Preface

This is not a mathematics text. We assume that the user has already had a formal introduction to most of the topics covered here, or is now studying them in a math class. It is possible, however, that attention to the intrinsic beauty of the formal structure has left little time for mundane applications. It turns out that simple mathematics is almost unreasonably successful in modeling phenomena of the real world. There is some point in learning to manipulate the model, even though its inner workings are not of personal concern. To be sure, somebody ought to study the model for its own sake if only to assure us that it does not have inner weaknesses or inconsistencies. But for many of our purposes, we need only know the range of validity of the model and how to work it.

In this book, which is part reference and part reminder, we are concerned with how to use math. We concentrate on those features that are most needed in the first two years of college science courses. That range is not rigorously defined, of course. A sophomore physics major at M.I.T. or Cal. Tech. must use differential equations routinely, while a general science major at some other place may still be troubled by logarithms. It is possible that even the Tech student has never really understood certain things about simple math. What, for instance, is natural about the natural logs? We have tried to cover a broad range of topics—all the things that a science student might want to know about math but has never dared ask.

The sections of the book are not sequential. Because of our assumption about the user’s prior exposure to math, we have felt free to make use of any topic or notation to help explain any other topic. One of our purposes is to demonstrate such cross-links. The book itself is extensively cross indexed and we have attempted to make the index at the back of the book as complete as possible.

The idea of writing a book like this arose many years ago when I discovered that a whole class of physics students, taking calculus concurrently, knew that sin θ ≃ θ for small θ. None of them, however, knew that the approximation is meaningful only if you use radians. When anyone learns new math he usually gleans only a superficial understanding. For math to be understood, it must be old and used.

I am delighted that the American Association of Physics Teachers has decided to re-issue Used Math. The original edition was published in 1973. Some changes have taken place since then. I corrected a few typos, and decided that it was no longer necessary to print pages of log and trig tables. However, I have added some extra comments about error analysis for students, a perennial subject of controversy at every level. The two M.I.T. students who helped in the first edition and were acknowledged on the title page, now have Ph.D.’s and are productive professionals, one an economist and the other a physicist.

I’ve used the book for reference a lot in the last 20 years. I hope that a new generation of teachers and students will also find it useful.

C.E.S.