an introduction to differential geometry with applications to elasticity - ciarlet

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1、ANINTRODUCTIONTODIFFERENTIALGEOMETRYWITHAPPLICATIONSTOELASTICITYPhilippeG.CiarletCityUniversityofHongKongContentsPreface51Three-dimensionaldifferentialgeometry9Introduction..............................91.1Curvilinearcoordinates.......................111.2Metrictensor.

2、............................131.3Volumes,areas,andlengthsincurvilinearcoordinates......161.4Covariantderivativesofavectorfield................191.5Necessaryconditionssatisfiedbythemetrictensor;theRiemanncurvaturetensor...........................241.6Existenceofanimmersi

3、ondefinedonanopensetinR3withaprescribedmetrictensor.......................251.7Uniquenessuptoisometriesofimmersionswiththesamemetrictensor.................................361.8Continuityofanimmersionasafunctionofitsmetrictensor..442Differentialgeometryofsurfaces59Introd

4、uction..............................592.1Curvilinearcoordinatesonasurface................612.2Firstfundamentalform.......................652.3Areasandlengthsonasurface...................672.4Secondfundamentalform;curvatureonasurface.........692.5Principalcurvatures;Ga

5、ussiancurvature..............732.6Covariantderivativesofavectorfielddefinedonasurface;theGaußandWeingartenformulas...................792.7Necessaryconditionssatisfiedbythefirstandsecondfundamen-talforms:theGaußandCodazzi-Mainardiequations;Gauß’TheoremaEgregium............

6、.............822.8Existenceofasurfacewithprescribedfirstandsecondfundamen-talforms................................852.9Uniquenessuptoproperisometriesofsurfaceswiththesamefundamentalforms..........................952.10Continuityofasurfaceasafunctionofitsfundamentalform

7、s..10034Contents3Applicationstothree-dimensionalelasticityincurvilinearcoordinates109Introduction..............................1093.1TheequationsofnonlinearelasticityinCartesiancoordinates..1123.2Principleofvirtualworkincurvilinearcoordinates........1193.3Equationsofe

8、quilibriumincurvilinearcoordinates;covariantderivativesofatensorfield.....................1273.4Constitutiveequationincurvili