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I was wondering out of these three which would you take

calculus by Spivak

calculus by Hughes-Hallet

calculus by Morris Kline

I'm taking calculus III next semester but I want a better book that's harder than calculus by Larson which I don't like all that much

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marked as duplicate by Eric Stucky, T. Bongers, Grigory M, Sami Ben Romdhane, Newb Dec 28 '13 at 7:34

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Relevant info: math.stackexchange.com/questions/114646/…. –  user87274 Dec 28 '13 at 5:05
A great one at a decent price that is more theoretical than the Larson text that covers the vector side of thing (which Larson ignores) is Vector Calculus by Colley. –  mathematics2x2life Dec 28 '13 at 5:34

3 Answers 3

None of them.

I know all the hard core mathematicians are going to recommend Spivak and they're all howling right now.But I can't in all good conscience recommend Spivak to a beginner in calculus, no matter how talented they are.

The problem with Spivak-and it is very beautiful,no question-is that it's just too austere and theoretical. The result is that even if a student is talented enough to master it, it'll only give half the story of calculus. Calculus is as much about the applications to the physical and social sciences as it is about the rigorous theory of limits and convergence of real valued functions. So a bizarre thing happens when a student learns calculus from this book-they can do epsilon-delta proofs with full rigor, but if you ask them to solve a simple projectile motion problem or to minimize an area, they look at you like you're speaking Martian. Choosing between a rigorous presentation of the theory of calculus and the study of it's applications to real life situations is no choice at all.

And the truth is-you shouldn't have to chose between them,there are now books that try and strive for a balance. Not many, but the ones that do exist are excellent.

An outstanding recent addition, which I'm dying to get my own copy of when I can, is Calculus With Applications by Peter Lax and Maria Shea Terell. It's a complete update and expansion of the calculus textbook Lax wrote in the 1970's and it's entire philosophy is to teach a course in single variable calculus that balances theory and applications in full measure. Not only does it prove all the major results of calculus fully and carefully, it contains an enormous number of applications and examples in mechanics, biology, chemistry, finance and the social sciences-as well as many major computer projects using computer algebra systems. If I could chose any text for an honors calculus class and could only pick one, this would be the one I'd pick.

Much more old fashioned but very much in this spirit is An Introduction to Calculus And Analysis by Richard Courant and Fritz John. In many ways, the Lax/Terrell book is an update of this one. Which is not really surprising since Lax first learned calculus in Germany from the original lectures of Courant at The University of Göttingen! It lacks computer exercises and doesn't quite have as many examples and applications as the Lax/Terrell book, but it's very similar otherwise and written by 2 masters. Still well worth reading after all this time.

Those would be my recommendations. If you insist on using Spivak, then I'd supplement it with the online version of Gilbert Strang's Calculus -which is the best applied calculus textbook that's ever been written.

Good luck!

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This is a really interesting point of view, one that I haven't encountered when looking up texts books...thank you –  Matty D Dec 28 '13 at 5:54
Mathemagician1234, I'd actually say that the newer 4th edition of Spivak does include some application (e.g. volumes of solids of rotation), but still does not address the multi-dimensional calculus included traditionally in a third-semester Calculus course. I'd agree with your assessment in general, but the author did ask for a more challenging text, and I think the second Calculus book by Apostol serves this purpose while providing excellent applications. –  Darrin Dec 30 '13 at 1:16
@Darrin Apostol is rather dry and the organization is a little bizarre. That being said-it IS an excellent text by a master and I really shouldn't have omitted it. And I excluded multivariable calculus because to me, a beginning text means single variable. If you want to include multivariable calculus texts, then the selection widens considerably and my suggestions can be found here: math.stackexchange.com/questions/265068/… –  Mathemagician1234 Dec 30 '13 at 4:41
I don't know much about Munkres' work, but I do know Spivak's Manifolds is well above a Calculus III level, and Spivak has expressed this to me himself via email! Buck would be an example of a text between Spivak's and Apostol II. For a student wanting to keep in touch with a rigorous text while going through this course, I can't see anything other than Apostol's second Calculus text that could achieve this goal. –  Darrin Dec 30 '13 at 4:49
@Darrin Take a look at the Hubbard and Hubbard book. And the Dover paperback by C.H. Edwards is a really good low price alternative if that's too expensive. I think you'll be converted to add these as alternatives to Apostol,trust me. –  Mathemagician1234 Dec 30 '13 at 8:08

Spivak, hands down. This text (along with Apostol's) is widely regarded as the Calculus text. In fact it is a valuable gem that many math students should have on their shelf.

Specifically, what in Larson did you not like?

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I feel like Larson trades quantity for quality in terms of problems and examples, and reading this book, to me, is like being in a lecture were the instructor throws problems on the board then mumbles a few words –  Matty D Dec 28 '13 at 5:22
Oh and you have the fact that the authors leave out huge chunks of the proof for say the Lim of Sin(x)/x and you are required to have the web portion of the book (which you have to buy) to see the full explanation...which to me is a cheap tactic to make you spend more money –  Matty D Dec 28 '13 at 5:26
So then I take it that you have taken an "applied" calculus course before then? If so, jumping into Spivak might actually do you some good in whipping you into shape for higher level math courses. It balances rigorous of calculus and problem-solving skills equally. Spivak will smooth talk you that you probably will not have to do that, but I cannot say the same for the exercises. –  Nameless Dec 28 '13 at 6:13

Spivak is not going to cover some of the material a traditional Calculus III text covers, e.g. partial derivatives, multiple integrals, vector calculus, etc. I would invest in Apostol's second volume of Calculus instead.

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