discusses functional (in)dependence (cf. Olver pp. 86ff. // PDF pp. 111ff.)
This book has been widely acclaimed for its clear, cogent presentation of the theory of partial differential equations, and the incisive application of its principal topics to commonly encountered problems in the physical sciences and engineering. It was developed and tested at Purdue University over a period of five years in classes for advanced undergraduate and beginning graduate students in mathematics, engineering and the physical sciences.
The book begins with a short review of calculus and ordinary differential equations, then moves on to explore integral curves and surfaces of vector fields, quasi-linear and linear equations of first order, series solutions and the Cauchy Kovalevsky theorem. It then delves into linear partial differential equations, examines the Laplace, wave and heat equations, and concludes with a brief treatment of hyperbolic systems of equations.
Among the most important features of the text are the challenging problems at the end of each section which require a wide variety of responses from students, from providing details of the derivation of an item presented to solving specific problems associated with partial differential equations. Requiring only a modest mathematical background, the text will be indispensable to those who need to use partial differential equations in solving physical problems. It will provide as well the mathematical fundamentals for those who intend to pursue the study of more advanced topics, including modern theory.
From the authors' preface: "In writing this introductory book on the old but still rapidly expanding field of mathematics known as partial differential equations, our objective has been to present an elementary treatment of the most important topics of the theory together with applications to problems from the physical sciences and engineering. The book should be accessible to students with a modest mathematical background and should be useful to those who will actually need to use partial differential equations in solving physical problems. At the same time we hope that the book will provide a good basis for those students who will pursue the study of more advanced topics including what is now known as the modern theory.
"Throughout the book, the importance of the proper formulation of problems associated with partial differential equations is emphasized. Methods of solution of any particular problem for a given partial differential equation are discussed only after a large collection of elementary solutions of the equation has been constructed.
"During the last five years, the book has been used in the form of lecture notes for a semester course at Purdue university. The students are advanced undergraduate or beginning graduate students in mathematics, engineering or one of the physical sciences. A course in advanced calculus or a strong course in calculus with extensive treatment of functions of several variables, and a very elementary introduction to ordinary differential equations constitute adequate preparation for the understanding of the book. In any case, the basic results of advanced calculus are recalled whenever needed.''
Table of Contents: Preface; I. Some concepts from calculus and ordinary differential equations; II. Integral curves and surfaces of vector fields; III. Theory and applications of quasi-linear and linear equations of first order; IV. Series solutions. The Cauchy-Kowalevsky theorem; V. Linear partial differential equations. Characteristics, classification and canonical forms; VI. Equations of mathematical physics; VII. Laplace's equation; VIII. The wave equation; IX. The heat equation; X. Systems of first order linear and quasi-linear equations; Guide to further study; Bibliography for further study; Answers to selected problems; Index.