Opening Doors

University of California, Davis

From
years of advising undergraduate students in the sciences, a favorite quotation
of mine is from James Caballero and appears in the epigraph to the book: “I
advise my students to listen carefully the moment they decide to take no more
mathematics courses. They might be able to hear the sound of closing doors.”
This book is written for those students. It is the outgrowth of a collection of
MathChapters from my *Physical Chemistry: A Molecular Approach*, which I wrote with John
Simon several years ago, and from my *Quantum
Chemistry*, which recently was published in its second edition. These
MathChapters consist of concise reviews of mathematical topics, discussing only
the minimum amount that you need to know to understand subsequent material. From
years of publishing scientific texts, my publisher says that physical chemistry
is difficult because of the mathematics, but it is impossibly difficult without
it. The point of the MathChapters is that by reading these reviews before the
mathematics is applied to physical chemistry topics, you will be able to spend
less time worrying about the math and more time learning the physical chemistry.
A number of people suggested that I expand these MathChapters into a single
volume, and this book is the result.

One
thing that makes mathematics courses difficult for many science students is that
they are taught by professional mathematicians, whose primary interest is to
develop mathematics for its own sake, often without regard to applications.
Consequently, they appreciate and recognize the need for rigor by carefully
specifying each and every condition for the validity of theorems and the use of
certain techniques. Most science students simply want to apply mathematics to
physical problems and bring a certain degree of physical intuition into their
mathematics courses and feel that the rigor is excessive. Unfortunately, this
intuition is not always correct. Since the development of calculus in the 17^{th}
and 18^{th} centuries, mathematicians have discovered many
counterintuitive examples of functions that are supposed to display certain
behavior. For example, there is a function that is continuous at every point but
has a derivative nowhere. In fact, there is an entire book on mathematical
counterexamples, *Counterexamples in*
*Analysis”,* by B.R. Gelbaum and J.M.H.
Olmsted, published by Dover. These
counterexamples rarely arise in physical problems but are of central importance
in mathematics. Thus, there is a natural and justifiable dichotomy between the
people who teach mathematics and many of the students who take mathematics. I
don’t know the source of this saying, but it goes “Applied mathematicians
don’t understand pure mathematicians and pure mathematicians don’t trust
applied mathematicians.”

There
are 23 chapters in the book. Each
one is fairly short, with the longest one being 20 pages, and is meant to be
read at a single sitting. The material is presented at a practical level with an
emphasis on applications to physical problems, although a few theorems along
with their conditions are given. Each chapter contains several Examples, which
serve to illustrate the techniques that are discussed. No one can learn physical
chemistry ( nor anything else in the physical sciences for that matter ) without
doing lots of problems. For this reason, I have included about 30 problems at
the end of each chapter. These problems range from filling in gaps to extending
the material presented in the chapter, but most illustrate applications to
physical problems. All told, there are over 600 problems, and I have provided
answers to most of them at the end of the book.

Throughout
the book, I encourage you to learn how to use one of a number of general
mathematics programs such as *Mathematica,
Maple*, or *MathCad*, which are
collectively called computer algebra systems ( CAS ). For a comparison of these
various CAS, see *http://en.wikipedia.org/wiki/Computer_algebra_system*.
These CAS make it easy to do calculations routinely that were formerly a
drudgery. Most chemistry departments have a license for one of these programs.
These programs not only perform numerical calculations but also can perform
algebraic manipulations. They are relatively easy to learn and to use, and every
serious science student should know how to use one of them. They allow you to
focus on the underlying physical ideas and free you from getting bogged down in
algebra. They also allow you to explore the properties of equations by varying
parameters and plotting the results. In fact, all the figures in the book were
produced by Professor Mervin Hanson of Humboldt State University using *Mathematica*.
There are a number of problems that require the use of one of these programs,
and many others are made much easier by using one of them.

Another
product of the computer age is the availability of websites for most any topic,
and that is true for mathematics as well. At one time, most physical chemistry
students owned some sort of math handbook such as the *CRC
Math* *Tables*, which contained
tables of trigonometric formulas, a table of integrals, and many other tables,
but these are all available online nowadays. I have suggested websites for many
topics throughout the book. Websites have the distressing habit of disappearing
now and then and I have tried to include only those that I think might still
exist when you go to them. If by any chance you can’t open one of them,
you’ll probably find it and many others by going to Google. There is a
complete list of the websites that I refer to in the References at the end of
the book.

It
always amazes me just how many people are involved in the production of a book.
Foremost are the reviewers, who frequently save me from promulgating my
misunderstandings and misconceptions.
I wish to thank Scott Feller of Wabash College, Mervin Hanson of Humboldt State
University, Helen Leung and Mark Marshall of Amherst College, and John Taylor of
the University of Colorado for very helpful reviews. I also wish to thank
Christine Taylor and her crew at Wilsted & Taylor Publishing Services and
particularly Jennifer Uhlich, who could not have done a more conscientious or
competent job of turning a manuscript into a beautiful -looking and inviting
book, Jennifer McClain for superb
copy editing, Yvonne Tsang for designing a great- looking book, and Bill Clark and Gunjan Chandola at ICC
Macmillan for one of the best jobs of composition that I have experienced. I
also wish to thank Mervin Hanson for
rendering and rerendering all the figures in *Mathematica* without one crusty word, Jane Ellis of University
Science Books for overseeing many of the production details and for suggesting
the cover that ties in so well with the epigraph by James Caballero, and Bruce
Armbruster and his wife and associate, Kathy, for being the best publishers
around and good friends in addition. Finally, I wish to thank my wife, Carole,
for preparing the manuscript in TeX, for reading the entire manuscript, and for
being my best critic in general ( in all things ).

There
are bound to be both typographical and conceptual errors in the book and I would
appreciate your letting me know about them so that they can be corrected in
subsequent printings. I would also welcome general comments, questions, and
suggestions at *mquarrie@mcn.org*,
or through the University Science Books website, *www.uscibooks.com*.