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.

For anyone who has never seen calculus before, this book is ABSOLUTELY NOT FOR YOU! This book as others have said is essentially an introduction to real analysis book. A lot of people in the more negative reviews say they aren't sure where this book fits in, well I'll tell you. If you just take some calculation heavy calculus course geared towards engineers and then sign up for a real analysis class expecting to do well you are going to get ABSOLUTELY SLAUGHTERED. Real analysis has virtually NO computational problems, nearly all of them are proofs. Thats where this book comes in, the proofs and exposition of this book are the absolute most explicit i have ever seen. Spivak goes to incredible pains to explain every detail of his proofs and give you intuition for how and why things work. I have never seen this in another mathematics text before ever. You will not be granted the same level of explicit detail in a real analysis class I can guarantee you that right now. THis book serves as a utensil to build your mathematical muscles and get accustomed to Real analysis proofs and how to think about calculus , if you study and work most of the problems in this book you will do very well in real analysis. You might say to me " well you just take a class on mathematical proofs and logic to get ready for upper level math classes" No you absolutely don't. The trivial nonsense proofs by induction and amateur proofs you do in a class like that are no where near the level of preparation you need to feel comfortable with proofs. You need a book like this as far as im concerned to get accustomed to how Analysis proofs work. This book taught me how to think. Let me make something clear. I bought this book initially based on the rave reviews on this site. I was just as frustrated and angry as all of the other reviewers who gave this book negative reviews. I took a couple calculus classes and then when I came back to Spivak it was a completely different book to me, Upon my second attempt at reading it I was absolutely astonished at how crystal clear he was making everything I had read in single variable calculus. He was destroying any and all confusion with topics i learned from reading Stewarts calculus one page at a time. And I wasn't intimidated by the problems, in fact it was the first time I'd ever appreciated difficult problems and was actually excited to work through them and see if I could prove them. I love this book. I hope it never goes out of print.