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I've already taken my calculus sequence and I'm interested in brushing up and staying sharp on the basics. So far, my calculus background is limited to single-variable calculus, which I applied in my physics sequence and engineering statistics. I learned a little bit of vector math in physics as well, but I'm not strong in it. My mathematical strengths lie in the discrete mathematics.

I'm particularly interested in studying for my IEEE Certified Software Development Associate exam - 10% of the exam is mathematics (calculus, differential equations, and statistics). Calculus is one of the mathematics knowledge areas that a software engineer should be competent in, so I'm looking for not only solidifying the courses that I've already taken, but moving forward with slightly more advanced topics, at an undergraduate mathematics level.

I'm currently using the textbook from my calculus sequence, but I'm interested to know what books other people used to learn calculus and if the book is potentially worth checking out.

What books are available to help me?

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    $\begingroup$ you could also go work in your school's math tutoring center, that way you'll get paid to re-learn every square inch of the calculus sequence. $\endgroup$ – Tom Stephens Aug 13 '10 at 20:02
  • $\begingroup$ Well, I'm in my final year of school. I'm also a volunteer mentor in my department (software engineering), which I find much more rewarding than a paid job. But I'm going back through and studying calculus, discrete mathematics, and statistics in preparation for the Certified Software Development Associate exam. $\endgroup$ – Thomas Owens Aug 13 '10 at 20:04
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    $\begingroup$ I think this question is too broad, and has the danger of generating a long list of answers of the form "I like textbook X". Could you say more about your specific needs and mathematical level? Is there some reason why just looking back at the textbook you used for the calculus classes you took would not be sufficient? $\endgroup$ – Pete L. Clark Aug 13 '10 at 21:02
  • $\begingroup$ Pete: I'll update the question. But I'm also looking for new books - I'm a fan of learning and sometimes, having a decent book to learn from beyond the one book I used in my one calculus sequence is better. $\endgroup$ – Thomas Owens Aug 13 '10 at 21:55
  • $\begingroup$ @Thomas: your revision is helpful, thanks. Yet more information would be even more helpful: what post-calculus math classes have you taken? $\endgroup$ – Pete L. Clark Aug 13 '10 at 22:09
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Thomas & Finney, Calculus and Analytic Geometry.

Spivak, Calculus.

Apostol, Calculus.

I recommend Apostol the most.

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  • $\begingroup$ Can you add links to the books? I found two volumes of the Apostol book on Amazon (are there more?) and it actually looks better than the Stewart book that I currently own and use (which is surprising - I've looked at a few other calc books in the library and so far have been most impressed by Stewart). $\endgroup$ – Thomas Owens Aug 13 '10 at 20:06
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    $\begingroup$ @Thomas: There are only two. Roughly, volume 1 is single-variable, and volume 2 is multivariable. It is a good book. They teach out of it at CalTech and MIT, among other places. $\endgroup$ – Larry Wang Aug 13 '10 at 20:51
  • $\begingroup$ OK. Thanks. If I were to purchase Apostol, I would probably want both of them. I've always wanted to learn multivariable calculus, just never had time to take the course. $\endgroup$ – Thomas Owens Aug 13 '10 at 20:56
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    $\begingroup$ The Thomas & Finney book is one of the old standard calc books (my father used the 3rd edition [just Thomas back then], I used the 7th, I later bought the 9th, and I've seen people teaching out of the 9th and 10th over the past 10 years). Solid, lots of great problems (a good teaching resource), and good exposition, but some newer books might be better (and I certainly prefer teaching out of some of the newer books). $\endgroup$ – Isaac Aug 14 '10 at 8:35
  • $\begingroup$ nice.............+1 $\endgroup$ – Bhaskara-III Dec 28 '15 at 0:01
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Elementary Analysis : The theory of calculus by Kenneth Ross is a beautiful written book. Link is here : http://www.springer.com/mathematics/analysis/book/978-0-387-90459-7

Diferential and Integral Calculus by R.Courant is again a very good book which one can learn. http://books.google.co.in/books?id=eyC1nk9-YjkC&printsec=frontcover&dq=Calculus+R.Courant&source=bl&ots=v0rLcDvVW_&sig=VPB6ZBXfMm6yj0vLG4YalVkpV7I&hl=en&ei=_6ZlTLXAN47IvQOhtZHyDA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CCEQ6AEwAQ#v=onepage&q=Calculus%20R.Courant&f=false

Advanced Calculus by J.M.H Ohmsted is also a very good book. Although title says advanced its not really advanced.

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Even though I took a course out of it more than 10 years ago, I still refer back to Spivak's Calculus all the time.

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  • $\begingroup$ To me, that's seems like it's a good sign that it's a solid, well written book. But how about for relearning? If I need to refresh my memory, it might be good, but if I've forgotten a topic and need to relearn it, will it also be good? $\endgroup$ – Thomas Owens Aug 13 '10 at 20:38
  • $\begingroup$ It depends what exactly you are looking for in a calculus book. Spivak's book is very rigorous, and the exercises (which are wonderful) are a mix of proofs and problems. $\endgroup$ – Jim Aug 14 '10 at 0:29
  • $\begingroup$ If you are looking for something more basic (i.e. how to differentiate and integrate - and not necessarily how to prove theorems), there is a free calculus book that you can download from the University of Wisconsin math dept. website. $\endgroup$ – Jim Aug 14 '10 at 0:32
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    $\begingroup$ @Thomas. I think Spivak's Calculus should be one of the best for self-studying, since it is not written in "definition-proposition-proof" style, but also "talks" about what is going on. For instance, Rudin's Principles of Mathematical Analysis is a great book, but I wouldn't recomend it for self-studying, since it is too much formal. Also Apostol's books seem a little hard to me for introductory calculus. $\endgroup$ – Agustí Roig Aug 14 '10 at 9:11
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If you can get a copy (which is improbable, I'm afraid),

  • Vladimir Smirnov, A Course of Higher Mathematics, VOLUME 1, Pergamon press 1964.

This is a translation of a beautiful treatise on mathematical analysis which served as a textbook for generations of engineering students in the USSR. The first volume, in particular, is one of the best introductions to calculus I've ever seen.

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My university uses Calculus: Early Transcendentals by James Stewart in the standard calculus 3 quarter sequence, the prolonged 4 quarter calculus sequence, as well as in multivariable calculus. This book covers topics such as functions and models, limits and derivatives, differentiation and applications of differentiation, integrals, techniques of integration, vectors, vector functions, partial derivatives, multiple integrals, vector calculus, and second-order differential equations.

I also used a different James Stewart book in high school during pre-calculus and calculus. I don't remember the name of that book - it might have even been an earlier edition of this book.

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  • $\begingroup$ This is probably the most widely used calculus textbook. I think the only difference between the 'early transcendentals' version and the regular version is that the former uses the (ε, δ) definition of limit. $\endgroup$ – Larry Wang Aug 13 '10 at 21:00
  • $\begingroup$ I'd call this a solid book, though it's not my favorite. $\endgroup$ – Isaac Aug 14 '10 at 8:32
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Calculus: Single Variable, Course Advantage Edition by Hughes-Hallett, Gleason, et al, aka the "Harvard Consortium" book, is perhaps the standard "reform calculus" book.

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  • $\begingroup$ What do you mean by "reform calculus"? I've never heard that phrase and am running off to look it up now, but I definition here would be useful for other people. $\endgroup$ – Thomas Owens Aug 14 '10 at 11:21
  • $\begingroup$ This summary page for a larger article is a reasonable explanation of reform calculus: math.duke.edu/~das/essays/trends/summary.html $\endgroup$ – Isaac Aug 14 '10 at 21:22
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Calculus from Graphical, Numerical, and Symbolic Points of View by Ostebee and Zorn is my favorite book from which to teach (er, actually, I have a slight preference for the 1st edition over the 2nd, but they're both good). It's aimed a little above average (i.e. harder than the Harvard Consortium book [Hughes-Hallett, Gleason, et al]), but it's much more of a modern reform-calculus than Spivak.

In particular, I think the conversational tone of the book makes it more readable and the variety of problems (especially the fact that it has challenging problems rather than just repetitive exercised) makes it a good book from which to study.

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I tried to work with Spivak's book. It was not very practical for selfstudy since the exercises are pretty tough. I could only solve every 10th problem in chapter one. I will come back to spivak sometime, but at the moment i simply can't afford to spend the time spivak takes. (I might have to mention at this point that I'm studying Engineering not Mathematics and that a student in mathematics might not have such difficulties with this text.)

Thus I decided to look for something else, that would be less time consuming. I tried Thomas' Calculus, a newer edition, which I do not recommend at all. The exercises are so simple you won't be challenged at all and thus not memorize and benefit much in the end. Simple Plug and Chug.

Then I found Marsden and Weinstein's Calculus 1-3. These books are just marvellous. You can find pdfs online on their webpage, I checked them and bought all the books. It's already 30 years ago that these books were published and they are not as much appreciated as they should be. Their approach to single variable and multivariavle calculus is unique. They made a clever choice of exercises with varying difficulty. Also there are a student guide and exams that help understanding the material and checking the progress which I found very helpful and practical.

If you are limited in time and if you are forced to do selfstudying these books will be a good choice.

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