An Introduction to Tensors for Students of Physics

JosephC.KoleckiGlennResearchCenter,Cleveland,OhioAnIntroductiontoTensorsforStudentsofPhysicsandEngineeringNASA/TM—2002-211716September2002TheNASASTIProgramOffice...inProfileSinceitsfounding,NASAhasbeendedicatedtotheadvancementofaeronauticsandspacescience.TheNASAScientificandTechnicalInformation(STI)ProgramOfficeplaysakeypartinhelpingNASAmaintainthisimportantrole.TheNASASTIProgramOfficeisoperatedbyLangleyResearchCenter,theLeadCenterforNASA’sscientificandtechnicalinformation.TheNASASTIProgramOfficeprovidesaccesstotheNASASTIDatabase,thelargestcollectionofaeronauticalandspacescienceSTIintheworld.TheProgramOfficeisalsoNASA’sinstitutionalmechanismfordisseminatingtheresultsofitsresearchanddevelopmentactivities.TheseresultsarepublishedbyNASAintheNASASTIReportSeries,whichincludesthefollowingreporttypes:•TECHNICALPUBLICATION.ReportsofcompletedresearchoramajorsignificantphaseofresearchthatpresenttheresultsofNASAprogramsandincludeextensivedataortheoreticalanalysis.Includescompilationsofsignificantscientificandtechnicaldataandinformationdeemedtobeofcontinuingreferencevalue.NASA’scounterpartofpeer-reviewedformalprofessionalpapersbuthaslessstringentlimitationsonmanuscriptlengthandextentofgraphicpresentations.•TECHNICALMEMORANDUM.Scientificandtechnicalfindingsthatarepreliminaryorofspecializedinterest,e.g.,quickreleasereports,workingpapers,andbibliographiesthatcontainminimalannotation.Doesnotcontainextensiveanalysis.•CONTRACTORREPORT.ScientificandtechnicalfindingsbyNASA-sponsoredcontractorsandgrantees.•CONFERENCEPUBLICATION.Collectedpapersfromscientificandtechnicalconferences,symposia,seminars,orothermeetingssponsoredorcosponsoredbyNASA.•SPECIALPUBLICATION.Scientific,technical,orhistoricalinformationfromNASAprograms,projects,andmissions,oftenconcernedwithsubjectshavingsubstantialpublicinterest.•TECHNICALTRANSLATION.English-languagetranslationsofforeignscientificandtechnicalmaterialpertinenttoNASA’smission.SpecializedservicesthatcomplementtheSTIProgramOffice’sdiverseofferingsincludecreatingcustomthesauri,buildingcustomizeddatabases,organizingandpublishingresearchresults...evenprovidingvideos.FormoreinformationabouttheNASASTIProgramOffice,seethefollowing:•AccesstheNASASTIProgramHomePageat•[e-mail]•FaxyourquestiontotheNASAAccessHelpDeskat301–621–0134•TelephonetheNASAAccessHelpDeskat301–621–0390•Writeto:NASAAccessHelpDeskNASACenterforAeroSpaceInformation7121StandardDriveHanover,MD21076JosephC.KoleckiGlennResearchCenter,Cleveland,OhioAnIntroductiontoTensorsforStudentsofPhysicsandEngineeringNASA/TM—2002-211716September2002NationalAeronauticsandSpaceAdministrationGlennResearchCenterAvailablefromNASACenterforAerospaceInformation7121StandardDriveHanover,MD21076NationalTechnicalInformationService5285PortRoyalRoadSpringfield,VA22100Availableelectronicallyat2002-2117161AnIntroductionToTensorsforStudentsofPhysicsandEngineeringJosephC.KoleckiNationalAeronauticsandSpaceAdministrationGlennResearchCenterCleveland,Ohio44135Tensoranalysisisthetypeofsubjectthatcanmakeeventhebestofstudentsshudder.Myownpost-graduateinstructorinthesubjecttookawaymuchofthefearbyspeakingofanimplicitrhythminthepeculiarnotationtraditionallyused,andhelpedmetoseehowthisrhythmplaysitswaythroughoutthevariousformalisms.Priortotakingthatclass,Ihadspentmanyyears“playing”onmyownwithtensors.Ifoundthegoingtobetremendouslydifficult,butwasable,overtime,tobackoutsomephysicalandgeometricalconsiderationsthathelpedtomakethesubjectalittlemoretransparent.Today,itissometimeshardnottothinkintermsoftensorsandtheirassociatedconcepts.Thisarticle,promptedandgreatlyenhancedbyMarlosJacob,whomI’vemetonlybye-mail,isanattempttorecordthoseearlynotionsconcerningtensors.Itisintendedtoserveasabridgefromthepointwheremostundergraduatestudents“leaveoff”intheirstudiesofmathematicstotheplacewheremosttextsontensoranalysisbegin.Abasicknowledgeofvectors,matrices,andphysicsisassumed.Asemi-intuitiveapproachtothosenotionsunderlyingtensoranalysisisgivenviascalars,vectors,dyads,triads,andsimilarhigher-ordervectorproducts.Thereadermustbepreparedtodosomemathematicsandtothink.Forthosestudentswhowishtogobeyondthishumblestart,Icanonlyrecommendmyprofessor’swisdom:findtherhythminthemathematicsandyouwillfareprettywell.BeginningsAttheheartofallmathematicsarenumbers.IfIweretoaskhowmanymarblesyouhadinabag,youmightanswer,“Three.”Iwouldfindyouranswerperfectlysatisfactory.The‘bare’number3,amagnitude,issufficienttoprovidetheinformationIseek.IfIweretoask,“Howfarisittoyourhouse?”andyouanswered,“Three,”however,Iwouldlookatyouquizzicallyandask,“Threewhat?”Evidently,forthisquestion,moreinformationisrequired.Thebarenumber3isnolongersufficient;Irequirea‘denominate’number–anumberwithaname.Supposeyourejoindered,“Threekm.”Thenumber3isnownamedasrepresentingacertainnumberofkm.Suchnumbersaresometimescalledscalars.Temperatureisrepresentedbyascalar.Thetotalenergyofathermodynamicsystemisalsorepresentedbyascalar.IfIwerenexttoask“ThenhowdoIgettoyo