偏微分方程（第1卷 第2版 英文版） 作者：（美）泰勒 著 出版时间：2014年版 内容简介 《偏微分方程（第1卷）（第2版）》是一套3卷集经典名著，第一版曾影印出版，广受好评。第2版新增内容312页（3卷），这是第1卷。本卷在引入连续统力学、电磁学和复分析和实例的基础上，介绍了许多解决实际问题的方法，如傅里叶分析、分布理论和索伯列夫空间，这些方法可用于解决线性偏微分方程的基本问题。书中涉及的线性偏微分方程有拉普拉斯方程、热方程、波动方程、一般椭圆方程、双曲方程和抛物方程等。目次：偏微分方程和向量场基本理论；拉普拉斯方程和波动方程；傅里叶分析、分布函数和常系数线性偏微分方程；索伯列夫空间；线性椭圆方程；线性发展方程；泛函分析概述；流形、向量丛和李群。 目录 Contents of Volumes II and III Preface 1Basic Theory of ODE and Vector Fields 1 The derivative 2 Fundamental local existence theorem for ODE 3 Inverse function and implicit function theorems 4 Constant-coefficientlinear systems; exponentiation of matrices 5 Variable-coefficientlinear systems of ODE: Duhamels principle 6 Dependence of solutions on initial data and on other parameters 7 Flows and vector fields 8 Lie brackets 9 Commuting flows; Frobeniuss theorem 10 Hamiltoniansystems 11 Geodesics 12 Variational problems and the stationary action principle 13 Differential forms N 14 The symplectic form and canonical transformations 15 First-order scalar nonlinear PDE 16 Completely integrable hamiltonian systems 17 Examples of integrable systems; central force problems 18 Relativistic motion 19 Topological applications of differential forms 20 Critical points and index of a vector field A Nonsmooth vector fields References 2 The Laplace Equation and Wave Equation 1 Vibrating strings and membranes 2 The divergence of a vector field 3The covariant derivative and divergence of tensor fields 4 The Laplace operator on a Riemannian manifold 5 The wave equation on a product manifold and energy conservation 6 Uniqueness and finite propagation speed 7 Lorentz manifolds and stress-energy tensors 8 More general hyperbolic equations; energy estimates 9 The symbol of a differential operator and a general Green-Stokes formula 10 The Hodge Laplacian on k-forms 11 Maxwells equations References 3 FourierAnalysisDistributions and Constant-Coefficient Linear PDE 1 Fourier series 2 Harmonic functions and holomorphic functions in the plane 3 The Fourier transform 4 Distributions and tempered distributions 5 The classical evolution equations 6 Radial distributions polar coordinates and Bessel functions 7 The method ofimages and Poissons summation formula 8 Homogeneous distributions and principal value distributions 9 Elliptic operators 10 Local solvability ofconstant-coefficientPDE 11 The discrete Fourier transform 12 The fast Fourier transform A The mighty Gaussian and the sublime gamma function References 4 SobolevSpaces 1 Sobolev spaces on Rn 2 The complex interpolation method 3 Sobolev spaces on compact manifolds 4 Sobolev spaces on bounded domains 5 The Sobolev spaces H50（Ω） 6 The Schwartzkerneltheorem 7 Sobolev spaces on rough domains References 5 Linear Elliptic Equations 1 Existence and regularity of solutions to the Dirichlet problem 2 The weak and strong maximum principles 3 The Dirichlet problem on the ba 4 The Riemann mapping theorem （smooth boundary） 5 The Dirichlet problem on a domain with a rough boundary 6 The Riemann mapping theorem （rough boundary） 7 The Neumann boundary problem 8 The Hodge decomposition and harmonic forms 9 Natural boundary problems for the Hodge Laplacian 10 Isothermal coordinates and conformal structures on surfaces 11 General elliptic boundary problems 12 Operator properties ofregular boundary problems …… 6 Linear Evolution Equations A Outline of FunctionaIAnalysis B Marufolds Vector Bundles and Lie Groups ……
