Preface

The major purposes of this book are to present partial differential equations

(PDEs) and vector analysis at an introductory level. As such, it could be con-

sidered a beginning text in mathematical physics. It is also designed to provide

a bridge from undergraduate mathematics to the first graduate mathematics

course in physics, applied mathematics, or engineering. In these disciplines, it

is not unusual for such a graduate course to cover topics from linear algebra,

ordinary and partial differential equations, advanced calculus, vector analysis,

complex analysis, and probability and statistics at a highly accelerated pace.

In this text we study in detail, but at an introductory level, a reduced list

of topics important to the disciplines above. In partial differential equations,

we consider Green’s functions, the Fourier and Laplace transforms, and how

these are used to solve PDEs. We also study using separation of variables to

solve PDEs in great detail. Our approach is to examine the three prototypical

second-order PDEs—Laplace’s equation, the heat equation, and the wave

equation—and solve each equation with each method. The premise is that in

doing so, the reader will become adept at each method and comfortable with

each equation.

The other prominent area of the text is vector analysis. While the usual

topics are discussed, an emphasis is placed on understanding concepts rather

than formulas. For example, we view the curl and gradient as properties of a

vector field rather than simply as equations. A significant—but optional—

portion of this area deals with curvilinear coordinates to reinforce the idea of

conversion of coordinate systems.

Reasonable prerequisites for the course are a course in multivariable cal-

culus, familiarity with ordinary differential equations including the ability to

solve a second-order boundary problem with constant coefficients, and some

experience with linear algebra.

In dealing with ordinary differential equations, we emphasize the linear

operator approach. That is, we consider the problem as being an eigenvalue/

eigenvector problem for a self-adjoint operator. In addition to eliminating

some tedious computations regarding orthogonality, this serves as a unifying

theme and an introduction to more advanced mathematics.

The level of the text generally lies between that of the classic encyclopedic

texts of Boas and Kreysig and the newer text by McQuarrie, and the partial

differential equations books of Weinberg and Pinsky. Topics such as Fourier

series are developed in a mathematically rigorous manner. The section on

completeness of eigenfunctions of a Sturm-Liouville problem is considerably

xi

more advanced than the rest of the text, and can be omitted if one wishes to

merely accept the result.

The text can be used as a self-contained reference as well as an introductory

text. There was a concerted effort to avoid situations where filling in details

of an argument would be a challenge. This is done in part so that the text

could serve as a source for students in subsequent courses who felt “I know I’m

supposed to know how to derive this, but I don’t.” A couple of such examples

are the fundamental solution of Laplace’s equation and the spectrum of the

Laplacian.

I want to give special thanks to Patricia Osborn of Elsevier Publishing

whose encouragement prompted me to turn a collection of disjointed notes

into what I hope is a readable and cohesive text, and also to Gene Wayne of

Boston University who provided valuable suggestions.

James Radford Kirkwood

xii Preface