This edition differs from the third mainly in the inclusion of additional problems, as well as a complete update of the Suggested Reading, together with some changes of exposition, mainly in Chapters 5 and 20.
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Spivak's celebrated Calculus combines leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Ideal for honours students and mathematics majors seeking an alternative to doorstop textbooks and more formidable introductions to real analysis.
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218 of 226 people found the following review helpful
Format:Hardcover
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.
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.
123 of 126 people found the following review helpful
Format:Hardcover
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 out there -- it was my text in 1956. They are best described as 'blink and think'. I don't think it would be possible to use the book for self-study with no teacher to talk to without getting the answer book. Spivak supplies answers and/or hints to only about 10% of the problems. The answer book does the rest (although I think it does contain a few typos).
The book would have been very difficult to use as a Freshman, with no calculus exposure while taking physics at the same time (which most of us did back then). Derivatives aren't introduced for a long time, and integrals even later. The Freshman physics course started right off with Newton's laws (this was John Wheeler after all). However, once Spivak gets you to derivatives and integrals you will understand them, rather than just mouthing formulas.
Spivak certainly isn't the only analysis book you'll need. Both Strichartz and Abbott go a lot farther. The book contains nothing about Fourier series or higher dimensions. However, Spivak is certainly the place to start if you want to understand "what's going on under the hood" as another reviewer put it. Hopefully I'll now be able to re-read Strichartz and Abbott having seen and pondered the mathematical flesh which is draped on the logical bones these two books discuss.
115 of 118 people found the following review helpful
Format:Hardcover
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.
Most Recent Customer Reviews
The Calculus Book I Always Wanted
This book is like a conversation with the Mathematics Professor you always wanted to hang out with. It explains the theoretical underpinnings of the Calculus and gently seduces you... Read more
Published 9 days ago by Dave the Math Guy
A religious experience
I've only read the first chapter of this book, but it's been completely fantastic so far. In just one chapter, Spivak's dispatched many of the nagging questions that I'd had about... Read more
Published 17 days ago by Joshua S. Infiesto
This approach can be hard on physics and engineering students
I used this book for a very, on reflection, theoretic introductory calculus course back in 1972. I guess it was a very novel approach in those days, but, as some other reviewers... Read more
Published 3 months ago by Michael E Ottlinger
Fantastic!!
Very good introduction to real analysis, but very difficult. Reads like a story with no loss of rigor and requires the reader to think about every statement. Read more
Published 4 months ago by Snowball
A great book, but who is its intended audience?
I will preface by saying that this is an excellent book that gives deep insight to Calculus. Like many stated here before, it can be a great bridge or introduction to Real... Read more
Published 6 months ago by El G
Great! But...
This is a fantastic book. Really. I loved being able to go to Spivak after having used Stewart's Calculus for so many years. Read more
Published 6 months ago by F. Pieterse
...like when artists first discover color.
On my professor's recommendation, I purchased this book (in 2002) as a supplement to my over-sized, overpriced, rather watered-down college text: Stewart's Calculus. Read more
Published 10 months ago by Bidit Mazumder
The best.
This is probably the best American textbook on Calculus ever written. More than merely a textbook, it could even be considered a piece of fine literature.
Published 14 months ago by weltschmerz
Not a Watered Down Calculus Book - Intense
Tried this book as part of my independent studying. It was way over my head as I really had trouble with the practice problems. Read more
Published 14 months ago by GS
Familiar with the circa 1975 edition
We used this text for an honors freshman calculus course that I took in 1975. It was extraordinary then. Read more
Published 17 months ago by tpzahm
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