《分析流形和物理学(第1卷):基础(修订版)》内容简介：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.
目录
Ⅰ. Review of Fundamental Notions of Analysis A. Set Theory, Definitions B. Algebraic Structures, Definitions C. Topology D. Integration E. Key Theorems in Linear Functional Analysis Problems and Exercises Problem 1: Clifford algebra; Spin(4) Exercise 2: Product topology Problem 3: Strong and weak topologies in Lz Exercise 4: Holder spaces See Problem VI 4: Application to the Schrtdinger equationⅡ. Differential Calculus on Banach Spaces A. Foundations B. Calculus of Variations C. Implicit Function Theorem. Inverse Function Theorem D. Differential Equations 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 fieldsⅢ. Differentiable Manifolds, Finite Dimensional Case A. Definitions B. Vector Fields; Tensor Fields C. Groups of Transformations D. Lie Groups 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 SO(3) and SU(2) Problem 6: The 2-sphereⅣ. Integration on Manifolds A. Exterior Differential Forms B. Integration C. Exterior Differential Systems Problems and Exercises Problem 1: Compound matrices Problem 2: Poincare 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 wavesⅤ. Riemannian Manifolds. Kahlerian Manifolds A. The Riemannian Structure B. Linear Connections C. Geodesics D. Almost Complex and Kahlerian Manifolds Problems and Exercises Problem I: 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 tensorⅤbis. Connections on a Principal Fibre Bundle A. Connections on a Principal Fibre Bundle B. Holonomy C. Characteristic Classes and Invariant Curvature Integrals Problems and Exercises Problem 1: The geometry of gauge fields Problem 2: Charge quantization Monopoles Problem 3: Instanton solution of eucfidean SU(2) Yang-Mills theory (connection on a non-trivial SU(2) bundle over S4) Problem 4: Spin structure, spinors, spin connectionsⅥ. Distributions A. Test Functions B. Distributions C. Sobolev Spaces and Partial Differential Equations Problems and Exercises Problem 1: Bounded distributions Problem 2: Laplacian of a discontinuous function Exercise 3: Regularized functions Problem 4: Application to the Schrodinger 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 Pv(I/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 oscillatorⅦ. Differentiable Manifolds, Infinite Dimensional Case A. Infinite-Dimensional Manifolds B. Theory of Degree; Leray-Schauder Theory C. Morse Theory D. Cylindrical Measures, 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 functionReferencesSymbolsIndex
Ⅰ. Review of Fundamental Notions of Analysis A. Set Theory, Definitions B. Algebraic Structures, Definitions C. Topology D. Integration E. Key Theorems in Linear Functional Analysis Problems and Exercises Problem 1: Clifford algebra; Spin(4) Exercise 2: Product topology Problem 3: Strong and weak topologies in Lz Exercise 4: Holder spaces See Problem VI 4: Application to the Schrtdinger equationⅡ. Differential Calculus on Banach Spaces A. Foundations B. Calculus of Variations C. Implicit Function Theorem. Inverse Function Theorem D. Differential Equations 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 fieldsⅢ. Differentiable Manifolds, Finite Dimensional Case A. Definitions B. Vector Fields; Tensor Fields C. Groups of Transformations D. Lie Groups 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 SO(3) and SU(2) Problem 6: The 2-sphereⅣ. Integration on Manifolds A. Exterior Differential Forms B. Integration C. Exterior Differential Systems Problems and Exercises Problem 1: Compound matrices Problem 2: Poincare 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 wavesⅤ. Riemannian Manifolds. Kahlerian Manifolds A. The Riemannian Structure B. Linear Connections C. Geodesics D. Almost Complex and Kahlerian Manifolds Problems and Exercises Problem I: 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 tensorⅤbis. Connections on a Principal Fibre Bundle A. Connections on a Principal Fibre Bundle B. Holonomy C. Characteristic Classes and Invariant Curvature Integrals Problems and Exercises Problem 1: The geometry of gauge fields Problem 2: Charge quantization Monopoles Problem 3: Instanton solution of eucfidean SU(2) Yang-Mills theory (connection on a non-trivial SU(2) bundle over S4) Problem 4: Spin structure, spinors, spin connectionsⅥ. Distributions A. Test Functions B. Distributions C. Sobolev Spaces and Partial Differential Equations Problems and Exercises Problem 1: Bounded distributions Problem 2: Laplacian of a discontinuous function Exercise 3: Regularized functions Problem 4: Application to the Schrodinger 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 Pv(I/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 oscillatorⅦ. Differentiable Manifolds, Infinite Dimensional Case A. Infinite-Dimensional Manifolds B. Theory of Degree; Leray-Schauder Theory C. Morse Theory D. Cylindrical Measures, 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 functionReferencesSymbolsIndex
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