Calculus by Michael Spivak - 4th Edition (Calculus 4th edition by Michael Spivak)

The prose in this book is perhaps the best writing on a technical subject I have ever seen. Calculus--a dry, boring, and mechanical subject for most other authors--is here presented as a beautiful and interesting intellectual achievement worthy of study on its own right. Even if you aren't a math for math's sake sort of person, it is always worthy of attention when an expert like Spivak explains what their world is like with such clarity and passion. And if any book may convince you that the people who say "math is beautiful" aren't nuts, it's this one. Spivak manages to deliver both an intuitive picture of a concept and the full mathematical rigor in a brilliant and playful style. He will often give a provisional definition of a tough concept to aid understanding first, but importantly and in contrast to more "accessible" math books, he signals very clearly that he is being intentionally imprecise. He then moves towards rigor by explaining exactly the way in which he has been imprecise, clearly driving the motivation for a more rigorous definition. The overall effect is that you rarely feel very lost and when he ultimately gives you the full picture, it often feels like an inevitability. A favorite example of this sort of style is at the start of Chapter 20: "The irrationality of e was so easy to prove that in this optional chapter we will attempt a more difficult feat, and prove that the number e is not merely irrational, but actually much worse. Just how a number might be even worse than irrational is suggested by a slight rewording of definitions..." Another impressive aspect of the book is the layout, where every relevant figure is only a glance away in the margin or directly inline with the text. It is the same style used in the Feynman lectures and Edward Tufte's books, and it is executed at its highest level here. Clear care went into the placement of each symbol in each equation and each figure. The exercises are quite hard, but there is a full solutions manual available for self-study (how I am working through the book). I will admit that I needed to bail out of this book at the very beginning, never having been exposed to doing proofs at this level before (formulaic high school geometry "proofs" don't count for much here). I used Velleman's "How To Prove It" and the first few chapters of Apostol's Calculus Volume I to get up to speed. Both these books are also recommended, and Apostol, in particular, gives an excellent and rigorous but more gentle on-ramp for the sort of thinking asked of you in Spivak Part I. In the long run, however, I think Spivak edges out Apostol for self-study because of the solutions manual. I picked up this book when I found that after 3 years of doing calculus in high school and college, I had forgotten most of it within a few years. I realized that while I could do the mechanics, I never really understood calculus in the first place. This book is probably a bit of overkill for just patching understanding, but I now have a much deeper appreciation and understanding of the mathematical way of thinking. It's not an easy book, but it is a wonderful one that will pay back dividends for hard work. But you don't have to do all the hard work just to appreciate what Spivak has done here. If you have an interest in good writing, this book is worth a look even if you aren't interested in learning the subject. I take special pleasure in reading great writing on any topic, and this book is up there with the best writing anywhere.

This book is a marvel for current (serious) math students, and for people who may one day find themselves teaching the subject, and want to choose best explanations for topics. First of all, this book should be called *Introduction to Analysis*, lest the prospective buyer get the impression that this will be an introductory book on calculus, which eschews rigor. On the contrary, the first words in the preface indicate its mission: "Every aspect of this book was influenced by the desire to present calculus not merely as a prelude to but as the first real encounter with mathematics." This book is the best I've seen for people who have a serious interest in learning and teaching mathematics, and maybe even want to be a mathematician. I wouldn't buy this book if you have never before seen the derivative and the integral. This is, however, a great book for the foundations of analysis. There quickly builds up a bond of trust between the discerning reader and the author: trust that the exercises will be worthwhile, and trust that topics you have seen before will be treated in such a way that you will want to read them again, just to remember the elegant explanation for teaching someday. The presentation is anything but austere. It is conversational, beautiful even. The reader will know what I mean when I say that this book was written with love. There are huge margins on the side, where, if you are like me, you will write the solutions to the exercises. The exercises are absolutely masterful. Besides maybe one or two warm-up problems at the beginning (and there are usually 20 and sometimes as many as 70 exercises after a chapter), the exercises are not so easy as to be trivial, nor do they fail to guide you toward a solution. Many exercises in lesser books ask you to prove rather arbitrary results; here, most of the exercises are something every mathematician should know. I am a graduate student, and I know that I would have passed my advanced calculus quals the first time if I had made a more careful reading of this book right off the bat. Buy, savor. Get the 4th edition, which has more exercises and more chapters.