张量几何

图书标准信息

• 作者 [英]多德森 著
• 出版社 世界图书出版公司
• 出版时间 2009-06
• 版次 1
• ISBN 9787510004797
• 定价 50.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 432页
• 正文语种 英语

【内容简介】

　　《张量几何(第2版)(英文版)》是Springer数学研究生丛书之一，是一部详细讲述张量几何的教程。书中对微分几何的处理方式，以及学习广义相对论需要的数学知识使得本教程对于稍微了解单变量基本微积分和一些向量代数的知识就可以完全读懂该书的内容。《张量几何(第2版)(英文版)》用以书的形式能够提供的三维或更多维的图的形式使得内容更加形象化，重点强调数学的几何。为了表达的流畅和增强可读性，许多证明都是以练习的形式展示给读者，而非长篇的列举方程。这样，读者只能亲自进行实际计算，而不是跳过现成的例子。
　　这本内容丰富的教程对微分几何在相对论研究中的应用是个巨大的贡献。

【目录】

Introduction
0.FundamentalNot(at)ions
1.Sets
2.Functions
3.PhysicalBackground
1.RealVectorSpaces
1.Spaces
Subspacegeometry，components
2.Maps
Linearity，singularity，matrices
3.Operators
Projections，eigenvMues，determinant，trace

Ⅱ.AffineSpaces
1.Spaces
Tangentvectors，parallelism，coordinates
2.CombinationsofPoints
Midpoints，convexity
3.Maps
Linearparts，translations，components

Ⅲ.DualSpaces
1.Contours，Co-andContravariance，DualBasis

Ⅳ.MetricVectorSpaces
1.Metrics
Basicgeometryandexamples，Lorentzgeometry
2.Maps
Isometries，orthogonalprojectionsandcomplements，adjoints
3.Coordinates
Orthonormalbases
4.DiagonalisingSymmetricOperators
Principaldirections，isotropy

Ⅴ.TensorsandMultilinearForms
1.MultilinearForms
TensorProducts，Degree，Contraction，RaisingIndices
ViiTopologicalVectorSpaces
1.Continuity
Metrics：topologies，homeomorphisms
2.Limits
Convergenceandcontinuity
3.TheUsualTopology
Continuityinfinitedimensions
4.CompactnessandCompleteness
IntermediateValueTheorem，convergence，extrema

Ⅶ.DifferentiationandManifolds
1.Differentiation
Derivativeaslocallinearapproxiamation
2.Manifolds
Charts，maps，diffeomorphisms
3.BundlesandFields
Tangentandtensorbundles，metrictensors
4.Components
HairyBallTheorem，transformationformulae，raisingindic
5.Curves
Parametrisation，length，integration
6.VectorFieldsandFlows
Firstorderordinarydifferentialequations
7.LieBrackets
Commutingvectorfieldsandflows

Ⅷ.ConnectionsandCovariantDifferentiation
1.CurvesandTangentVectors
Representingavectorbyacurve
2.RollingWithoutTurning
Differentiationalongcurvesinembeddedmanifolds
3.DifferentiatingSections
Connectionshorizontalvectors，Christoffelsymbols
4.ParallelTransport
Integratingaconnection
5.TorsionandSymmetry
Torsiontensorofaconnection
6.MetricTensorsandConnections
Levi-Civitaconnection
7.CovariantDifferentiationofTensors
Paralleltransport，RiccisLemma，components，constancy

Ⅸ.Geodesics
1.LocalCharacterisation
Undeviatingcurves
2.GeodesicsfromaPoint
Completeness，exponentialmap，normalcoordinates
3.GlobalCharacterisation
Criticalityoflengthandenergy，FirstVariationFormula
4.Maxima，Minima，Uniqueness
Saddlepoints，mirages，TwinsParadox
5.GeodesicsinEmbeddedManifolds
Characterisation，examples
6.AnExampleofLieGroupGeometry
2x2matricesasapseudo-Riemannianmanifold

Ⅹ.Curvature
1.FlatSpaces
Intrinsicdescriptionoflocalflatness
2.TheCurvatureTensor
PropertiesandComponents
3.CurvedSurfaces
Ganssiancurvature，Gauss-BonnetTheorem
4.GeodesicDeviation
Tidaleffectsinspacetime
5.SectionalCurvature
SchursTheorem，constantcurvature
6.RicciandEinsteinTensors
Signs，geometry，Einsteinmanifolds，conservationequation
7.TheWeylTensor

Ⅺ.SpecialRelativity
1.OrientingSpacetimes
Causality，particlehistories
2.MotioninFlatSpacetime
Inertialframes，momentum，restmass，mass-energy
3.Fields
Mattertensor，conservation
4.Forces
Noscalarpotentials
5.GravitationalRedShiftandCurvature
Measurementgivesacurvedmetrictensor

Ⅻ.GeneralRelativity
I.HowGeometryGovernsMatter
Equivalenceprinciple，freefall
2.WhatMatterdoestoGeometry
Einsteinsequation，shapeofspacetime
3.TheStarsinTheirCourses
Geometryofthesolarsystem，Schwarzschildsolution
4.FarewellParticleAppendix.ExistenceandSmoothnessofFlows
1.Completeness
2.TwoFixedPointTheorems
3.SequencesofFunctions
4.IntegratingVectorQuantities
5.TheMainProof
6.InverseFunctionTheorem
Bibliography
IndexofNotations
Index