Note: Front Cover -- Partial Differential Equations of Mathematical Physics -- Copyright Page -- Table of Contents -- TRANSLATON EDITOR'S PREFACE -- AUTHOR'S PREFACES TO THE FIRST AND THIRD EDITIONS -- LECTURE 1. DERTVATON OF THE FUNDAMENTAL EQUATONS -- 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 Waves -- LECTURE 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 Example -- LECTURE 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. Characteristics -- LECTURE 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 Conditions -- LECTURE 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 Formula -- LECTURE 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 Theorem -- LECTURE 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 Parameter -- LECTURE 8. THE EQUATION OF HEAT CONDUCTION -- 1. Principal Solution -- 2. The Solution of Cauchy's Problem -- LECTURE 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 Layer -- LECTURE 10. SOME GENERAL CONSEQUENCES OF GREEN'S FORMULA -- 1. The Mean-Value Theorem for a Harmonic Function -- 2. Behaviour of a Harmonic Function near a Singular Point -- 3. Behaviour of a Harmonic Function at Infinity. Inverse Points -- LECTURE 11. POISSON'S EQUATION IN AN UNBOUNDED MEDIUM. NEWTONIAN POTENTIAL -- LECTURE 12. THE SOLUTION OF THE DIRICHLET PROBLEM FOR A SPHERE -- LECTURE 13. THE DIRICHLET PROBLEM AND THE NEUMANN PROBLEM FOR A HALF-SPACE -- LECTURE 14. THE WAVE EQUATION AND THE RETARDED POTENTIAL -- 1. The Characteristics of the Wave Equation -- 2. Kirchhoff's Method of Solution of Cauchy's Problem -- LECTURE 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. Behaviour of the Potentials at Infinity -- LECTURE 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 Problems -- LECTURE 17. LAPLACE'S EQUATON AND POISSON'S EQUATION IN A PLANE -- 1. The Principal Solution -- 2. The Basic Problems -- 3. The Logarithmic Potential -- LECTURE 18. THE THEORY OF INTEGRAL EQUATIONS -- 1. General Remarks -- 2. The Method of Successive Approximations -- 3. Volterra Equations -- 4. Equations with Degenerate Kernel -- 5. A Kernel of Special Type. Fredhohn's Theorems -- 6. Generalization of the Results -- 7. Equations with Unbounded Kernels of a Special Form -- LECTURE 19. APPLICATION OF THE THEORY OF FREDHOLM EQUATIONS TO THE SOLUTION OF THE DIRICHLET AND NEUMANN PROBLEMS -- 1. Derivation of the Properties of Integral Equations -- 2. Investigation of the Equations -- LECTURE 20. GREEN'S FUNCTION -- 1. The Difíerential Operator with One Independent Variable -- 2. Adjoint Operators and Adjoint Families -- 3. The Fundamental Lemma on the Integrals of Adjoint Equations -- 4. The Influence Function -- 5. Definition and Construction of Green's Function -- 6. The Generalized Green's Function for a Linear Second-Order Equation -- 7. Examples -- LECTURE 21. GREEN'S FUNCTION FOR THE LAPLACE OPERATOR -- 1. Green's Function for the Dirichlet Problem -- 2. The Concept of Green's Function for the Neumann Problem -- LECTURE 22. CORRECTNESS OF FORMULATION OF THE BOUNDARY-VALUE PROBLEMS OF MATHEMATICAL PHYSICS -- 1. The Equation of Heat Conduction -- 2. The Concept of the Generalized Solution -- 3. The Wave Equation -- 4. The Generalized Solution of the Wave Equation -- 5. A Property of Generalized Solutions of Homogeneous Equations -- 6. Bunyakovski's Inequality and Minkovski's Inequality -- 7. The Riesz-Fischer Theorem -- LECTURE 23. FOURIER'S METHOD. , 1. Separation of the Variables -- 2. The Analogy between the Problems of Vibrations of a Continuous Medium and Vibrations of Mechanical Systems with a Finite Number of Degrees of Freedom -- 3. The Inhomogeneous Equation -- 4. Longitudinal Vibrations of a Bar -- LECTURE 24. INTEGRAL EQUATONS WTIH REAL, SYMMETRIC KERNELS -- 1. Elementary Properties. Completely Continuous Operators -- 2. Proof of the Existence of an Eigenvalue -- LECTURE 25. THE BILINEAR FORMULA AND THE HILBERT-SCHMIDT THEOREM -- 1. The Bilinear Formula -- 2. The Hilbert-Schmidt Theorem -- 3. Proof of the Fourier Method for the Solution of the Boundary-Value Problems of Mathematical Physics -- 4. An Application of the Theory of Integral Equations with Symmetric Kernel -- LECTURE 26. THE INHOMOGENEOUS INTEGRAL EQUATION WTTH A SYMMETRIC KERNEL -- 1. Expansion of the Resolvent -- 2. Representation of the Solution by means of Analytical Functions -- LECTURE 27. VIBRATIONS OF A RECTANGULAR PARALLELEPIPED -- LECTURE 28. LAPLACE'S EQUATON IN CURVILINEAR COORDINATES. EXAMPLES OF THE USE OF FOURIER'S METHOD -- 1. Laplace's Equation in Curvilinear Coordinates -- 2. Bessel Functions -- 3. Complete Separation of the Variables in the Equation V2u= O in Polar Coordinates -- LECTURE 29. HARMONIC POLYNOMIALS AND SPHERICAL FUNCTIONS -- 1. Definition of Spherical Functions -- 2. Approximation by means of Spherical Harmonics -- 3. The Dirichlet Problem for a Sphere -- 4. The Differential Equations for Spherical Functions -- LECTURE 30. SOME ELEMENTARY PROPERTIES OF SPHERICAL FUNCTIONS -- 1. Legendre Polynomials -- 2. The Generating Function -- 3. Laplace's Formula -- INDEX -- OTHER VOLUMES IN THIS SERIES.