分析流形和物理学 pdf epub mobi txt 电子书 下载 2025

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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