Title Page; Copyright Page; PREFACE; Table of Contents; Chapter 1 --
ORDINARY DIFFERENTIAL EQUATIONS IN MORE THAN TWO VARIABLES; 1. Surfaces and Curves in Three Dimensions; 2. Simultaneous Differential Equations of the First Order and the First Degree in Three Variables; 3. Methods of Solution of dx/P = dy/Q = dz/R; 4. Orthogonal Trajectories of a System of Curves on a Surface; 5. Pfaffian Differential Forms and Equations; 6. Solution of Pfaffian Differential Equations in Three Variables; 7. Carathéodory's Theorem; 8. Application to Thermodynamics. Chapter 2 --
PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER1. Partial Differential Equations; 2. Origins of First-order Partial Differential Equations; 3. Cauchy's Problem for First-order Equations; 4. Linear Equations of the First Order; 5. Integral Surfaces Passing through a Given Curve; 6. Surfaces Orthogonal to a Given System of Surfaces; 7. Nonlinear Partial Differential Equations of the First Order; 8. Cauchy's Method of Characteristics; 9. Compatible Systems of First-order Equations; 10. Charpit's Method; 11. Special Types of First-order Equations. 12. Solutions Satisfying Given Conditions13. Jacobi's Method; 14. Applications of First-order Equations; Chapter 3 --
PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER; 1. The Origin of Second-order Equations; 2. Second-order Equations in Physics; 3. Higher-order Equations in Physics; 4. Linear Partial Differential Equations with Constant Coefficients; 5. Equations with Variable Coefficients; 6. Characteristic Curves of Second-order Equations; 7. Characteristics of Equations in Three Variables; 8. The Solution of Linear Hyperbolic Equations; 9. Separation of Variables. 10. The Method of Integral Transforms11. Nonlinear Equations of the Second Order; Chapter 4 --
LAPLACE'S EQUATION; 1. The Occurrence of Laplace's Equation in Physics; 2. Elementary Solutions of Laplace's Equation; 3. Families of Equipotential Surfaces; 4. Boundary Value Problems; 5. Separation of Variables; 6. Problems with Axial Symmetry; 7. Kelvin's Inversion Theorem; 8. The Theory of Green's Function for Laplace's Equation; 9. The Relation of Dirichlet's Problem to the Calculus of Variations; 10. "Mixed" Boundary Value Problems; 11. The Two-dimensional Laplace Equation. 12. Relation of the Logarithmic Potential to the Theory of Functions13. Green's Function for the Two-dimensional Equation; Chapter 5 --
THE WAVE EQUATION; 1. The Occurrence of the Wave Equation in Physics; 2. Elementary Solutions of the One-dimensional Wave Equation; 3. The Riemann-Volterra Solution of the One-dimensional Wave Equation; 4. Vibrating Membranes: Application of the Calculus of Variations; 5. Three-dimensional Problems; 6. General Solutions of the Wave Equation; 7. Green's Function for the Wave Equation; 8. The Nonhomogeneous Wave Equation; 9. Riesz's Integrals.