Calculus (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.

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.

The Calculus fourth edition by Spivak sat on my desk for a while. When I first purchased it, I tried some of the beginning problems, then I felt that they were a little more rigorous and abstract, meaning more mastery of mathematics for me to do before I can do the exercises. So, there it remained for a couple of years. Until suddenly a few weeks ago, when I was working in Lebesgue Integration, that I needed to bridge the gap from Calculus to Analysis to Integration and Measure theory. I found a chapter devoted to Riemann-Darboux Integration in Spivak's "Calculus," with a picture in particular of inf and sup of a function in a particular interval. Nowhere have I found an explanation for Lim Sup and Lim Inf that satisfied definitively what I need to grasp for advanced mathematics. I began to understand Lim Sup and Lim Inf by inferring that they are the intersection of the union and the union of the intersection respectively of arbitrary sets where the puzzling thing is how to write the notation correctly to discuss these two concepts without resorting to pictures to describe what I am talking about? I must say the section on Riemann integration is superior to many other texts, and this is where I found my understanding of Lim Sup and Lim Inf increased, Spivak did not make the connection explicitly for me. But it was with most mathematical books, what is stated in print, I have to make plausible inferences or interpretations to squeeze more out of what is symbolically stated. It is the usual case that when looking for Lim Sup and Lim Inf, I would assume to look in introductory analysis textbooks and not in a Calculus textbook like Spivak's. Thus, he does present concepts further along the journey of a mathematician's training, addressed to so fundamental ideas as to enlighten them. Using highly developed concepts to redirect our attention to the fundamental ideas we started out with that were unanswered or seemed unsatisfying, is the approach found here. It gives the vague ideas more definitive and detailed accounts of what is going on in the assumptions being made when these ideas were first introduced. For Example, the supremum of a sequence is applied to functions and also the infemum of a sequence applied to functions here, is used to clarify the Riemann Sum both Upper and Lower Sums beautifully. Now, I would say that I am trying now to comprehend the notations used in Measure theory which is written in Set theoretic notation, with a lot of Lim Inf and Lim Sup, in other words Spivak provided me with an answer to my vexing fundamental question as a key to the higher and more advanced mathematics of Integration and Measure theory. I am thankful and grateful for this key.

CALCULUS Fourth Edition Michael Spivak As new condition and still had the plastic shrink wrap on. I've read reviews that the binding is not as durable as the earlier editions.

I disagree with the people who say this book is not for people who have never seen calculus before. Quite the contrary. This is the book you want so that you don't have to unlearn what you were taught in calculus to move on to higher math. Furthermore, I do not believe it is possible to truly understand what a limit is if you don't read something like this. I am appalled at calculus curricula at colleges and universities that just expect you to memorize rules and drill problems over and over; I prefer knowing exactly why something is true before doing any problems. I also prefer fully 100% understanding every single reason a theory works, so this book is perfect for me, though it doesn't construct the real numbers from the rationals. This is a prerequisite to understanding limits because if you don't understand what irrational numbers are first, then you'll never understand what is meant by "infinitely close," period. So I recommend you buy "Understanding Analysis" by Stephen Abbott with this book, and read ch1 of Abbott before this. That way you'll know what a real number actually is, by the way, it's the limit of a sequence of rational numbers, but don't worry, Abbott ch 1 is incredibly well written and you'll get it if you just read it and think about why it makes sense. If you want a little deeper of an understanding than Abbott (which is splitting hairs, and few could pull it off, but the man I'm about to name is one of the few who can) then take a look at "Analysis 1" by Terrance Tao. I'd read either of these analysis books before or concurrently with Spivak, and I might also suggest grabbing a copy of "Single Variable Calculus: Concepts and Contexts" 4th edition, by James Stewart in order to see the applications of things like the integral and in order to get a better idea of how modern courses have a working understanding of the subject in general. Sometimes it's just good to be up to date with things.

This is by no means a Calculus textbook in the sense of being a book that would help you get through Calculus 1, 2 or 3. It's some sort of ill-considered intro to Real Analysis, though it falls far short of being that either. Spivak is probably a brilliant man, but he's a very poor teacher. There is little relation between the problems presented and the text itself (even with his own answer book, it's not so much a stretch as a leap when you undertake the exercises). I passed standard Calculus courses, and was looking at this as a review text, with some enrichment. Not even good as that. I would recommend Stewart's books as the most workmanlike available for a price, and the free online textbook by Paul Dawkins as a particularly helpful and useful aid. I'm not throwing out my copy of Spivak yet: I mean to look at it again after I get through some R. Anal. But as a book to learn or review calculus from, this was rather a waste. I too was mislead by the rave reviews.

After 7 years of math in high school, decided to get deeper in the subject. Bought this book and studied it in combination with Mit Ocw lectures (calculus 1 and calculus revisited) and it was awesome. It is a great book to learn everything about calculus of a single variable, and may not be so good for students who just want formulas and recipes for calculus. The book is completely self contained, teaching everything you need to know, starting with the basic operation properties. It makes you derive a lot of properties by yourself and many times a prior exercise is needed for the development of a harder one. Since I didn't have anyone to guide me through which exercises were more important, I decided to do all of them. It was a lot of work, but very rewarding. By the end of the book, you will be able to do any dervative or integral (as long it is doable) with security. With the bonus of knowing where it came from and how to derive it, making memorization of formulas unnecessary. If you intend to take analysis any time afterwards, this book will give you all the initial knowledge and experience in profs, theorems ande definitions to tackle it.