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九五品
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作者[美]朗 著
出版社世界图书出版公司
出版时间2010-09
版次2
装帧平装
上书时间2023-05-28
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图书标准信息
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作者
[美]朗 著
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出版社
世界图书出版公司
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出版时间
2010-09
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版次
2
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ISBN
9787510027468
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定价
35.00元
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装帧
平装
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开本
24开
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纸张
胶版纸
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页数
250页
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正文语种
英语
- 【内容简介】
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thisbookisanoutgrowthofmyintroductiontodifferentiablemanifolds(1962)anddifferentialmanifolds(1972).bothiandmypublishersfeltitworthwhiletokeepavailableabriefintroductiontodifferentialmanifolds.
thebookgivesanintroductiontothebasicconceptswhichareusedindifferentialtopology,differentialgeometry,anddifferentialequations.indifferentialtopology,onestudiesforinstancehomotopyclassesofmapsandthepossibilityoffindingsuitabledifferentiablemapsinthem(immersions,embeddings,isomorphisms,etc.).onemayalsousedifferentiablestructuresontopologicalmanifoldstodeterminethetopologicalstructureofthemanifold(forexample,alasmale[sm67]).indifferentialgeometry,oneputsanadditionalstructureonthedifferentiablemanifold(avectorfield,aspray,a2-form,ariemannianmetric,adlib.)andstudiespropertiesconnectedespeciallywiththeseobjects.formally,onemaysaythatonestudiespropertiesinvariantunderthegroupof.differentiableautomorphismswhichpreservetheadditionalstructure.indifferentialequations,onestudiesvectorfieldsandtheirintegralcurves,singularpoints,stableandunstablemanifolds,etc.acertainnumberofconceptsareessentialforallthree,andaresobasicandelementarythatitisworthwhiletocollectthemtogethersothatmoreadvancedexpositionscanbegivenwithouthavingtostartfromtheverybeginnings.theconceptsareconcernedwiththegeneralbasictheoryofdifferentialmanifolds.myfundamentalsofdifferentialgeometry(1999)canthenbeviewedasacontinuationofthepresentbook.
- 【目录】
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Foreword
Acknowledgments
CHAPTERI
DifferentialCalculus
1.Categories
2.FiniteDimensionalVectorSpaces
3.DerivativesandCompositionofMaps
4.IntegrationandTayiorsFormula
5.TheInverseMappingTheorem
CHAPTERII
Manifolds
1.Atlases,Charts,Morphisms
2.Submanifolds,Immersions,Submersions
3.PartitionsofUnity
4.ManifoldswithBoundary
CHAPTERIII
VectorBundles
l.Definition,PullBacks
2.TheTangentBundle
3.ExactSequencesofBundles
4.OperationsonVectorBundles
5.SplittingofVectorBundles
CHAPTERIV
VectorFieldsandDifferentialEquations
1.ExistenceTheoremforDifferentialEquations
2.VectorFields,Curves,andFlows
3.Sprays
4.TheFlowofaSprayandtheExponentialMap
5.ExistenceofTubularNeighborhoods
6.UniquenessofTubularNeighborhoods
CHAPTERV
OiretionsonVectorFieldsendDifferentialForms
1.VectorFields,DifferentialOperators,Brackets
2.LieDerivative
3.ExteriorDerivative
4.ThePoincareLemma
5.ContractionsandLieDerivative
6.VectorFieldsandl-FormsUnderSelfDuality
7.TheCanonical2-Form
8.DarbouxsTheorem
CHAPTERVI
TheTheoremofFrobenius
1.StatementoftheTheorem
2.DifferentialEquationsDependingonaParameter
3.ProofoftheTheorem
4.TheGlobalFormulation
5.LieGroupsandSubgroups
CHAPTERVII
Metrics
1.DefinitionandFunctoriality
2.TheMetricGroup
3.ReductiontotheMetricGroup
4.MetricTubularNeighborhoods
5.TheMorseLemma
6.TheRiemannianDistance
7.TheCanonicalSpray
CHAPTERVIII
IntegretionofDifferentialForms
1.SetsofMeasure0
2.ChangeofVariablesFormula
3.Orientation
4.TheMeasureAssociatedwithaDifferentialForm
CHAPTERIX
StokesTheorem
1.StokesTheoremforaRectangularSimplex
2.StokesTheoremonaManifold
3.StokesTheoremwithSingularities
CHAPTERX
ApplicationsofStokesTheorem
1.TheMaximaldeRhamCohomology
2.VolumeformsandtheDivergence
3.TheDivergenceTheorem
4.CauchysTheorem
5.TheResidueTheorem
Bibliography
Index
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