Introduction to Mathematical Fluid Dynamics_Univ_of_Michigan

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1、Math654IntroductiontoMathematicalFluidDynamicsProfessorCharlieDoeringTranscriptionbyIanTobascoUniversityofMichigan,Winter2011Math654,Lecture11/5/11–1Lecture1:Vectors,Tensors,andOperators1Vectors:NotationandOperationsGivenavectorx∈R3,wecanwriteitwithr

2、especttothecanonicalbasis{ˆi,ˆj,kˆ}asx=xˆi+yˆj+zkˆ.Inthismanner,wecandefinevectorfieldsasv(x,y,z)=u(x,y,z)ˆi+v(x,y,z)ˆj+w(x,y,z)kˆ.Notethatsometimesthecanonicalbasisiswrittenas{eˆ1,eˆ2,eˆ3},andsimilarlyx=x1eˆ1+x2eˆ2+x3eˆ3,v(x1,x2,x3)=v1(x1,x2,x3)eˆ1+v2

3、(x1,x2,x3)eˆ2+v3(x1,x2,x3)eˆ3.InthiswayweeasilygeneralizetoRd,withavectorfieldbeingXdv(x1,...,xd)=vi(x1,...,xd)eˆii=1orjustv(x)=vi(x)eˆiusing“Einsteinnotation.”1Thefunctionviscommonlyreferredtoasthe“ithcomponent”ofthevectorifield.Wehavethefollowingoper

4、ationsonpairsofvectors.Definition1.Thedotproduct(orinnerproduct)ofv,w∈Rdisdefinedasv·w=viwi.Wecanarriveatthiswiththefollowingformalism.First,definethedotproductonthecanonicalbasisas(1i=jeˆi·eˆj=δij=.0i6=jThenwritev=vieˆiandw=wjeˆj,anddefinev·w=viwj(eˆi·e

5、ˆj).Carryingouttheimpliedsummationyieldstheearlierdefinition.Definition2.Theouterproductofv,w∈Rdisalinearself-mappingofRddefinedviavw=viwjeˆieˆj,whereeˆieˆjisthelinearself-mappingofRdhavingasmatrixrepresentationinthecanonicalbases(eˆieˆj)mn=δimδjn.So,th

6、eouterproductofv,w∈Rdproducesthelinearmapwiththecanonicalmatrixrepresentation(vw)mn=vmwn,aso-called“dyadictensor.”Thisbringsustothenexttopic.1Ahandynotationwhererepeatedindicesimplysummation.Math654,Lecture11/5/11–22TensorsDefinition3.A2-tensoronRdisa

7、bilinearformonRd.Specifically,T:Rd×Rd→Risa2-tensorifitsatisfies1.Component-wiseadditivity:T(v+v0,w)=T(v,w)+T(v0,w)T(v,w+w0)=T(v,w)+T(v,w0),2.Component-wisehomogeneity:T(αv,w)=αT(v,w)T(v,βw)=βT(v,w),givenα,β∈R.Proposition1.Thesetof2-tensorsonRdisisomorp

8、hictothesetoflinearself-mapsofRd.Inotherwords,2-tensorsarematrices;wepursuethisideathroughouttherestofthissection.First,justasthesetoflinearself-mapsofRdformsalinearspace,thesetof2-tensorsonRdformsalinearspace.Moreover,onecaneasilyshowthat{eˆieˆj}1≤i