Lectures on Partial Differential Equations

by Arnold, V. I.; Cooke, Roger

• ISBN13: 9783540404484
• ISBN10: 3540404481
• Format: Paperback
• Copyright: 2004-02-01
• Publisher: Springer Verlag

Summary

Choice Outstanding Title! (January 2006)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.

Table of Contents

Preface to the Second Russian Edition
1. The General Theory for One First-Order Equation	1	(10)
Literature	10	(1)
2. The General Theory for One First-Order Equation (Continued)	11	(10)
Literature	20	(1)
3. Huygens' Principle in the Theory of Wave Propagation	21	(6)
4. The Vibrating String (d'Alembert's Method)	27	(8)
4.1. The General Solution	27	(1)
4.2. Boundary-Value Problems and the Cauchy Problem	28	(1)
4.3. The Cauchy Problem for an Infinite String. d'Alembert's Formula	29	(2)
4.4. The Semi-Infinite String	31	(1)
4.5. The Finite String. Resonance	31	(1)
4.6. The Fourier Method	32	(3)
5. The Fourier Method (for the Vibrating String)	35	(6)
5.1. Solution of the Problem in the Space of Trigonometric Polynomials	35	(1)
5.2. A Digression	36	(1)
5.3. Formulas for Solving the Problem of Section 5.1	36	(1)
5.4. The General Case	36	(1)
5.5. Fourier Series	37	(1)
5.6. Convergence of Fourier Series	37	(2)
5.7. Gibbs' Phenomenon	39	(2)
6. The Theory of Oscillations. The Variational Principle	41	(10)
Literature	50	(1)
7. The Theory of Oscillations. The Variational Principle (Continued)	51	(14)
8. Properties of Harmonic Functions	65	(12)
8.1. Consequences of the Mean-Value Theorem	67	(6)
8.2. The Mean-Value Theorem in the Multidimensional Case	73	(4)
9. The Fundamental Solution for the Laplacian. Potentials	77	(16)
9.1. Examples and Properties	78	(1)
9.2. A Digression. The Principle of Superposition	79	(10)
9.3. Appendix. An Estimate of the Single-Layer Potential	89	(4)
10. The Double-Layer Potential	93	(12)
10.1. Properties of the Double-Layer Potential	94	(11)
11. Spherical Functions. Maxwell's Theorem. The Removable Singularities Theorem	105	(16)
12. Boundary-Value Problems for Laplace's Equation. Theory of Linear Equations and Systems	121	(14)
12.1. Four Boundary-Value Problems for Laplace's Equation	121	(4)
12.2. Existence and Uniqueness of Solutions	125	(2)
12.3. Linear Partial Differential Equations and Their Symbols	127	(8)
A. The Topological Content of Maxwell's Theorem on the Multifield Representation of Spherical Functions	135	(14)
A.1. The Basic Spaces and Groups	136	(1)
A.2. Some Theorems of Real Algebraic Geometry	137	(2)
A.3. From Algebraic Geometry to Spherical Functions	139	(2)
A.4. Explicit Formulas	141	(3)
A.5. Maxwell's Theorem and CP2/conj = S4	144	(1)
A.6. The History of Maxwell's Theorem	145	(4)
Literature	146	(3)
B. Problems	149
B.1. Material from the Seminars	149	(7)
B.2. Written Examination Problems	156