現代數學物理教程

現代數學物理教程

《現代數學物理教程》是一本斯澤克雷斯編制,由世界圖書出版公司在2011年6月1日出版的書籍。

基本信息

內容簡介

《現代數學物理教程》是一部學習數學物理入門書籍,也是一部教程,讓讀者在物理的背景下建立現代數學概念,重點強調微分幾何。寫作風格上保持了作者一貫的特點,清晰,透徹,引人入勝。大量的練習和例子是《現代數學物理教程》的一大亮點,擴展索引對初學者也是十分有用。內容涵蓋了張量代數,微分幾何,拓撲,李群和李代數,分布理論,基礎分析和希爾伯特空間。目次:幾何與結構;群;向量空間;線性運算元和矩陣;內積空間;代數;張量;外代數;狹義相對論;拓撲學;測度論和積分;分布;希爾伯特空間;量子力學;微分幾何;微分形式;流形上的積分;聯絡和曲率;李群和李代數。讀者對象:數學、物理專業的本科生,研究生和相關的科研人員。

作者簡介

編者:(澳大利亞)斯澤克雷斯(PeterSzekeres)

目錄

acknowledgements

1 sets and structures

1.1 sets and logic

1.2 subsets, unions and intersections of sets

1.3Cartesianproducts 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

9.4electrodynamics

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,cotangentand 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 thePoincarelemma

18 connections and curvature

18.1 linear connections andgeodesics

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

18.7caftanformalism

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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