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1、Math654IntroductiontoMathematicalFluidDynamicsProfessorCharlieDoeringTranscriptionbyIanTobascoUniversityofMichigan,Winter2011Math654,Lecture11/5/11–1Lecture1:Vectors,Tensors,andOperators1Vectors:NotationandOperationsGivenavectorx∈R3,wecanwriteitwithr
2、especttothecanonicalbasis{?i,?j,k?}asx=x?i+y?j+zk?.Inthismanner,wecande?nevector?eldsasv(x,y,z)=u(x,y,z)?i+v(x,y,z)?j+w(x,y,z)k?.Notethatsometimesthecanonicalbasisiswrittenas{e?1,e?2,e?3},andsimilarlyx=x1e?1+x2e?2+x3e?3,v(x1,x2,x3)=v1(x1,x2,x3)e?1+v2
3、(x1,x2,x3)e?2+v3(x1,x2,x3)e?3.InthiswayweeasilygeneralizetoRd,withavector?eldbeingXdv(x1,...,xd)=vi(x1,...,xd)e?ii=1orjustv(x)=vi(x)e?iusing“Einsteinnotation.”1Thefunctionviscommonlyreferredtoasthe“ithcomponent”ofthevectori?eld.Wehavethefollowingoper
4、ationsonpairsofvectors.De?nition1.Thedotproduct(orinnerproduct)ofv,w∈Rdisde?nedasv·w=viwi.Wecanarriveatthiswiththefollowingformalism.First,de?nethedotproductonthecanonicalbasisas(1i=je?i·e?j=δij=.0i6=jThenwritev=vie?iandw=wje?j,andde?nev·w=viwj(e?i·e
5、?j).Carryingouttheimpliedsummationyieldstheearlierde?nition.De?nition2.Theouterproductofv,w∈Rdisalinearself-mappingofRdde?nedviavw=viwje?ie?j,wheree?ie?jisthelinearself-mappingofRdhavingasmatrixrepresentationinthecanonicalbases(e?ie?j)mn=δimδjn.So,th
6、eouterproductofv,w∈Rdproducesthelinearmapwiththecanonicalmatrixrepresentation(vw)mn=vmwn,aso-called“dyadictensor.”Thisbringsustothenexttopic.1Ahandynotationwhererepeatedindicesimplysummation.Math654,Lecture11/5/11–22TensorsDe?nition3.A2-tensoronRdisa
7、bilinearformonRd.Speci?cally,T:Rd×Rd→Risa2-tensorifitsatis?es1.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{e?ie?j}1≤i
