In the mid-twentieth century the theory of partial differential equations was considered the summit of mathematics, both because of the difficulty and significance of the problems it solved and because it came into existence later than most areas of mathematics.

The cause of this degeneration of an important general mathematical theory into an endless stream of papers bearing titles like "On a property of a solution of a boundary-value problem for an equation" is most likely the attempt to create a unified, all-encompassing, superabstract "theory of everything."

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 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 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.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' Phenomenon

6.The Theory of Oscillations.The Variational 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-Value Theorem in the Multidimensional 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 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 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/con≈S4

　A.6.The History of Maxwell's Theorem

　Literature

B.Problems

　B.1.Material frorn the Seminars

　B.2.Written Examination Problems