Calculus

There are hundreds, even thousands, of books on Calculus. The good thing is that you can always find one that is best suited for your way of learning, and teachers always know which are best for their courses. The bad thing is that there are so many to choose that one ocasionally doesn't know how to start. I will not say that Spivak's book is the greatest or the worst (as many reviews do), because this is mostly a matter of taste. I will, instead, say what is different in Spivak's book from the other many calculus books that I know. Spivak's book reads as a romance. Indeed, you can be completely taken by his prose, and see the battle for arriving at the concepts in a rigorous, yet flexible, manner. Spivak sees calculus as the evolution of one idea - as he says in the preface to the first edition. And everyone who reads this book with attention understand what is this idea - I will not tell, not say who is the murderer of this romance. Will you finish this book and be a master of calculus? Will the exercises help? This depends mostly on you, as mathematics is, for the most, only understood when you create it in your mind, when you retrace every important step of this human endeavour inside of you. There are easier books, that's true. For those who want just to be able to use calculus, perhaps there are better ways. But Spivak succeeds to do one very difficult thing, that is, to explain mathematics, all its beauty and all its harmony, to a reader that possibly never really understood it. For one thing is to make calculations, learn how to solve equations and the like, but another thing very different is to understand what is mathematics about. To my knowledge, only Courant wrote in such passionately manner - and Spivak is clearly a follower of Courant's way of thinking. The difference between Courant and Spivak is that Courant is more worried about the applications of calculus, and Spivak is more worried about the essence of it (it doesn't make too big a difference, as both are, in a sense, the same thing told in two different ways). There is something that is really fantastic in this book: Spivak usually discuss with a lot of care the evolution of one idea since scratch up to its refined description in mathematical language - see how he arrives at the rigorous definition of a function. He also criticises the notations, and this is pretty useful to understand the concept behind the notation. If you are able to love mathematics (many people aren't), you will with the greatest probability appreciate this book.

Some reviewers have been puzzled as to the style of this book, deep mathematics for the unsophisticated reader. This is explained by its origin in the 1960's when many bright high school students were not offered calculus until college. Hence some top colleges experimented with very high level introductions to calculus aimed at gifted and committed students who had never seen calculus. Possibly Spivak took such a course, but certainly his book was used as the text for one at Harvard, and was still used more recently at a few schools still offering this course, such as University of Chicago. Unfortunately today, due to the somewhat misguided AP movement, which is oriented to standardized test performance rather than understanding, almost all mathematically talented high school students take calculus before college, receiving significantly inferior preparation to what they would receive in college. The result is that many top colleges where the Spivak type course originated, no longer see the need to offer it. This means that gifted freshmen at schools such as Harvard and Stanford are now asked to begin with an advanced honors calculus course for which Spivak is the ideal prerecquisite, although those same schools do not offer that prerecquisite. Thus if you are a high school student hoping to become a mathematician and planning to attend many elite colleges, almost the only way to be adequately prepared for an honors level mathematics program is to read this book first. It may be that a book like Stewart or even Calculus Made Easy, is useful as a first introduction to calculus, but it will not get you to the level you need for a course out of Apostol vol. 2, or Loomis and Sternberg. Although some seasoned mathematicians eventually come to prefer other treatments such as those of Apostol or Courant, the appeal for a young student of Spivak's light hearted, clear, and enthusiastic presentation is probably unmatched by any other work. For forty years there has been a cult of fanatical adherents to the books of Spivak, and they still deliver the goods to the audience for which they are intended. There is only one caveat that may even be unimportant for beginners, but it is that Spivak is almost too clear and thorough in his explanations. I.e. a future researcher needs to learn to question, to discover and work things out on his own, whereas Spivak explains everything for you. So every now and then resist reading his explanation and try working out a proof or a generalization on your own, and see if it doesn't become even clearer afterwards. Understanding how a proof was thought of is different from just knowing how it goes.

Calculus wasn't taught in public high schools 50 years ago -- our brains were thought too soft to encounter it under the age of 18. The 17 year old supermarket checkout girl where I live in rural New York will take it this fall. I've always been fascinated with math, particularly the inevitability of it all (how mathematicians were dragged kicking and screaming into imaginary numbers, non-Euclidean geomtery and the completed infinite). Despite minimal background, I was able to get A's in calculus, differential equations, and complex variables at Princeton back then. It was like my Bar Mitzvah, getting through by mumbling incantations in a language I didn't understand. Figuring that I'd received a decent mathematical background, I tried studying math at a higher level 5 years ago when I left medical practice. Strichartz was dense and I spent hours puzzling over notation in the first edition (until I found that some of the most confusing parts were actually errors not all of which were corrected in the second paperback edition). I made it about half way through -- it just seemed too abstract. Abbott's book was also quite good, but again pure analysis is about the logical structure underneath mathematics (something I certainly was trying to understand). Having read the rave reviews of Spivak's book in this forum, I bought it (along with the answer book), and have spent the last 8 months going through it, and doing about 3/4 of the problems. It is marvellous. The exposition is clear and friendly (as are Strichartz and Abbott) -- something not seen in the math books of the 50s (although Spivak's first edition goes back to 1968). Almost nothing is assumed (except the properties of the rational numbers). Everything is derived from them and clearly (including one construction of the real numbers from the rational numbers at the end and (typically) two more constructions of the reals in the problems at the end of that chapter). Even better -- the book doesn't just show you how mathematical consequences follow logically from inscrutible definitions. It shows you why the definitions must be the way they are. The chapter on the definitions and properties of the logarithmic and the exponential functions is aparticularly fine example of this technique and of Spivak's teaching style. One does not study human anatomy by studying only the bones which hold up the physical structure, although without bones we are a just pile of goo. To learn anatomy one must also study the flesh draped on the (physical/logical) skeleton. This is also where Spivak excels. There are plenty of examples sprinkled throughout the text in addition to the logical structure -- several whole chapters are devoted to applications of the theory just developed. The real mathematical flesh is in the exercises. None of them are trivial. None are of the 'plug and chug' variety seen in Thomas-- which amazingly is still o

I agree with some of the previous reviewers that Spivak's book is a bit much for any but the brightest first-year calculus students. It would be quite the uncommon 17-18 year old who's disciplined and mature enough to rise to the challenge. Maybe with the guidance of an outstanding lecturer...On the other hand, Spivak serves as excellent preparation for one's first real analysis course: ideally, read through this book (and, crucially, do ALL the exercises) the summer before introductory analysis, and you'll be in great shape to tackle the likes of Rudin's "Principles of Mathematical Analysis." In the process, you'll also build a much deeper understanding and appreciation of the material covered in first-year calculus.Spivak's book is also a wonderful re-introduction to mathematics for those who've been away for a while. It's very well suited to independent study, and Spivak is an excellent teacher.The book is carefully written, chatty but not informal, conversational but not overly long-winded. The exercises are challenging, but provide additional insight into the material and, more importantly, deepen your understanding and build your problem-solving and proof-writing skills. With patience and diligence they're all quite solvable by anyone who has, or who is serious about cultivating, a little mathematical maturity.

According to my knowledge, this is one of the best calculus books in English for honors freshman. I personally rank the following star-calculus texts in such an order: T.M Apostol's > Michael Spivak's > R. Cournat's > Hijab's > James Stewar's. Apostol's texts provide a deeper understanding of the subject than do any other books. One must have a firm knowledge of elementary calculus if he wants to penetrate this "brightest" gem of calculus. Dr. Spivak's book, I think, is most suitable for those who desire challeges. This book contains many typical examples and problems, they are neither easy or single-step, and, therefore tends to always motivate students' minds. The other three are equally good. Hijab's has rigorous proofs and a lot of computational problems; Courant's has an excellent balance between theory and applications. While Dr. Stewart's tell you in more detail how to solve calculus problems, especially those somewhat challenging ones. All these are the "kings" of the world of calculus! Pick the best one for yourself.