偏微分方程讲义

出版时间：2009-8  出版社：世界图书出版公司  作者：Vladimir I. Arnold  页数：157  
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前言

　　In the mid-twentieth century the theory of partial differential equations wasconsidered the summit of mathematics， both because of the difficulty andsignificance of the problems it solved and because it came into existence laterthan most areas of mathematics.　　Nowadays many are inclined to look disparagingly at this remarkable areaof mathematics as an old-fashioned art of juggling inequalities or as a testingground for applications of functional analysis. Courses in this subject haveeven disappeared from the obligatory program of many universities （for ex-ample， in Paris）. Moreover， such remarkable textbooks as the classical three-volume work of Goursat have been removed as superfluous from the library ofthe University of Paris-7 （and only through my own intervention was it possi-ble to save them， along with the lectures of Klein， Picard， Hermite， Darboux，Jordan　）　　The cause of this degeneration of an important general mathematical the-ory into an endless stream of papers bearing titles like "On a property ofa solution of a boundary-value problem tor an equation" is most likely theattempt to create a unified， all-encompassing， superabstract "theory of every-thing."　　The principal source of partial differential equations is found in thecontinuous-medium models of mathematical and theoretical physics. Attemptsto extend the remarkable achievements of mathematical physics to systemsthat match its models only formally lead to complicated theories that aredifficult to visualize as a whole， just as attempts to extend the geometry ofsecond-order surfaces and the algebra of quadratic forms to objects of higherdegrees quickly leads to the detritus of algebraic geometry with its discourag-ing hierarchy of complicated degeneracies and answers that can be computedonly theoretically.　　The situation is even worse in the theory of partial differential equations：here the difficulties of conunutative algebraic geometry are inextricably boundup with noncomnutative differential algebra， in addition to which the topo-logical and analytic problems that arise arc profoundly nontrivial.

内容概要

　　In the mid-twentieth century the theory of partial differential equations wasconsidered the summit of mathematics， both because of the difficulty andsignificance of the problems it solved and because it came into existence laterthan most areas of mathematics.　　Nowadays many are inclined to look disparagingly at this remarkable areaof mathematics as an old-fashioned art of juggling inequalities or as a testingground for applications of functional analysis. Courses in this subject haveeven disappeared from the obligatory program of many universities （for ex-ample， in Paris）. Moreover， such remarkable textbooks as the classical three-volume work of Goursat have been removed as superfluous from the library ofthe University of Paris-7 （and only through my own intervention was it possi-ble to save them， along with the lectures of Klein， Picard， Hermite， Darboux，Jordan　）

书籍目录

Preface to the Second Russian Edition1.  The General Theory for One First-Order Equation Literature  2.  The General Theory for One First-Order Equation(Continued)Literature3.  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 Ca'uchy Problem　4.3. The Cauehy Problem for an Infinite Strifig. d'Alembert's Formula  　4.4. The Semi-Infinite String  　4.5. The Finite String. Resonance　4.6. The Fourier Method5.  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. Formulas for Solving the Problem of Section 5.1　5.4. The General Case　5.5. Fourier Series　5.6. Convergence of Fourier Series  　5.7. Gibbs' Phenomenon6.  The Theory of Oscillations. The Variational Principle Literature7.  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-Value Theorem in the Multidimensional Case9.  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 Potential10. The Double-Layer Potential　10.1. Properties of the Double-Layer Potential11 Spherical Functions. Maxwell's Theorem. The Removable　Singularities Theorem12. Boundary-Value Problems for Laplaee's Equation. Theory of Linear Equations and Systems　12.1. Four Boundary-Value Problems for Laplace's Equation　12.2. Existence and Uniqueness of Solutions　12.3. Linear Partial Differential Equations and Their SymbolsA. 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/con≈S4　A.6. The History of Maxwell's Theorem　LiteratureB. Problems　B.1. Material frorn the Seminars 　B.2. Written Examination Problems.

用户评论 (总计8条)

•   非常新，内容不错。ODE也是作者的。
•   很薄的一本书，风格也很独特。
•   经典的教材。
•   阿诺尔德的经典著作
•   很不错的书，大家之作，值得一读，想必Anold的书是相当的难啃！
•   Arnold的书非常重视几何思路
  而且在直观的同时也丝毫不会丧失其严谨性
  这本书虽然很薄，但确实是读一页就有一页的收获
  比起通常PDE课本大量的堆积公式以及繁琐的范数不等式，这本书确实有超凡的高度
  不过还应注意，这本书虽然能提供高层次思路，但如果需要扎实的PDE功底，其他经典教材还是不能少
•   这是Anorld 的经典图书之一，秉承了他一贯的风格，尽管页码很少，可是高屋建瓴，还是很难理解的。不过，由于他本人已经辞世，所以，值得收藏
•   这本书还是比较基础的 适合入门使用 我还是最喜欢这种书 有读下去的兴趣 要不太难了就不想看了