vector calculus - theodore voronov

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1、VectorCalculus(Maths214)TheodoreVoronovJanuary20,2003Contents1RecollectionofdifferentialcalculusinRn31.1Pointsandvectors........................31.2Velocityvector..........................61.3Differentialofafunction.....................91.4Changesofcoordinates......................152Lineintegralsan

2、d1-forms203Algebraofforms243.1Jacobian..............................243.2Rulesofexteriormultiplication.................264Exteriorderivative274.1Dependenceoflineintegralsonpaths..............274.2Exteriorderivative:construction.................274.3Mainpropertiesandexamplesofcalculation..........2

3、85Stokes’stheorem295.1Integrationofk-forms......................295.2Stokes’stheorem:statementandexamples...........345.3Aproofforasimplecase.....................396Classicalintegraltheorems416.1Formscorrespondingtoavectorfield..............416.2TheOstrogradski–GaussandclassicalStokestheorems....46

4、IntroductionVectorcalculusdevelopsonsomeideasthatyouhavelearnedfromelementarymultivariatecalculus.Ourmaintaskistodevelopthegeometrictools.Thecentralnotionofthiscourseisthatofadifferentialform(shortly,form).THEODOREVORONOVExample1.Theexpressions2dx+5dy¡dzandxdxdy+edydzareexamplesofdifferentialforms.I

5、nfact,theformerexpressionaboveisanexampleofwhatiscalleda“1-form”,whilethelatterisanexampleofa“2-form”.(Youcanguesswhat1and2standfor.)Youwilllearntheprecisedefinitionofaformprettysoon;meanwhileIwillgivesomemoreexamplesinordertodemonstratethattoacertainextentthisobjectisalreadyfamiliar.Example2.Inthe

6、usualintegraloverasegmentinR,e.g.,Z2¼sinxdx;0theexpressionsinxdxisa1-formon[0;2¼](oronR).Example3.ThetotaldifferentialofafunctioninR3(ifyouknowwhatitis),@f@f@fdf=dx+dy+dz;@x@y@zisa1-forminR3.Example4.Whenyouintegrateafunctionoveraboundeddomainintheplane:Zf(x;y)dxdyDtheexpressionundertheintegral,f(x

7、;y)dxdy,isa2-forminD.Wecanconcludethataformisalinearcombinationofdifferentialsortheirproducts.Ofcourse,weneedtoknowthealgebraicrulesofhandlingtheseproducts.Thiswillbediscussedinduetime.Whenwewilllearnhowtohandlefo