Partial Differential Equations

This book is an attempt to make available to the student a coherent modern view of the theory of partial differential equations. Here equations of the first order and linear second order equations are treated with the tensor calculus, which combines generality and insight, in mind. Since the book is self-contained, much of the material is classical, but an effort has been made to achieve a modern outlook on these topics. A number of significant recent developments are introduced, and treated in relation to the natural background formed by geometry and physics. Special features of the exposition are: (a) the simplified general treatment of first order equations; (b) the geometrical foundations of the theory of linear second order equations (c) unified treatment of boundary value problems and related topics by integral equations; (d) the theory of generalized hyperbolic potentials. Preface 7 Contents 9 I. Differential Equations and Their Solutions 11 II. Linear Equations of the First Order 34 III. Non-Linear Equations of the First Order 57 IV. Linear Equations of the Second Order 80 V. Self-Adjoint Elliptic Equations 108 VI. Linear Integral Equations 135 VII. Boundary Value Problems 157 VIII. Eigenfunctions 179 IX. Normal Hyperbolic Equations 205 X. Integration of the Wave Equation 227 Bibliography 254 Index 256