分析流形和物理学

出版时间：2010-9  出版社：世界图书出版公司  作者：许凯布里哈特  页数：630  
Tag标签：无  

前言

All too often in physics familiarity is a substitute for understanding, andthe beginner who lacks familiarity wonders which is at fault: physics orhimself. Physical mathematics provides well defined concepts and techni-ques for the study of physical systems. It is more than mathematicaltechniques used in the solution of problems which have already beenformulated; it helps in the very formulation of the laws of physicalsystems and brings a better understanding of physics. Thus physicalmathematics includes mathematics which gives promise of being useful inour analysis of physical phenomena. Attempts to use mathematics for thispurpose may fail because the mathematical tool is too crude; physics maythen indicate along which lines it should be sharpened. In fact, theanalysis of physical systems has spurred many a new mathematicaldevelopment.Considerations of relevance to physics underlie the choice of materialincluded here. Any choice is necessarily arbitrary; we included first thetopics which we enjoy most but we soon recognized that instant gratifica-tion is a short range criterion. We then included material which can beappreciated only after a great deal of intellectual asceticism but which maybe farther reaching. Finally, this book gathers the starting points of somegreat currents of contemporary mathematics. It is intended for anadvanced physical mathematics course.

内容概要

we are happy that the success of the first edition gave us a chance to prepare a revised edition. we have made numerous changes and added exercises with their solutions to ease the study of several chapters. the major addition is a chapter "connections on principal fibre bundles" which includes sections on holonomy, characteristic classes, invariant curvature integrals and problems on the geometry of gauge fields, mono poles, instantons, spin structure and spin connections. other additions include a section on the second fundamental form, a section on almost complex and kiihlerian manifolds, and a problem on the method of stationary phase. more than 150 entries have been added to the index. 　　can this book, now polished by usage, serve as a text for an advanced physical mathematics course? this question raises another question: what is the function of a text book for graduate studies? in our times of rapidly expanding knowledge, a teacher looks for a book which will provide a broader base for future developments than can be covered in one or two semesters of lectures and a student hopes that his purchase will serve him for many years. a reference book which can be used as a text is an answer to their needs. this is what this book is intended to be, and thanks to a publishing company which keeps it moderately priced, this is what we hope it will he.

作者简介

作者：（法国）许凯布里哈特

书籍目录

i. review of fundamental notions of analysis a. set theory, definitions 1. sets 2. mappings 3. relations 4. orderings b. algebraic structures, definitions 1. groups 2. rings 3. modules 4. algebras 5. linear spaces c. topology 1. definitions 2. separation 3. base 4. convergence 5. covering and compactness 6. connectedness 7. continuous mappings 8. multiple connectedness 9. associated topologies 10. topology related to other structures 11. metric spaces metric spaces cauchy sequence; completeness 12. banach spaces normed vector spaces banach spaces strong and weak topology; compactedness 13. hilbert spaces d. integration 1. introduction 2. measures 3. measure spaces 4. measurable functions 5. lntegrable functions 6. integration on locally compact spaces 7. signed and complex measures 8. integration of vector valued functions 9. l1 space 10. l1 space e. key theorems in linear functional analysis 1. bounded linear operators 2. compact operators 3. open mapping and closed graph theorems problems and exercises problem 1: clifford algebra; spin(4) exercise 2: product topology problem 3: strong and weak topologies in l2 exercise 4: htlder spaces see problem vi 4: application to the schrtdinger equation ii. differential calculus on banach spaces a. foundations 1. definitions. taylor expansion 2. theorems 3. diffeomorphisms 4. the euler equation 5. the mean value theorem 6. higher order differentials b. calculus of variations 1. necessary conditions for minima 2. sufficient conditions 3. lagrangian problems c. implicit function theorem. inverse function theorem 1. contracting mapping theorems 2. inverse function theorem 3. implicit function theorem 4. global theorems d. differential equations 1. first order differential equation 2. existence and uniqueness theorems for the lipschitzian case problems and exercises problem 1: banach spaces, first variation, linearized equation problem 2: taylor expansion of the action; jacobi fields; the feynman-green function; the van vleck matrix; conjugate points; caustics problem 3: euler-lagrange equation; the small disturbance equation; the soap bubble problem; jacobi fields iii. differentiable manifolds, finite dimensional case a. definitions 1. differentiable manifolds 2. diffeomorphisms 3. lie groups b. vector fields; tensor fields 1. tangent vector space at a point tangent vector as a derivation tangent vector defined by transformation properties tangent vector as an equivalence class of curves images under differentiable mappings 2. fibre bundles definition bundle morphisms tangent bundle frame bundle principal fibre bundle 3. vector fields vector fields moving frames images under cliffeomorphisms 4. covariant vectors; cotangent bundles dual of the tangent space space of differentials cotangent bundle reciprocal images 5. tensors at a point tensors at a point tensor algebra 6. tensor bundles; tensor fields c. groups of transformations i. vector fields as generators of transformation groups 2. lie derivatives 3. invariant tensor fields d. lie groups 1. definitions; notations 2. left and right translations; lie algebra; structure constants 3. one-parameter subgroups 4. exponential mapping; taylor expansion; canonical coordinates 5. lie groups of transformations; realization 6. adjoint representation 7. canonical form, maurer--cartan form problems and exercises problem 1: change of coordinates on a fiber bundle, configuration space, phase space problem 2: lie algebras of lie groups problem 3: the strain tensor problem 4: exponential map; taylor expansion; adjoint map; left and right differentials; haar measure problem 5: the group manifolds of soo) and su(2) problem 6: the 2-sphere iv. integration on manifolds a. exterior differential forms 1. exterior algebra exterior product local coordinates; strict components change of basis 2. exterior differentiation 3. reciprocal image of a form (pull back) 4. derivations and antiderivations definitions interior product 5. forms defined on a lie group invariant forms maurer--cartan structure equations 6. vector valued differential forms b. integration 1. integration orientation odd forms integration of n-forms in r" partitions of unity properties of integrals 2. stokes' theorem p-chains integrals of p-forms on p-chains boundaries mappings of chains proof of stokes' theorem 3. global properties homology and cohomology o-forms and o-chains betti numbers poincar6 lemmas de rham and poincare duality theorems c exterior differential systems 1. exterior equations 2. single exterior equation 3. systems of exterior equations ideal generated by a system of exterior equations algebraic equivalence solutions examples 4. exterior differential equations integral manifolds associated pfaff systems generic points closure 5. mappings of manifolds introduction immersion embedding submersion 6. pfaff systems complete integrability frobenius theorem integrability criterion examples dual form of the frobenius theorem 7. characteristic system characteristic manifold example: first order partial differential equations complete integrability construction of integral manifolds cauchy problem examples 8. invariants invariant with respect to a pfaff system integral invariants 9. example: integral invariants of classical dynamics liouville theorem canonical transformations 10. symplectic structures and hamiltonian systems problems and exercises problem 1: compound matrices problem 2:poincar6 lemma, maxwell equations, wormholes problem 3: integral manifolds problem 4: first order partial differential equations, hamilton-jacobi equations, lagrangian manifolds problem 5: first order partial differential equations, catastrophes problem 6: darboux theorem problem 7: time dependent hamiltonians see problem vi 11 paragraph c: electromagnetic shock waves v. riemannian manifolds. kahlerian manifolds a. the riemannian structure 1. preliminaries metric tensor hyperbolic manifold 2. geometry of submanifolds, induced metric 3. existence of a riemannian structure proper structure hyperbolic structure euler-poincare characteristic 4. volume element. the star operator volume element star operator 5. isometries b. linear connections 1. linear connections covariant derivative connection forms parallel translation affine geodesic torsion and curvature 2. riemannian connection definitions locally flat manifolds 3. second fundamental form 4. differential operators exterior derivative operator divergence laplacian c. geodesics 1. arc length 2. variations euler equations energy integral 3. exponential mapping definition normal coordinates 4. geodesics on a proper riemannian manifold properties geodesic completeness 5. geodesics on a hyperbolic manifold d. almost complex and kahlerian manifolds problems and exercises problem 1 maxwell equation; gravitational radiation problem 2: the schwarzschild solution problem 3: geodetic motion; equation of geodetic deviation; exponentiation; conjugate points problem 4: causal structures; conformal spaces; weyl tensor vbis. connections on a principal fibre bundle a. connections on a principal fibre bundle 1. definitions 2. local connection l-forms on the base manifold existence theorems section canonically associated with a trivialization potentials change of trivialization examples 3. covariant derivative associated bundles parallel transport covariant derivative examples 4. curvature definitions cartan structural equation local curvature on the base manifold field strength bianchi identities 5. linear connections definition soldering form, torsion form torsion structural equation standard horizontal (basic) vector field curvature and torsion on the base manifold bundle homomorphism metric connection b. hoionomy 1. reduction 2. holonomy groups c. characteristic classes and invariant curvature integrals 1. characteristic classes 2. gauss-bonnet theorem and chern numbers 3. the atiyah-singer index theorem problems and exercises problem 1: the geometry of gauge fields problem 2: charge quantization. monopoles problem 3: instanton solution of euclidean su(2) yang-mills theory (connection on a non-trivial su(2) bundle over s4) problem 4: spin structure; spinors; spin connections vi. distributions a. test functions 1. seminorms definitions hahn-banach theorem topology defined by a family of seminorms 2. d-spaces definitions inductive limit topology convergence in dm(u) and d(u) examples of functions in truncating sequences density theorem b. distributions 1. definitions distributions measures; dirac measures and leray forms distribution of order p support of a distribution distributions with compact support 2. operations on distributions sum product by c function direct product derivations examples inverse derivative 3. topology on d' weak star topology criterion of convergence 4. change of variables in rn change of variables in rn transformation of a distribution under a diffeomorphism invariance 5. convolution convolution algebra l1(rn) convolution algebra d'+ and d'- derivation and translation of a convolution product regularization support of a convolution equations of convolution differential equation with constant coefficients systems of convolution equations kernels 6. fourier transform fourier transform of integrable functions tempered distributions fourier transform of tempered distributions paley-wiener theorem fourier transform of a convolution 7. distribution on a c∞ paracompact manifold 8. tensor distributions c. sobolev spaces and partial differential equations i. sobolev spaces properties density theorems w? spaces fourier transform plancherel theorem sobolev's inequalities 2. partial differential equations definitions cauchy-kovalevski theorem classifications 3. elliptic equations; laplacians elementary solution of laplace's equation subharmonic distributions potentials energy integral, green's formula, unicity theorem liouville's theorem boundary-value problems green function introduction to hilbertian methods; generalized dirichlet problem hilbertian methods example: neumann problem 4. parabolic equations heat diffusion 5. hyperbolic equation; wave equations elementary solution of the wave equation cauchy problem energy integral, unicity theorem existence theorem 6. leray theory of hyperbolic systems 7. second order systems; propagators problems and exercises problem 1: bounded distributions problem 2: laplacian of a discontinuous function exercise 3: regularized functions problem 4: application to the schrbdinger equation exercise 5: convolution and linear continuous responses problem 6: fourier transforms of exp (-x2) and exp (ix2) problem 7: fourier transforms of heaviside functions and pr(l/x) problem 8: dirac bitensors problem 9: legendre condition problem 10: hyperbolic equations; characteristics problem 11: electromagnetic shock waves problem 12: elementary solution of the wave equation problem 13: elementary kernels of the harmonic oscillator vii. differentiable manifolds, infinite dimensional case a. infinite-dimensional manifolds 1. definitions and general properties e-manifolds differentiable functions tangent vector vector and tensor field differential of a mapping submanifold immersion, embedding, submersion flow of a vector field differential forms 2. symplectic structures and hamiltonian systems definitions complex structures canonical symplectic form symplectic transformation hamiltonian vector field conservation of energy theorem riemannian manifolds b. theory of degree; leray-schauder theory i. definition for finite dimensional manifolds degree integral formula for the degree of a function continuous mappings 2. properties and applications fundamental theorem borsuk's theorem brouwer's fixed point theorem product theorem 3. leray-schauder theory definitions compact mappings degree of a compact mapping schauder fixed point theorem leray-schauder theorem c. morse theory 1. introduction 2. definitions and theorems 3. index of a critical point 4. critical neck theorem d. cylindrical measures, wiener integral 1. introduction 2. promeasures and measures on a locally convex space projective system promeasures image of a promeasure integration with respect to a promeasure of a cylindrical function fourier transforms 3. gaussian promeasures gaussian measures on rn gaussian promeasures gaussian promeasures on hilbert spaces 4. the wiener measure wiener integral sequential wiener integral problems and exercises problem a: the klein-gordon equation problem b: application of the leray-schauder theorem problem c1: the reeb theorem problem c2: the method of stationary phase problem d1: a metric on the space of paths with fixed end points problem d2: measures invariant under translation problem d3: cylindrical σ-field of c([a, b]) problem d4: generalized wiener integral of a cylindrical function references symbols index

章节摘录

插图：

编辑推荐

《分析流形和物理学(第1卷):基础(修订版)》由世界图书出版公司出版。

用户评论 (总计3条)

•   东西不错，但书的包装薄膜有破损
•   法国人写的。内容非常丰富，几乎涵盖了量子场论和GR用到的所有数学工具。
•   是研究量子引力的数学储备书，值得拥有！很全！