偏微分方程

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

Partial differential equations equally appear in physics and geometry. Within mathematics they unite the areas of complex analysis, differential geome-try and calculus of variations. The investigation of partial differential equa-tions has substantially contributed to the development of functional analysis.Though a relatively uniform treatment of ordinary differential equations is possible, quite multiple and diverse methods are available for partial differen-tial equations. With this two-volume textbook we intend to present the entire domain PARTIAL DIFFERENTIAL EQUATIONS - so rich in theories and applica-tions - to students at the intermediate level.