I've been working through this book and it's top-notch. I have the second edition which made significant improvements to the first edition:

1. more exercises were added

2. a chapter devoted to an introduction to differential equations was added

3. the book was divided into smaller chapters

4. several sections were revised with the help of other professors

5. an enormous introduction to linear algebra was added (over 170 pages!).

This book was published 40 years ago, and it's a shame that universities across the country continue to use dumbed down textbooks by authors such as Thomas with a dozen different unnecessary versions.

When working through Apostol, the student can actually pick up a distinct tone instead of seeing large blocks of examples, theorems, and then a list of predictable exercises. With most new concepts, he briefly talks about the historical developments dating back anywhere from Leibniz to the Greeks. I smiled when he introduced the philosophical problem of Zeno's paradox on the chapter Sequences, Infinite Series, Improper Integrals.

He develops proofs rigorously and I have to disagree with someone at amazon.com that said Apostol is the master of obscure proofs. His proofs are complete and don't cut corners. All of his examples are difficult (as they should be) but he offers plenty of explanation so that the reader has no excuse to get lost.

The format of the exercises was a breath of fresh air. The usual formatting of the average textbook is for example, in a chapter on integration using natural logarithms, a subheading "Finding the Derivative" then "Logarithmic Differentiation". Wow, thanks for giving away the answer since you do the exact same thing every time. Apostol's exercises are pleasantly varied with theory and applications and he'll often throw in a review problem or two that incorporates old material to keep you on your toes. You are forced to think about what to do since everything you've gone over up to that point is fair game. Also, he often asks you to prove or show something which will scare away the modern math student familiar with simply doing a massive list of evaluations.

Unfortunately, there are no elegant pictures that are contained in a modern textbook since they didn't have computers with Mathematica 6.0 and laser printers in the 1960s. All the pictures are sufficient however and give you enough spatial understanding.

I found out later that Vol. I and Vol. II (multivariable calculus portion) are basically a complete course in linear algebra along with the calculus. The linear algebra book by Apostol doesn't have much new material except for the applications to differential equations.

Working through this book may confuse you because you begin to wonder whether you're learning single-variable calculus or taking a course in real analysis, linear algebra, and differential equations all at the same time. He writes conversationally, so although he sprints ahead non-traditionally, the student will feel comfortable tackling material in later courses early in their studies.

So if you want to learn calculus rather than learning how to do problems involving calculus, pick up Tom M. Apostol.

EDIT: I want to add a warning that this book is

extremely difficult. You may want to pick up "Math Proofs Demystified" (you can get this at any major book store in the math section or at

www.amazon.com for cheap). It can be hard to follow the theory if you're used to being spoon fed.