Preface IXconstructions to a minimum, we could not avoid them altogether: some
ideas, such as the separation of variables for the heat and wave equations,
just ask to be generalized, and we hope the reader will appreciate the
benefits of these generalizations. As a consolation to the reader who is not
comfortable with abstract constructions, we mention that everything in this
book, no matter how abstract it might look, is nowhere near the level of
abstraction to which one can take it.
The inevitable consequence of unifying mathematics and physics, as we
do here, is a possible confusion with notations. For example, it is customary
in mathematics to denote a generic region in the plane or in space by G,
from the German word Gebiet, meaning "territory." On the other hand,
the same letter is used in physics for the universal gravitational constant;
in our book, we use G to denote this constant (notice a slight difference
between G and G). Since these two symbols never appear in the same
formula, we hope the reader will not be confused.
We are not including the usual end-of-section exercises, and instead
incorporate the exercises into the main presentation. These exercises act
as speed-bumps, forcing the reader to have a pen and pencil nearby. They
should also help the reader to follow the presentation better and, once
solved, provide an added level of confidence. Each exercise is rated with a
super-script A, B, or C; sometimes, different parts of the same exercise have
different ratings. The rating is mostly the subjective view of the authors
and can represent each of the following: (a) The level of difficulty, with C
being the easiest; (b) The degree of importance for general understanding
of the material, with C being the most important; (c) The aspiration of the
student attempting the exercise. Our suggestion for the first reading is to
understand the question and/or conclusion of every exercise and to attempt
every C-rated exercise, especially those that ask to verify something. The
problems are at the very end, in the chapter called "Further Developments,"
and are not rated. These problems provide a convenient means to give brief
extensions of the subjects treated in the text (see, for example, the problems
on special relativity).
A semester-long course using this book would most likely emphasize
the chapters on complex numbers, Fourier analysis, and partial differential
equations, with the chapters on vectors, mechanics, and electromagnetic
theory covered only briefly while reviewing vectors and vector analysis.
The chapters on complex numbers and Fourier analysis are short enough to
be covered more or less completely, each in about ten 50-minute lectures.
The chapter on partial differential equations is much longer, and, beyond