多维实分析（第1卷）

Thisbook，whichisintwoparts，providesanintroductiontothetheoryofvector-valuedfunctionsonEuclideanspace.Wefocusonfourmainobjectsofstudyandinadditionconsidertheinteractionsbetweenthese.VolumeIisdevotedtodifferentiation.DifferentiablefunctionsonRncomefirst，inChapters1through3.Next，differentiablemanifoldsembeddedinRarediscussed，inChapters4and5.InVolume11wetakeupintegration.Chapter6dealswiththetheoryofn-dimensionalintegrationoverR.Finally，inChapters7and8lower-dimensionalintegrationoversubmanifoldsofRnisdeveloped;particularattentionispaidtovectoranalysisandthetheoryofdifferentialforms，whicharetreatedindependentlyfromeachother.Generallyspeaking，theemphasisisongeometricaspectsofanalysisratherthanonmattersbelongingtofunctionalanalysis.

VolumeⅠ
Preface
Acknowledgments
Introduction
1Continuity
1.1Innerproductandnorm
1.2Openandclosedsets
1.3Limitsandcontinuousmappings
1.4Compositionofmappings
1.5Homeomorphisms
1.6Completeness
1.7Contractions
1.8Compactnessanduniformcontinuity
1.9Connectedness

2Differentiation
2.1Linearmappings
2.2Differentiablemappings
2.3Directionalandpartialderivatives
2.4Chainrule
2.5MeanValueTheorem
2.6Gradient
2.7Higher-orderderivatives
2.8Taylor'sformula
2.9Criticalpoints
2.10Commutinglimitoperations

3InverseFunctionandImplicitFunctionTheorems
3.1Diffeomorphisms
3.2InverseFunctionTheorems
3.3ApplicationsoflnverseFunctionTheorems
3.4Implicitlydefinedmappings
3.5ImplicitFunctionTheorem
3.6ApplicationsoftheImplicitFunctionTheorem
3.7ImplicitandInverseFunctionTheoremsonC

4Manifolds
4.1Introductoryremarks
4.2Manifolds
4.3ImmersionTheorem
4.4Examplesofimmersions
4.5SubmersionTheorem
4.6Examplesofsubmersions
4.7Equivalentdefinitionsofmanifold
4.8Morse'sLemma

5TangentSpaces
5.1Definitionoftangentspace
5.2Tangentmapping
5.3Examplesoftangentspaces
5.4MethodofLagrangemultipliers
5.5Applicationsofthemethodofmultipliers
5.6Closerinvestigationofcriticalpoints
5.7Gaussiancurvatureofsurface
5.8CurvatureandtorsionofcurveinR3
5.9One-parametergroupsandinfinitesimalgenerators
5.10LinearLiegroupsandtheirLiealgebras
5.11Transversality
Exercises
ReviewExercises
ExercisesforChapter1
ExercisesforChapter2
ExercisesforChapter3
ExercisesforChapter4
ExercisesforChapter5
Notation
Index
VolumeⅡ
Preface
Acknowledgments
Introduction

6Integration
6.1Rectangles
6.2Riemannintegrability
6.3Jordanmeasurability
6.4Successiveintegration
6.5Examplesofsuccessiveintegration
6.6ChangeofVariablesTheorem:formulationandexamples
6.7Partitionsofunity
6.8ApproximationofRiemannintegrablefunctions
6.9ProofofChangeofVariablesTheorem
6.10AbsoluteRiemannintegrability
6.11Applicationofintegration:Fouriertransformation
6.12Dominatedconvergence
6.13Appendix:twootherproofsofChangeofVariablesTheorem

7IntegrationoverSubmanifolds
7.1Densitiesandintegrationwithrespecttodensity
7.2AbsoluteRiemannintegrabilitywithrespecttodensity
7.3Euclideand-dimensionaldensity
7.4ExamplesofEuclideandensities
7.5Opensetsatonesideoftheirboundary
7.6Integrationofatotalderivative
7.7Generalizationsoftheprecedingtheorem
7.8Gauss'DivergenceTheorem
7.9ApplicationsofGauss'DivergenceTheorem

8OrientedIntegration
8.1Lineintegralsandpropertiesofvectorfields
8.2Antidifferentiation
8.3Green'sandCauchy'sIntegralTheorems
8.4Stokes'IntegralTheorem
8.5ApplicationsofStokes'IntegralTheorem
8.6Apotheosis:differentialformsandStokes'Theorem.
8.7Propertiesofdifferentialforms
8.8Applicationsofdifferentialforms
8.9HomotopyLemma
8.10Poincare'sLemma
8.11Degreeofmapping
Exercises
ExercisesforChapter6
ExercisesforChapter7
ExercisesforChapter8
Notation
Index