| 词条 | 物理学家用的微分几何和李群 |
| 释义 | 图书信息出版社: 世界图书出版公司; 第1版 (2008年11月1日) 平装: 697页 正文语种: 英语 开本: 16 ISBN: 9787506292672 条形码: 9787506292672 尺寸: 25.6 x 18.2 x 3.4 cm 重量: 1.2 Kg 作者简介作者：(英国)斯洛伐 费茨科 (Fecko.M.) 内容简介《物理学家用的微分几何和李群》以一种非正式的形式写作，作者给出了1000多例子重在强调对一般理论的深刻理解。微分几何在现代理论物理和应用数学中扮演着越来越重要的角色。《物理学家用的微分几何和李群》给出了在理论物理和应用数学中很重要的几何知识的引入，包括，流形、张量场、微分形式、联络、辛几何、李群作用、族以及自旋。《物理学家用的微分几何和李群》将要为读者很好的学习拉格郎日现代处理方法、哈密顿力学、电磁、规范场，相对论以及万有引力做充足的准备。《物理学家用的微分几何和李群》很适合作为物理、数学以及工程专业的高年级本科生以及研究生的教程，也是一本很难得自学教程。 目录Preface Introduction 1 The concept of a manifold 1.1 Topology and continuous maps 1.2 Classes of smoothness of maps of Cartesian spaces 1.3 Smooth structure, smooth manifold 1.4 Smooth maps of manifolds 1.5 A technical description of smooth surfaces in Rn Summary of Chapter 1 2 Vector and tensor fields 2.1 Curves and functions on M 2.2 Tangent space, vectors and vector fields 2.3 Integral curves of a vector field 2.4 Linear algebra of tensors (multilinear algebra) 2.5 Tensor fields on M 2.6 Metric tensor on a manifold Summary of Chapter 2 3 Mappings of tensors induced by mappings of manifolds 3.1 Mappings of tensors and tensor fields 3.2 Induced metric tensor Summary of Chapter 3 4 Lie derivative 4.1 Local flow of a vector field 4.2 Lie transport and Lie derivative 4.3 Properties of the Lie derivative 4.4 Exponent of the Lie derivative 4.5 Geometrical interpretation of the commutator [V, W], non-holonomic frames 4.6 Isometries and conformal transformations, Killing equations Summary of Chapter 4 5 Exterior algebra 5.1 Motivation: volumes of paraUelepipeds 5.2 p-forms and exterior product 5.3 Exterior algebra AL* 5.4 Interior product iv 5.5 Orientation in L 5.6 Determinant and generalized Kronecker symbols 5.7 The metric volume form 5.8 Hodge (duality) operator* Summary of Chapter 5 6 Differential calculus of forms 6.1 Forms on a manifold 6.2 Exterior derivative 6.3 Orientability, Hodge operator and volume form on M 6.4 V-valued forms Summary of Chapter 6 7 Integral calculus of forms 7.1 Quantities under the integral sign regarded as differential forms 7.2 Euclidean simplices and chains 7.3 Simplices and chains on a manifold 7.4 Integral of a form over a chain on a manifold 7.5 Stokes' theorem 7.6 Integral over a domain on an orientable manifold 7.7 Integral over a domain on an orientable Riemannian manifold 7.8 Integral and maps of manifolds Summary of Chapter 7 8 Particular cases and applications of Stokes' theorem 8.1 Elementary situations 8.2 Divergence of a vector field and Gauss' theorem 8.3 Codifferential and LaPlace-deRhana operator 8.4 Green identities 8.5 Vector analysis in E3 8.6 Functions of complex variables Summary of Chapter 8 9 Poincare lemma and cohomologies 9.1 Simple examples of closed non-exact forms 9.2 Construction of a potential on contractible manifolds 9.3* Cohomologies and deRham complex Summary of Chapter 9 10 Lie groups: basic facts 10.1 Automorphisms of various structures and groups 10.2 Lie groups: basic concepts Summary of Chapter 10 11 Differential geometry on Lie groups 11.1 Left-invariant tensor fields on a Lie group 11.2 Lie algebra g of a group G 11.3 One-parameter subgroups 11.4 Exponential map 11.5 Derived homomorphism of Lie algebras 11.6 Invariant integral on G 11.7 Matrix Lie groups: enjoy simplifications Summary of Chapter 11 12 Representations of Lie groups and Lie algebras 12.1 Basic concepts 12.2 Irreducible and equivalent representations, Schur's lemma 12.3 Adjoint representation, Killing-Cartan metric 12.4 Basic constructions with groups, Lie algebras and their representations 12.5 Invariant tensors and intertwining operators 12.6* Lie algebra cohomologies Summary of Chapter 12 13 Actions of Lie groups and Lie algebras on manifolds 13.1 Action of a group, orbit and stabilizer 13.2 The structure of homogeneous spaces, G/H 13.3 Covering homomorphism, coverings SU(2) →SO(3) andSL(2, C)→ L↑+ 13.4 Representations of G and g in the space of functions on a G-space, fundamental fields 13.5 Representations of G and g in the space of tensor fields of type p Summary of Chapter 13 14 Hamiltonian mechanics and symplectic manifolds 14.1 Poisson and symplectic structure on a manifold 14.2 Darboux theorem, canonical transformations and symplectomorphisms 14.3 Poincare-Cartan integral invariants and Liouville's theorem 14.4 Symmetries and conservation laws 14.5* Moment map 14.6* Orbits of the coadjoint action 14.7* Symplectic reduction Summary of Chapter 14 15 Parallel transport and linear connection on M 15.1 Acceleration and parallel transport 15.2 Parallel transport and covariant derivative 15.3 Compatibility with metric, RLC connection 15.4 Geodesics 15.5 The curvature tensor 15.6 Connection forms and Cartan structure equations 15.7 Geodesic deviation equation (Jacobi's equation) 15.8* Torsion, complete parallelism and fiat connection Summary of Chapter 15 16 Field theory and the language of forms 16.1 Differential forms in the Minkowski space E1'3 16.2 Maxwell's equations in terms of differential forms 16.3 Gauge transformations, action integral 16.4 Energy-momentum tensor, space-time symmetries and conservation laws due to them 16.5* Einstein gravitational field equations, Hilbert and Cartan action 16.6* Non-linear sigma models and harmonic maps Summary of Chapter 16 17 Differential geometry on TM and T*M 17.1 Tangent bundle TM and cotangent bundle T*M 17.2 Concept of a fiber bundle 17.3 The maps Tf and T*f 17.4 Vertical subspace, vertical vectors 17.5 Lifts on TM and T*M 17.6 Canonical tensor fields on TM and T*M 17.7 Identities between the tensor fields introduced here Summary of Chapter 17 18 Hamiltonian and Lagrangian equations 18.1 Second-order differential equation fields 18.2 Euler-Lagrange field 18.3 Connection between Lagrangian and Hamiltonian mechanics, Legendre map 18.4 Symmetries lifted from the base manifold (configuration space) 18.5 Time-dependent Hamiltonian, action integral Summary of Chapter 18 19 Linear connection and the frame bundle 19.1 Frame bundle π : LM→M 19.2 Connection form on LM 19.3 k-dimensional distribution D on a manifold.M 19.4 Geometrical interpretation of a connection form: horizontal distribution on LM 19.5 Horizontal distribution on LM and parallel transport on M 19.6 Tensors on M in the language of LM and their parallel transport Summary of Chapter 19 20 Connection on a principal G-bundle 20.1 Principal G-bundles 21 Gauge theories and connections 22* Spinor fields and the Dirac operator Appendix A Some relevant algebraic structures A.I Linear spaces A.2 Associative algebras A.3 Lie algebras A.4 Modules A.5 Grading A.6 Categories and functors Appendix B Starring Bibliography Index of (frequently used) symbols Index |
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