Quick introduction to tensor analysis

arXiv:math/0403252v1[math.HO]16Mar20042MSC97U20PACS01.30.PpR.A.Sharipov.QuickIntroductiontoTensorAnalysis:lecturenotes.Freelydistributedon-line.Isfreeforindividualuseandeducationalpurposes.Anycommercialusewithoutwrittenconsentfromtheauthorisprohibited.ThisbookwaswrittenaslecturenotesforclassesthatItaughttoundergraduatestudentsmajoringinphysicsinFebruary2004duringmytimeasaguestinstructoratTheUniversityofAkron,whichwassupportedbyDr.SergeiF.Lyuksyutov’sgrantfromtheNationalResearchCouncilundertheCOBASEprogram.These4classeshavebeentaughtintheframeofaregularElectromagnetismcourseasanintroductiontotensorialmethods.Iwrotethisbookina”do-it-yourself”stylesothatIgiveonlyadraftoftensortheory,whichincludesformulatingdefinitionsandtheoremsandgivingbasicideasandformulas.Allotherworksuchasprovingconsistenceofdefinitions,derivingformulas,provingtheoremsorcompletingdetailstoproofsislefttothereaderintheformofnumerousexercises.Ihopethatthisstylemakeslearningthesubjectreallyquickandmoreeffectiveforunderstandingandmemorizing.IamgratefultoDepartmentChairProf.RobertR.MallikfortheopportunitytoteachclassesandthustobeinvolvedfullyintheatmosphereofanAmericanuniversity.IamalsogratefultoMr.M.Boiwka([e-mail])Mr.A.Calabrese([e-mail])Mr.J.Comer([e-mail])Mr.A.Mozinski([e-mail])Mr.M.J.Shepard([e-mail])forattendingmyclassesandreadingthemanuscriptofthisbook.IwouldliketoespeciallyacknowledgeandthankMr.JeffComerforcorrectingthegrammarandwordinginit.Contactstoauthor.Office:MathematicsDepartment,BashkirStateUniversity,32Frunzestreet,450074Ufa,RussiaPhone:7-(3472)-23-67-18Fax:[phone]Home:5Rabochayastreet,450003Ufa,RussiaPhone:7-(917)-75-55-786E-mails:[e-mail],[e-mail],[e-mail],URL:§1.Geometricalandphysicalvectors............................................................4.§2.Boundvectorsandfreevectors...............................................................5.§3.Euclideanspace.....................................................................................8.§4.BasesandCartesiancoordinates............................................................8.§5.Whatifweneedtochangeabasis?......................................................12.§6.Whathappenstovectorswhenwechangethebasis?............................15.§7.Whatisthenoveltyaboutvectorsthatwelearnedknowingtransformationformulafortheircoordinates?.......................................17.CHAPTERII.TENSORSINCARTESIANCOORDINATES......................18.§8.Covectors............................................................................................18.§9.Scalarproductofvectorandcovector...................................................19.§10.Linearoperators................................................................................20.§11.Bilinearandquadraticforms..............................................................23.§12.Generaldefinitionoftensors...............................................................25.§13.Dotproductandmetrictensor...........................................................26.§14.Multiplicationbynumbersandaddition..............................................27.§15.Tensorproduct..................................................................................28.§16.Contraction.......................................................................................28.§17.Raisingandloweringindices...............................................................29.§18.Somespecialtensorsandsomeusefulformulas....................................29.CHAPTERIII.TENSORFIELDS.DIFFERENTIATIONOFTENSORS....31.§19.TensorfieldsinCartesiancoordinates.................................................31.§20.ChangeofCartesiancoordinatesystem...............................................32.§21.Differentiationoftensorfields.............................................................34.§22.Gradient,divergency,androtor.Laplaceandd’Alambertoperators......35.CHAPTERIV.TENSORFIELDSINCURVILINEARCOORDINATES......38.§23.Generalideaofcurvilinearcoordinates................................................38.§24.AuxiliaryCartesiancoordinatesystem................................................38.§25.Coordinatelinesandthecoordinategrid.............................................39.§26.Movingframeofcurvilinearcoordinates..............................................41.§27.Dynamicsofmovingframe.................................................................42.§28.FormulaforChristoffelsymbols..........................................................42.§29.Tensorfieldsincurvilinearcoordinates................................................43.§30.Differentiationoftensorfieldsincurvilinearcoordinates.......................44.§31.Concordanceofmetricandconnec