現(xiàn)代數(shù)學(xué)物理教程

內(nèi)容概要

　　本書是一部學(xué)習(xí)數(shù)學(xué)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(xiàn)代數(shù)學(xué)概念，重點(diǎn)強(qiáng)調(diào)微分幾何。寫作風(fēng)格上保持了作者一貫的特點(diǎn)，清晰，透徹，引人入勝。大量的練習(xí)和例子是本書的一大亮點(diǎn)，擴(kuò)展索引對初學(xué)者也是十分有用。內(nèi)容涵蓋了張量代數(shù)，微分幾何，拓?fù)?，李群和李代?shù)，分布理論，基礎(chǔ)分析和希爾伯特空間。目次：幾何與結(jié)構(gòu)；群；向量空間；線性算子和矩陣；內(nèi)積空間；代數(shù)；張量；外代數(shù)；狹義相對論；拓?fù)鋵W(xué)；測度論和積分；分布；希爾伯特空間；量子力學(xué)；微分幾何；微分形式；流形上的積分；聯(lián)絡(luò)和曲率；李群和李代數(shù)。
　　讀者對象：數(shù)學(xué)、物理專業(yè)的本科生，研究生和相關(guān)的科研人員。

作者簡介

編者：(澳大利亞)斯?jié)煽死姿?(Peter Szekeres)

書籍目錄

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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