Partial Differential Equations of Mathematical Physics

Translation Editor's PrefaceAuthor's Prefaces to the First and Third EditionsLecture 1. Derivation of the Fundamental Equations § 1. Ostrogradski's Formula § 2. Equation for Vibrations of a String § 3. Equation for Vibrations of a Membrane § 4. Equation of Continuity for Motion of a Fluid. Laplace's Equation § 5. Equation of Heat Conduction § 6. Sound WavesLecture 2. The Formulation of Problems of Mathematical Physics. Hadamard's Example § 1. Initial Conditions and Boundary Conditions § 2. The Dependence of the Solution on the Boundary Conditions. Hadamard's ExampleLecture 3. The Classification of Linear Equations of the Second Order § 1. Linear Equations and Quadratic Forms. Canonical Form of an Equation § 2. Canonical Form of Equations in Two Independent Variables § 3. Second Canonical Form of Hyperbolic Equations in Two Independent Variables § 4. CharacteristicsLecture 4. The Equation for a Vibrating String and its Solution by d'Alembert's Method § 1. D'Alembert's Formula. Infinite String § 2. String with Two Fixed Ends § 3. Solution of the Problem for a Non-Homogeneous Equation and for More General Boundary ConditionsLecture 5. Riemann's Method § 1. The Boundary-Value Problem of the First Kind for Hyperbolic Equations § 2. Adjoint Differential Operators § 3. Riemann's Method § 4. Riemann's Function for the Adjoint Equation § 5. Some Qualitative Consequences of Riemann's FormulaLecture 6. Multiple Integrals: Lebesgue Integration § 1. Closed and Open Sets of Points § 2. Integrals of Continuous Functions on Open Sets § 3. Integrals of Continuous Functions on Bounded Closed Sets § 4. Summable Functions § 5. The Indefinite Integral of a Function of One Variable. Examples § 6. Measurable Sets. Egorov's Theorem § 7. Convergence in the Mean of Summable Functions § 8. The Lebesgue-Fubini TheoremLecture 7. Integrals Dependent on a Parameter § 1. Integrals which are Uniformly Convergent for a Given Value of Parameter § 2. The Derivative of an Improper Integral with respect to a ParameterLecture 8. The Equation of Heat Conduction § 1. Principal Solution § 2. The Solution of Cauchy's ProblemLecture 9. Laplace's Equation and Poisson's Equation § 1. The Theorem of the Maximum § 2. The Principal Solution. Green's Formula § 3. The Potential due to a Volume, to a Single Layer, and to a Double LayerLecture 10. Some General Consequences of Green's Formula § 1. The Mean-Value Theorem for a Harmonic Function § 2. Behavior of a Harmonic Function near a Singular Point § 3. Behavior of a Harmonic Function at Infinity. Inverse PointsLecture 11. Poisson's Equation in an Unbounded Medium. Newtonian PotentialLecture 12. The Solution of the Dirichlet Problem for a SphereLecture 13. The Dirichlet Problem and the Neumann Problem for a Half-SpaceLecture 14. The Wave Equation and the Retarded Potential § 1. The Characteristics of the Wave Equation § 2. Kirchhoff's Method of Solution of Cauchy's ProblemLecture 15. Properties of the Potentials of Single and Double Layers § 1. General Remarks § 2. Properties of the Potential of a Double Layer § 3. Properties of the Potential of a Single Layer § 4. Regular Normal Derivative § 5. Normal Derivative of the Potential of a Double Layer § 6. Behavior of the Potentials at InfinityLecture 16. Reduction of the Dirichlet Problem and the Neumann Problem to Integral Equations § 1. Formulation of the Problems and the Uniqueness of their Solutions § 2. The Integral Equations for the Formulated ProblemsLecture 17. Laplace's Equation and Poisson's Equation in a Plane § 1. The Principal Solution § 2. The Basic Problems § 3. The Logarithmic PotentialLecture 18. The Theory of Integral Equations § 1. General Remarks § 2. The Method of Successive Approximations § 3.