《现代数学物理教程》(作者斯泽克雷斯)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本书的一大亮点，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。

"Modern mathematical physics"(Author Sizekeleisi)provides an introduction to the major mathematical structures used in physics today. It covers the concepts and techniques needed for topics such as group theory, Lie algebras, topology, Hilbert spaces and differential geometry. Important theories of physics such as classical and quantum mechanics, thermodynamics, and special and general relativity are also developed in detail, and presented in the appropriate mathematical language.

"Modern mathematical physics" is suitable for advanced undergraduate and beginning graduate students in mathematical and theoretical physics. It includes numerous exercises and worked examples to test the reader's understanding of the various concepts, as well as extending the themes covered in the main text. The only prerequisites are elementary calculus and linear algebra.No prior knowledge of group theory, abstract vector spaces or topology is required.PETER SZEKERES received his Ph.D. from King's College London in 1964, in the area of general relativity. He subsequently held research and teaching positions at Cornell University, King's College and the University of Adelaide, where he stayed from 1971 till his recent retirement. Currently he is a visiting research fellow at that institution. He is well known internationally for his research in general relativity and cosmology, and has an excellent reputation for his teaching and lecturing.

preface

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