这本《分析流形和物理学(第1卷基础修订版)》由法国许凯布里哈特所著，内容是：All too often in physics familiarity is a substitute for understanding, and the beginner who lacks familiarity wonders which is at fault: physics or himself. Physical mathematics provides well defined concepts and techniques for the study of physical systems. It is more than mathematical techniques used in the solution of problems which have already been formulated; it helps in the very formulation of the laws of physical systems and brings a better understanding of physics. Thus physical mathematics includes mathematics which gives promise of being useful in our analysis of physical phenomena. Attempts to use mathematics for this purpose may fail because the mathematical tool is too crude; physics may then indicate along which lines it should be sharpened. In fact, the analysis of physical systems has spurred many a new mathematical development.

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 equation

Ⅱ. 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 fields

Ⅲ. 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-sphere

Ⅳ. 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 waves

Ⅴ. 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 tensor

Ⅴbis. 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 connections

Ⅵ. 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 oscillator

Ⅶ. 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 function

References

Symbols

Index