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Michael Spivak
Publish or Perish, Inc.
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
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OK, boys and girls, listen up, because I'm only going to say it once:

This is the best Calculus textbook ever written.

Got that? If any of you want to champion some other book, you're itching for a fight.

Perhaps you'd like me to expand on that. Let me begin with some autobiography. I first studied calculus during my first year at the University of São Paulo, in Brazil. At that time, we didn't use a textbook; instead, we used references, and we used notes. The notes, prepared by one of the teachers, were small booklets (probably mimeographed — remember that?). The references were a variety of books that our professors felt could be consulted with profit. Among them was Spivak's Calculus. I didn't buy a copy, because I was fairly happy with the notes and with the books I already had. But the name stuck in my head.

A couple of months into the course, I was trolling the aisles of my favorite bookstore, and saw there a copy of the book. I think it was the first edition. It was definitely in soft covers, perhaps an edition for sale in third world markets. I remembered the name, bought the book, took it home, and set to reading it.

It was love at first sight. OK, I'll admit that I wasn't doing all — or even most — of the problems. (There are lots of very hard problems; instructors will be glad that there is an answer book.) But Spivak's account of what Calculus is all about, his careful but precise account of the theoretical underpinnings of the material, his chapter on the "hard theorems" (the ones that require an understanding of the completeness of the reals), his pictures, even his asides... this was calculus as an intellectual adventure, deep, compelling, and beautiful. I read the whole book, making my way far beyond what we had covered in class by that point. I even read, with some small level of understanding, the sections on complex analysis and the explanation of Dedekind cuts.

It will give you a measure of Spivak's impact on me to note that I took quite seriously the annotated suggested reading list given in the back, sought out many of those books, and even managed to read some of them.

I never forgot some bits of the book. Spivak's comment that "mathematicians like to pretend that they can't even add, but most of them can when they have to" (on page 179) stuck in my mind. I've used it in class many times. His chapter on integration in finite terms is another I remembered, and especially problem 7 in that chapter (described as "Potpourri. No holds barred."). The treatment of power series also stuck to me, with the result that I've been a fan of series ever since.

This third edition is better than the one I read. Spivak has added, in particular, a careful account of how one can deduce Kepler's laws of planetary motion from Newton's laws of physics. This was, of course, one of the first applications of the calculus, and it is still one of the most impressive ones. Its inclusion is a definite improvement. Beyond that, there are very few changes, mostly corrections of small mistakes. The suggested reading list has not, alas, been updated, with the result that it now has a slightly antiquarian flavor: some of the books are out of print, and some, like tables of integrals, are best replaced by appropriate software tools.

But the core of the text is still the same. If you want a calculus textbook that does everything honestly but gently enough so that good first-year students can follow it, this is the one.

There is no multivariable calculus, which is a pity; I'd have loved to see what Spivak could do with that at this level. (His Calculus on Manifolds is, of course, a classic, but it is so terse as to be impenetrable for most students.)

I have used this text for the Honors Calculus course at Colby. Students find it very hard, as one might expect. But they also understand that they are getting the real thing, that they are being treated as intellectual adults. When I get to teach that course again, I'm sure I'll use it again. And maybe, if I'm lucky, a student or two will fall in love.

Fernando Q. Gouvêa is professor of mathematics at Colby College. He got his undergraduate degree from the University of São Paulo, in Brazil.

Part I    Prologue

                 1  Basic Properties of Numbers
                 2  Numbers of Various Sorts

Part II    Foundations

                 3 Functions
                    Appendix. Ordered Pairs
                 4  Graphs
                   Appendix 1. Vectors
                    Appendix 2. The Conic Sections
                    Appendix 3. Polar Coordinates

                 5  Limits
                 6  Continuous Functions
                 7  Three Hard Theorems
                 8  Least Upper Bounds
                   Appendix. Uniform Continuity

Part III    Derivatives and Integrals

                 9  Derivatives
                10 Differentiation
                11 Significance of the Derivative
                    Appendix. Convexity and Concavity
                12 Inverse Functions
                   Appendix. Parametric Representation of Curves
                13 Integrals
                   Appendix. Riemann Sums
                14 The Fundamental Theorem of Calculus
                15 The Trigonometric Functions
              *16 Pi is Irrational
              *17 Planetary Motion
                18 The Logarithm and Exponential Functions
                19 Integration in Elementary Terms
                   Appendix. The Cosmopolitan Integral

Part IV    Infinite Sequences and Infinite Series

                20 Approximation by Polynomial Functions
               *21 e is Transcendental
                22 Infinite Sequences
                23 Infinite Series
                24 Uniform Convergence and Power Series
                25 Complex Numbers
                26 Complex Functions
                27 Complex Power Series

Part V    Epilogue

                28 Fields
                29 Construction of the Real Numbers
                30 Uniqueness of the Real Numbers

                    Suggested Reading