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.
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