雅可比行列式的意义及推导.pdf

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1、Lecture5:Jacobians•In1Dproblemsweareusedtoasimplechangeofvariables,e.g.fromxtou1DJacobianmapsstripsofwidthdxtostripsofwidthdu•Example:Substitute2DJacobian•Foracontinuous1-to-1transformationfrom(x,y)to(u,v)•Then•WhereRegion(inthexyplane)mapsontoregioni

2、ntheuvplane2DJacobianmapsareasdxdytoareasdudv•HereaftercallsuchtermsetcWhythe2DJacobianworks•TransformationTyielddistortedgridoflinesofconstantuandconstantv•Forsmallduanddv,rectanglesmapontoparallelograms•ThisisaJacobian,i.e.thedeterminantoftheJacobia

3、nMatrixRelationbetweenJacobians•TheJacobianmatrixistheinversematrixofi.e.,•Because(andsimilarlyfordy)•ThismakessensebecauseJacobiansmeasuretherelativeareasofdxdyanddudv,i.e•SoSimple2DExampleAreaofcircleA=rHarder2DExamplewhereRisthisregionofthexyplane,

4、whichmapstoR’here8419AnImportant2DExample•Evaluatea•Firstconsider-aa-a•Put•as3DJacobian•mapsvolumes(consistingofsmallcubesofvolume•........tosmallcubesofvolume•Where3DExample•TransformationofvolumeelementsbetweenCartesianandsphericalpolarcoordinatesys

5、tems(seeLecture4)