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
……
