《偏微分方程(第1卷)》是一部两卷集的偏微分方程教材。多变量椭圆，抛物和双曲方程是研究的主要对象，解决了PDE和多变量方法之间的关系。第一卷中集中研究了流形上的积分和微分，泛函解析基础，映射的Brouwer度，广义解析函数和圆周同调这些议题，在这一卷中通过积分表示论解决偏微分方程问题，第二卷中讲述函数解析解法。书中各章的独立性较强，有一定偏微分方程基本知识的读者可以独立阅读各章。本书由索维尼著。

1 Differentiation and Integration on Manifolds

The Weierstrai

Parameter-invariant integrals and differential forms

The exterior derivative of differential forms

The Stokes integral theorem for manifolds

The integral theorems of Gauand Stokes

Curvilinear integrals

The lemma of Poincare

Co-derivatives and the Laplace-Beltrami operator

Some historical notices to chapter I

2 Foundations of Functional Analysis

Daniell's integral with examples

Extension of Daniell's integral to Lebesgue's integral

Measurable sets

Measurable functions

Riemann's and Lebesgue's integral on rectangles

Banach and Hilbert spaces

The Lebesgue spaces LP（X）

Bounded linear functionals on LP（X） and weak convergence .

Some historical notices to chapter II

3 Brouwer's Degree of Mapping with Geometric Applications

The winding number

The degree of mapping in Rn

Geometric existence theorems

The index of a mapping

The product theorem

Theorems of Jordan-Brouwer

4 Generalized Analytic Functions

The Cauchy-Riemann differential equation

Holomorphic functions in Cn

Geometric behavior of holomorphic functions in C

Isolated singularities and the general residue theorem

The inhomogeneous Cauchy-Riemann differential equation

Pseudoholomorphic functions

Conformal mappings

Boundary behavior of conformal mappings

Some historical notices to chapter IV

5 Potential Theory and Spherical Harmonics

Poisson's differential equation in Rn

Poisson's integral formula with applications

Dirichlet's problem for the Laplace equation in Rn

Theory of spherical harmonics: Fourier series

Theory of spherical harmonics in n variables

6 Linear Partial Differential Equations in Rn

The maximum principle for elliptic differential equations

Quasilinear elliptic differential equations

The heat equation

Characteristic surfaces

The wave equation in Rn for n = 1, 3, 2

The wave equation in Rn for n _> 2

The inhomogeneous wave equation and an initial-boundary-

value problem

Classification, transformation and reduction of partial

differential equations

Some historical notices to the chapters V and VI

References

Index