泰勒编著《偏微分方程（第1卷第2版）》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。

Contents of Volumes II and III

Preface

1 Basic 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

 A Spaces of generalized functions on manifolds with boundary

 B The Mayer-Vietoris sequ6nce in deRham cohomology

 References

6 Linear Evolution Equations

 1 The heat equation and the wave equation on bounded domains

 2 The heat equation and wave equation on unbounded domains

 3 Maxwell's equations

 4 TheCauchy-Kowalewsky theorem

 5 Hyperbolic systems

 6 Geometrical optics

 7 The formation of caustics

 8 Boundary layer phenomena for the heat semigroup

 A Some Banach spaces of harmonic functions

 B The stationary phase method

 References

A Outline of Functional Analysis

 1 Banach spaces

 2 Hilbert spaces

 3 Fr6chet spaces; locally convex spaces

 4 Duality

 5 Linear operators

 6 Compact operators

 7 Fredholm operators

 8 Unbounded operators

 9 Semigroups

 References

B Manifolds, Vector Bundles, and Lie Groups

 1 Metric spaces and topological spaces

 2 Manifolds

 3 Vector bundles

 4 Sard's theorem

 5 Lie groups

 6 The Campbell-Hausdorffformula

 7 Representations of Lie groups and Lie algebras

 8 Representations of compact Lie groups

 9 Representations of SU(2) and related groups

 References

Index