# Math Toolbox: Searching for the right Calculus book

Categories: Math Toolbox

A lot of my research and development work revolves around mathematics, and to that end I have a large collection of math books in my personal library. They're focused mostly on applied mathematics, but I have a few on pure mathematics as well (analysis, algebra, topology). However, my library is lacking in one important area of mathematics that bridges pure and applied math. That area is calculus.

A glance at the calendar reminds me that this week marks 20 years since I took the Advanced Placement Calculus AB exam. Calculus was one of those courses that seemed so foreboding and intimidating from a distance, but once engaged actually came to me quite quickly. I think it helped that I had a great book: Calculus, 3rd edition by Larson, Hostetler, and Edwards. It was a big book, with its leafy cover and a white spine with "Calculus" written in large lettering. I liked the introductory chapter which showed what you could do with calculus that was difficult to do with arithmetic, the explanations of differential and integral calculus were so clear, the applications realistic, and the illustrations descriptive but not overpowering by too much color or text. This book is now in its 7th edition, and the description on Amazon says that it is "the only calculus text ever to increase its sales and market share in each edition through seven editions". I wanted to use that book as a reference during college, but I had borrowed my brother's book for Calculus AB/BC (he had used the same book for his AP Calculus course four years prior!) and my brother wanted it back. That book spoiled me, and I became very picky about the type of Calculus book I would use and keep. My search for the right Calculus book began.

When I was in college (undergrad) I used Gilbert Strang's Calculus book, and I hated it. Gilbert Strang is a very well-known mathematician and teacher at MIT, best known for his work on linear algebra. I haven't read his linear algebra books, but other people who I trust tell me that they're really good. I did not like his calculus book at all — it was too conversational and shallow to be of any use for class, much less as a reference. I sold it as soon as I was done with the Calculus sequence. Most contemporary Calculus textbooks are awful and I have not been impressed with any of them. Important concepts are too often watered down, elaborate color pictures are added just because, the book is integrated with CDROMs or websites "just because" while missing the actual content, and the beauty of the Calculus is lost. Several times I've walked into a university bookstore with the intention of purchasing a Calculus book and walked out disgusted by what they had on offer.

As a result, I've turned to the older books that presented Calculus in a different way before the modern educational trends started taking over. The challenge of writing a Calculus book is presenting concepts so that they can be used in applications, yet provide a foundation for deeper study in pure mathematics. I have one volume of a two-volume set in Calculus written by Tom Apostol, former math professor at Caltech. Apostol's book is different from so many other calculus books in that he covers integral calculus first before differential calculus, which takes some getting used to, but is an approach that he claims is historically and pedagogically sound. Richard Courant, who founded the Institute of Mathematical Sciences at NYU, wrote a three-volume set on Calculus that handles differential and integral calculus in the same order as Apostol. It's a challenging book, but if you master the concepts and the exercises you will be very well prepared for deeper mathematics. Michael Spivak's Calculus book, written in the late 60s/early 70s, is very much sought after. Spivak has a more conversational style than Courant and Apostol, but he is also very thorough (extremely thorough in fact; his books on Differential Geometry span at least six volumes!). None of these books are cheap; Apostol's two-volume set costs more than $200, Courant's three-volume set over $150, while Spivak's book costs $75 (the solutions manual, which I recommend, sells for $50). The book by Larson et al. also sells at triple-digits. Either book will be a valuable and consulted reference for the rest of your career, but it behooves you to think of how you will use the book before you purchase it.

If anyone else has a Calculus book that they particularly enjoyed and still keep as a reference, please mention it in the comments. I'll add other books as readers recommend them.

**POSTSCRIPT:** I forgot to mention whether I found the Calculus book I wanted. I decided to order Spivak's Calculus book a few days ago. I like his writing and I like the fact that he cares deeply about teaching mathematics in a rigorous and fundamental way. It was less expensive than the other options, so that helped my decision as well.