xii Preface
- allows the text to be used for a one-semester course without the need
to jump around.
A typical one-semester course would cover the core Chapters 1–6. Beyond
that, one might consider doing Chapter 7, or the beginnings of Chapters 9
and 10, or Chapter 11. Alternatively, if the students already are familiar with
special functions, one may wish to cover Chapter 8 or most of Chapter 9.
Motivation
The author believes that it is essential to provide the students with motivation
(other than grade) for each of the various topics. We have tried, as far as
possible, to provide such motivation, both physical and mathematical (so, for
example, the Fourier series is introduced only after the need for it, through
solving the heat equation via separation of variables, has been established).
Further, we begin by considering PDEs on bounded domains before looking
at unbounded domains, because
- This approach allows us to get to Fourier series early on.
- Problems on bounded domains are more natural than those on un-
bounded domains, at least in one dimension.
Further, and in this same vein, we have provided a Prelude to each chapter,
the purpose of which is to describe the topics to be covered in the chapter,
so as to tell the student what is coming and why it is coming, and to put the
material into its historical setting, as well.
Exercises
Of course, mathematics is not a spectator sport, and can only be learned by
doing. Thus, it goes without saying that the exercises are a key part of the
text. Basically, they are of four types:
- “solve-the-problem” exercises,
- proofs,
- “extend-the-material” exercises,
- graphical exercises.