Preface

The purpose of this book is to introduce students of the physical sciences to several mathematical methods often essential to the successful solution of real problems. The methods chosen are those most frequently used in typical physics and engineering applications. The treatment is not intended to be exhaustive; the subject of each chapter can be found as the title of a book that treats the material in much greater depth. The reader is encouraged to consult such a book should more study be desired in any of the areas introduced.

Perhaps it would be helpful to discuss the motivation that led to the writing of this text. Undergraduate education in the physical sciences has become more advanced and sophisticated with the advent of the space age and computers, with their demand for the solution of very difficult problems. During the recent past, mathematical topics usually reserved for graduate study have become part of the undergraduate program. It is now common to find an applied mathematics course, usually covering one topic, that follows differential equations in engineering and physical science curricula. Choosing the content of this mathematics course is often difficult. In each of the physical science disciplines, different phenomena are investigated that result in a variety of mathematical models. To be sure, a number of outstanding textbooks exist that present advanced and comprehensive treatment of these methods. However, these texts are usually written at a level too advanced for the undergraduate student, and the material is so exhaustive that it inhibits the effective presentation of the mathematical techniques as a tool for the analysis of some of the simpler problems encountered by the undergraduate. This book was written to provide for an additional course, or two, after a course in differential equations, to permit more than one topic to be introduced in a term or semester, and to make the material comprehensive to the undergraduate. However, rather than assume a knowledge of differential equations, we have included all of the essential material usually found in a course on that subject, so that this text can also be used in an introductory course on differential equations or in a second applied course on differential equations. Selected sections from several of the chapters would constitute such courses.

Ordinary differential equations, including a number of physical applications, are reviewed in Chapter 1. The use of series methods is presented in Chapter 2. Subsequent chapters present Laplace transforms, matrix theory and applications, vector analysis, Fourier series and transforms, partial differential equations, numerical methods using finite differences, complex variables, and wavelets. The material is presented so that more than one subject, perhaps four subjects, can be covered in a single course, depending on the topics chosen and the completeness of coverage. The style of presentation is such that the reader, with a minimum of assistance, may follow the step-by-step derivations from the instructor. Liberal use of examples and homework problems should aid the student in the study of the mathematical methods presented.

Incorporated in this new edition is the use of certain computer software packages. Short tutorials on Maple, demonstrating how problems in advanced engineering mathematics can be solved with a computer algebra system, are included in most sections of the text. Problems have been identified at the end of sections to be solved specifically with Maple, and there are also computer laboratory activities, which are longer problems designed for Maple. Completion of these problems will contribute to a deeper understanding of the material. There is also an appendix devoted to simple Maple commands. In addition, Matlab and Excel have been included in the solution of problems in several of the chapters. Excel is more appropriate than a computer algebra system when dealing with discrete data (such as in the numerical solution of partial differential equations).

At the same time, problems from the previous edition remain, placed in the text specifically to be done without Maple. These problems provide an opportunity for students to develop and sharpen their problem-solving skills—to be human algebra systems.1 Ignoring these sorts of exercises will hinder the real understanding of the material.