现代数学物理教程

章节摘录

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内容概要

编者：(澳大利亚)斯泽克雷斯 (Peter Szekeres)

书籍目录

acknowledgements
1 sets and structures
　1.1 sets and logic
　1.2 subsets, unions and intersections of sets
　1.3 cartesian products and relations
　1.4 mappings
　1.5 infinite sets
　1.6 structures
　1.7 category theory
2 groups
　2.1 elements of group theory
　2.2 transformation and permutation groups
　2.3 matrix groups
　2.4 homomorphisms and isomorphisms
　2.5 normal subgroups and factor groups
　2.6 group actions
　2.7 symmetry groups
3 vector spaces
　3.1 rings and fields
　3.2 vector spaces
　3.3 vector space homomorphisms
　3.4 vector subspaces and quotient spaces
　3.5 bases ofavector space
　3.6 summation convention and transformation of bases
　3.7 dual spaces
4 linear operators and matrices
　4.1 eigenspaces and characteristic equations
　4.2 jordan canonical form
　4.3 linear ordinary differential equations
　4.4 introduction to group representation theory
5 inner product spaces
　5.1 real inner product spaces
　5.2 complex inner product spaces
　5.3 representations of finite groups
6 algebras
　6.1 algebras and ideals
　6.2 complex numbers and complex structures
　6.3 quaternions and clifford algebras
　6.4 grassmann algebras
　6.5 lie algebras and lie groups
7 tensors
　7.1 free vector spaces and tensor spaces
　7.2 multilinear maps and tensors
　7.3 basis representation of tensors
　7.4 operations on tensors
8 exterior algebra
　8.1 r-vectors and r-forms
　8.2 basis representation of r-vectors
　8.3 exterior product
　8.4 interior product
　8.5 oriented vector spaces
　8.6 the hodge dual
9 special relativity
　9.1 minkowski space-time
　9.2 relativistic kinematics
　9.3 particle dynamics
　9.4 electrodynamics
　9.5 conservation laws and energy-stress tensors
10 topology
　10.1 euclidean topology
　10.2 general topological spaces
　10.3 metric spaces
　10.4 induced topologies
　10.5 hausdorff spaces
　10.6 compact spaces
　10.7 connected spaces
　10.8 topological groups
　10.9 topological vector spaces
11 measure theory and integration
　11.1 measurable spaces and functions
　11.2 measure spaces
　11.3 lebesgue integration
12 distributions
　12.1 test functions and distributions
　12.2 operations on distributions
　12.3 fourier transforms
　12.4 green's functions
13 hilbert spaces
　13.1 definitions and examples
　13.2 expansion theorems
　13.3 linear functionals
　13.4 bounded linear operators
　13.5 spectral theory
　13.6 unbounded operators
14 quantum mechanics
　14.1 basic concepts
　14.2 quantum dynamics
　14.3 symmetry transformations
　14.4 quantum statistical mechanics
15 differential geometry
　15.1 differentiable manifolds
　15.2 differentiable maps and curves
　15.3 tangent, cotangent and tensor spaces
　15.4 tangent map and submanifolds
　15.5 commutators, flows and lie derivatives
　15.6 distributions and frobenius theorem
16 differentiable forms
　16.1 differential forms and exterior derivative
　16.2 properties of exterior derivative
　16.3 frobenius theorem: dual form
　16.4 thermodynamics
　16.5 classical mechanics
17 integration on manifolds
　17.1 partitions of unity
　17.2 integration of n-forms
　17.3 stokes' theorem
　17.4 homology and cohomology
　17.5 the poincare lemma
18 connections and curvature
　18.1 linear connections and geodesics
　18.2 covariant derivative of tensor fields
　18.3 curvature and torsion
　18.4 pseudo-riemannian manifolds
　18.5 equation of geodesic deviation
　18.6 the riemann tensor and its symmetries
　18.7 caftan formalism
　18.8 general relativity
　18.9 cosmology
　18.10 variation principles in space-time
19 lie groups and lie algebras
　19.1 lie groups
　19.2 the exponential map
　19.3 lie subgroups
　19.4 lie groups of transformations
　19.5 groups of isometrics
bibliography
index

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《现代数学物理教程》由世界图书出版公司出版。

作者简介

《现代数学物理教程》是一部学习数学物理入门书籍，也是一部教程，让读者在物理的背景下建立现代数学概念，重点强调微分几何。写作风格上保持了作者一贯的特点，清晰，透彻，引人入胜。大量的练习和例子是《现代数学物理教程》的一大亮点，扩展索引对初学者也是十分有用。内容涵盖了张量代数，微分几何，拓扑，李群和李代数，分布理论，基础分析和希尔伯特空间。目次:几何与结构;群;向量空间;线性算子和矩阵;内积空间;代数;张量;外代数;狭义相对论;拓扑学;测度论和积分;分布;希尔伯特空间;量子力学;微分几何;微分形式;流形上的积分;联络和曲率;李群和李代数。

读者对象:数学、物理专业的本科生，研究生和相关的科研人员。

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