BC)B(CA)C()ABvvv=-A()CBvvv=-B()ACvvv=-C()BA2)VectorTripleProduct（knownas“back-cab”rule）vvvvvvvvvABCBACCAB3.6.2Relationsbetweenbasevectorsindiffere

2、ntcoordinatesvyayvvvvavaxayazarvfarcosfsinf0vvaxa-sinfcosf0vaz001foxvzazvvvvaraazaRvqaRsinq0cosqvvaracosq0-sinqqvva010qaorvvvaxayazvaRsinqcosfsinqsinfcosqv?acoscoscosqsinf-sinqva-sinfcosf0vvvvvvaacosasin,aaasincosrxyxyorvvvvvvaacosasin,aaasincosxryr

3、UnitvectorchangeswithvarvvvaxysinaacosvaavcosaavvsinxyrvvvPositionvectorrarzraz0vvvvvvvdrdarrdazzardrrdarazzdzzdavvvadrardadzrzHowaboutSphericalcoordinates？vraRˆRReviewR-5.1DirectionalderivativeVVVVcoscoscoslxyz2

4、22coscoscos1R-5.2GradientofascalarFieldVVVVau1aauu23l1ll23VVVVau1aauu23h1u1hhu2hu33R-5.3RelationsbetweenDirectionalderivativeandGradientofascalarFieldVVgaˆllOutline•DivergenceofaVectorField•CurlofaVectorfield5.1DivergenceofaVectorField5.1.1FluxFlu

5、xisusuallytheintegralofavectorquantity,fluxdensity,overafinitesurface.Suchasmagneticflux,electricfluxvvEdSSThefluxofavectorfieldisanalogoustotheflowofanincompressiblefluidsuchaswater.F=0(nosource)F>0(source)F<0(sink)5.1.2DivergenceThedivergenceofavectorfieldAatapointisthenet

6、outwardfluxofAperunitvolumeasthevolumeaboutthepointtendstozero(spatialderivativesofavectorfield):rrrdivA@lim1AdSzvv0SDzØDerivationoftheexpressionfordivAinCartesiancoordinatesP(aboutdetails,pleasereferto马冰然P23)DyDxrrrroyAdSfrontbackrightlefttopbottomAgdSS

7、facefacefacefacefacefacexrrrrrfrontAdSAfrontSAfrontaxyzfacefacefacexAxx0,,y00zyz2Taylorseriesexpansion0xxAxAxxx0,y0,z0Ax0,yz00,higher-orderterms22xx0,,yz00rAAAxzydivAxyzWiththevectordifferentialoperatordelrrdivAAIng

8、eneralorthogonalcoordinatesu1,,uu23.r1