Lectures on Partial Differential Equations

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.

1. The General Theory for One First-Order Equation.- 2. The General Theory for One First-Order Equation (Continued).- 3. Huygens’ Principle in the Theory of Wave Propagation.- 4. The Vibrating String (d’Alembert’s Method).- 5. The Fourier Method (for the Vibrating String).- 6. The Theory of Oscillations. The Variational Principle.- 7. The Theory of Oscillations. The Variational Principle (Continued).- 8. Properties of Harmonic Functions.- 9. The Fundamental Solution for the Laplacian. Potentials.- 10. 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.- 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.6. The History of Maxwell’s Theorem.- Literature.- B. Problems.- B.1. Material from the Seminars.- B.2. Written Examination Problems.