雅可比行列式的意义及推导

Lecture5:Jacobians•In1Dproblemsweareusedtoasimplechangeofvariables,e.g.fromxtou•Example:Substitute1DJacobianmapsstripsofwidthdxtostripsofwidthdu2DJacobian•Foracontinuous1-to-1transformationfrom(x,y)to(u,v)•Then•WhereRegion(inthexyplane)mapsontoregionintheuvplane•Hereaftercallsuchtermsetc2DJacobianmapsareasdxdytoareasdudv•TransformationTyielddistortedgridoflinesofconstantuandconstantv•Forsmallduanddv,rectanglesmapontoparallelograms•ThisisaJacobian,i.e.thedeterminantoftheJacobianMatrixWhythe2DJacobianworks•TheJacobianmatrixistheinversematrixofi.e.,•Because(andsimilarlyfordy)•ThismakessensebecauseJacobiansmeasuretherelativeareasofdxdyanddudv,i.e•SoRelationbetweenJacobiansSimple2DExamplerAreaofcircleA=Harder2DExamplewhereRisthisregionofthexyplane,whichmapstoR’here1489AnImportant2DExample•Evaluate•Firstconsider•Put•asa-a-aa3DJacobian•mapsvolumes(consistingofsmallcubesofvolume•........tosmallcubesofvolume•Where3DExample•TransformationofvolumeelementsbetweenCartesianandsphericalpolarcoordinatesystems(seeLecture4)