Lectures on partial differential equations偏微分方程讲座
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作者： Vladimir I. Arnold著
出 版 社： 湖南文艺出版社
出版时间： 2004-1-1字数：版次： 1页数： 157印刷时间： 2004/04/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540404484包装： 平装内容简介
Like all of Vladimir Arnold’s books，this book is full of geometric insight。Arnold illustrates every principle with a figure。This book aims to cover the most basic parts of the subject and confines itself largely to the Cauchy and Neumann problems for the classical linear equations of mathematical physics， especially Laplace’s equation and the wave equation， although the heat equation and the Korteweg-de Vries equation are also discussed。 Physical intuition is emphasized。 A large number of problems are sprinkled throughout the book， and a full set of problems from examinations given in Moscow are included at the end。 Some of these problems are quite challenging！
What makes the book unique is Arnold’s particular talent at holding a topic up for examination from a new and fresh perspective。 He likes to blow away the fog of generality that obscures so much mathematical writing and reveal the essentially simple intuitive ideas underlying the subject。 No other mathematical writer does this quite so well as Arnold。
目录
Preface to the Second Russian Edition
1.The General Theory for One First-Order Equation.
Literature
2.The General Theory for One First-Order Equation（Continued）
Literature
3.Huygens’Principle in the Theory of Wave Propagation.
4.The Vibrating String（d’Alembert’S Method）
4.1.The General Solution
4.2.Boundary—Value Problems and the Cauchy Problem
4.3.The Cauchy Problem for an Infinite String.d’Alembert’S Formula
4.4.The Semi—Infinite String
4.5.The Finite String.Resonance
4.6.The Fclurier Method
5.The Fourier Method（for the Vibrating String）
5.1.Solution of the Problem in the Space of Trigonometric Polynomials.
5.2.A Digression
5.3.F0rmulas for Solving the Problem of Section
5.4.The General C8.se
5.5.Fourier Series
5.6.Convergence of Fourier Series
5.7.Gibbs’Phenomenon
6.The Theory of Oscillations.The Vlariational Principle
Literature
7.The Theory of Oscillations.The Variational Principle （Continued）
8.Properties of Harmonic Functions
8.1.Consequences of the Mean—Value Theorem
8.2.The Mean.VaJue Theorem in the MultidimensionaI Case
9.The Fundamental Solution for the Laplacian.Potentials
9.1.Examples and Properties.
9.2.A Digression.The Principle of Superposition
9.3.Appendix.An Estimate of the Single—Layer Potential
10.The Double-Layer Potential.
10.1.Properties of the Double—Layer Potential、
11.Spherical Functions.Maxwell’S Theorem.The Removable
Singularities Theorem.
12.Boundary—Value Problems for Laplace’S Equation.Theory
of Linear Equations and Systems
12.1.F0ur Boundary—Value Problems for Laplace’S Equation.
12.2.Existence and Uniqueness of Solutions
12.3.Linear Partial Differential Equations and Their Symbols
A.The Topological Content of Maxwell’S Theorem on the
Multifield Representation of Spherical Functions
A.1.The Basic Spaces and Groups
A.2.Some Theorems of Real Algebraic Geometry
A.3.From Algebraic Geometry to Spherical Functions
A.4.Explicit Formulas.
A.5.Maxwell’S Theorem and CP2/conj≈S4
A.6.The History of Maxwell’S Theorem
Literature
B.Problems
B.1.Material from the Sereinars
B.2.Written Examination Problems