IntroductionToTensorCalculus(1).pdf

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1、IntroductiontoTensorCalculusandContinuumMechanicsbyJ.H.HeinbockelDepartmentofMathematicsandStatisticsOldDominionUniversityPREFACEThisisanintroductorytextwhichpresentsfundamentalconceptsfromthesubjectareasoftensorcalculus,differentialgeometryandcontinuummechanics.Thematerialpresentedissuitabl

2、eforatwosemestercourseinappliedmathematicsandisflexibleenoughtobepresentedtoeitherupperlevelundergraduateorbeginninggraduatestudentsmajoringinappliedmathematics,engineeringorphysics.Thepresentationassumesthestudentshavesomeknowledgefromtheareasofmatrixtheory,linearalgebraandadvancedcalculus.

3、Eachsectionincludesmanyillustrativeworkedexamples.Attheendofeachsectionthereisalargecollectionofexerciseswhichrangeindifficulty.Manynewideasarepresentedintheexercisesandsothestudentsshouldbeencouragedtoreadalltheexercises.Thepurposeofpreparingthesenotesistocondenseintoanintroductorytextthebas

4、icdefinitionsandtechniquesarisingintensorcalculus,differentialgeometryandcontinuummechanics.Inparticular,thematerialispresentedto(i)developaphysicalunderstandingofthemathematicalconceptsassociatedwithtensorcalculusand(ii)developthebasicequationsoftensorcalculus,differentialgeometryandcontinuum

5、mechanicswhichariseinengineeringapplications.Fromthesebasicequationsonecangoontodevelopmoresophisticatedmodelsofappliedmathematics.Thematerialispresentedinaninformalmannerandusesmathematicswhichminimizesexcessiveformalism.Thematerialhasbeendividedintotwoparts.Thefirstpartdealswithanintroduc-

6、tiontotensorcalculusanddifferentialgeometrywhichcoverssuchthingsastheindicialnotation,tensoralgebra,covariantdifferentiation,dualtensors,bilinearandmultilinearforms,specialtensors,theRiemannChristoffeltensor,spacecurves,surfacecurves,cur-vatureandfundamentalquadraticforms.Thesecondpartemphasiz

7、estheapplicationoftensoralgebraandcalculustoawidevarietyofappliedareasfromengineeringandphysics.Theselectedapplicationsarefromtheareasofdynamics,elasticity,fluidsandelectromag-netictheory.Thecontinuummechanicsportionfocusesonanintroductionofthebasicconcep