For financial reasons, I dropped out my senior year of college as a piano performance major. I will be returning to college to dual major in mathematics and computer science. I've taught myself to program out of SICP and would like to develop my mathematical chops. I got a 5 on the Calc BC test my junior year of high school and have always considered myself decent at math.

I prefer learning from rigorous material, so Spivak's Calculus seemed to be a logical choice. I made it through the first chapter, but was blindsided by the difficulty of the first chapter's problems.

Question 1. So, after solving a few problems, I put Spivak aside and picked up How to Prove It to brush up my proof writing skills. I've long ways away from finishing that, but what prerequisite material is recommended for Spivak?

I remember feeling similarly overwhelmed when starting "Structure and Interpretation of Computer Programs" however, there's a lot of supplementary information available for that book and I now feel I mastered that material. "Helper" material for Spivak seems sparse (excepting the solutions manual). Question 2. Does anyone know of any video lectures that cover Spivak?

To my uninformed mind, Spivak seemed like a good way to "brush up" on Calculus while improving my general math skills. Obviously, I horribly underestimated its difficulty.

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    $\begingroup$ +1 for choosing such good books. They are hard but rewarding. $\endgroup$ – lhf Oct 4 '11 at 20:24
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    $\begingroup$ Thanks, I've always enjoyed rigorous work. I enjoy actual mastery more than being able to say "I can program derp derp." $\endgroup$ – Josh Infiesto Oct 4 '11 at 20:38
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    $\begingroup$ I've never been embarassed to tell anyone I first learned calculus from "Calculus for Dummies" before I learned it from Courant. Depending on your age/mathematical maturity, it's asking a bit much to develop decent intuition simultaneously to proving things rigorously. First develop the basic skills from an easier book, so when you go through Spivak you can focus on the subtleties instead of the basics. $\endgroup$ – Ragib Zaman Oct 5 '11 at 1:53
  • $\begingroup$ I wouldn't be ashamed of learning from an easier source either. However, I learned calculus in High School. I had an excellent teacher, and was very comfortable with the material. It just wasn't very rigorous. $\endgroup$ – Josh Infiesto Oct 5 '11 at 18:11
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    $\begingroup$ I'll never forget all the time I spent figuring out one of Spivak's problems: finding the antiderivative of $\sqrt{\tan(x)}$. Nowadays, Mathematica gets that one automatically. $\endgroup$ – Brian B Nov 20 '14 at 16:54

I have read through the first few chapters of Spivak, however my personal preference is for Apostol's Calculus. It's also a very rigorous approach, and a very well respected book, however it starts more gently than Spivak's. With Spivak's book, the problems start out extremely hard, and get easier as the book goes on (mostly by getting used to his style, not objectively). With Apostol I was able to understand and answer all the questions in the first few chapters much more easily, and then I saw the difficulty increase a bit; however it increases progressively throughout the book. Many of the problems in the introduction of Apostol are exactly the same as those in Spivak, however the order and context that they are presented in leads you to the correct method for proving them, whereas Spivak's are more isolated.

There are many great discussions about calculus books on other forums, such as The Should I Become a Mathematician? Thread on Physics Forums.

I agree wholeheartedly with mathwonk's statement that, although the books are difficult, reading different approaches and going over them multiple times is really what gives you a deeper understanding of calculus. Mathwonk also mentions that most students find Apostol very dry and scholarly, where Spivak is more fun; however, I have not found this to be the case. I have worked through every problem in Apostol's Calculus through chapter 10 so far, and it has been a joy (most times). As an added bonus, Apostol's Calculus covers linear algebra as well, and the second volume covers multivariable calculus. Spivak's analogous book, "Calculus on Manifolds", is known as an extremely difficult text, and is commonly used as an introduction to differential geometry (indeed, his comprehensive volumes on differential geometry mention Calculus on Manifolds as a prerequisite).

The choice of book should also reflect your future interests. I am a computer programmer currently, and am looking to go into mathematics exclusively. It sounds like you are still melding the two. I would say that Apostol's book might serve you a little better in this respect as well, as it is slightly tilted towards analysis, whereas Spivak's is tilted towards differential geometry. For instance, Apostol introduces "little-o" notation, a cousin of "big-O" notation which is used extensively in computer science. That being said, Spivak has been described by some as a deep real-analysis text more than a calculus book, so you would still deeply cover all the fundamentals.

Another set of calculus books which I own and are held in high regard are Courant's. My brief skim of them, as well as other's comments, suggest that they are more focused on applications perhaps than some of the other books. Apostol's is still, in my opinion, very well peppered throughout with applications; many chapters contain a specific "applications of ..." section which links the theoretical concepts you just learned with the applied use of those concepts.

My only exposure to Courant's expository style comes from his excellent book What is Mathematics. This is a book I would strongly recommend reading regardless of what calculus book you choose. I cannot praise Courant's lucid writing highly enough, and look forward to working though his Calculus texts in the future.

I think that you would find Apostol's book sufficiently rigorous, as well as extremely intuitive. I also am a musician, and coupled with my computer programming experience it seems that perhaps we think alike. Whatever book you choose, recognize before you start it that you are running a marathon, not a sprint.

  • $\begingroup$ I couldn't add more than two links, so here are the google books references to the books I mentioned: Apostol's Calculus Vol. I Apostol's Calculus Vol. II What Is Mathematics - Courant $\endgroup$ – process91 Oct 9 '11 at 16:19
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    $\begingroup$ +1 I also recommend Apostol's books wholeheartedly. Apostol is an excellent writer and explains the material clearly, usually motivates very well the discussion to come and presents a good deal of examples which is always welcome in a math textbook. $\endgroup$ – Adrián Barquero Oct 9 '11 at 16:28
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    $\begingroup$ Thanks, I'll buy Apostol volume 1 then! $\endgroup$ – Josh Infiesto Oct 9 '11 at 16:38
  • $\begingroup$ Great! I don't think you'll regret it. Get What Is Mathematics while you're at it, it's only $15 and you will have some of the most profound realizations while reading it. No prerequisites either, jump right into both of them. $\endgroup$ – process91 Oct 9 '11 at 16:45
  • $\begingroup$ The link to Vol I in my comment above actually links to Vol II. Here's the proper link: Apostol's Calculus Vol. I $\endgroup$ – process91 Oct 10 '11 at 2:10

About a year later, here's what I'd wished I'd known before starting Spivak. The proof stuff is not so hard. Mostly, I wished I'd known more about inequalities before starting. Specifically, AM-GM, the trivial inequality, and the Cauchy inequality. They get used in the problems A LOT.

  • $\begingroup$ Those are both introduced in Chap. 1.. $\endgroup$ – user70962 Jun 10 '13 at 6:12
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    $\begingroup$ I know. They are covered better elsewhere. $\endgroup$ – Josh Infiesto Jun 24 '13 at 2:12
  • $\begingroup$ I don't fully agree. I've looked through a lot of texts and the only way I've seen these inequalities covered is simply by stating them and proving them. I wouldn't really consider this as being better covered. I think the problem possibly is that he left them to the excercises so for someone who isn't familiar with them at all thy might be overlooked, but the detail in which they're covered seems just right to me. $\endgroup$ – user70962 Jun 24 '13 at 13:55
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    $\begingroup$ I should rephrase. There are other books that helped me develop a "repertoire" of techniques for using those inequalities in proofs. None of the books I'm referring to are textbooks. I have a smorgasbord of contest problem solving books, which I found to be far more helpful than Spivak in the inequality department. $\endgroup$ – Josh Infiesto Jun 25 '13 at 9:58
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    $\begingroup$ If someone has the same concerns about inequalities, I highly recommend Bellman's Introduction to Inequalities. $\endgroup$ – user59083 Nov 5 '13 at 16:53

You might take a look at "Introduction to Mathematical Reasoning" by Peter Eccles. It is a "nuts and bolts" book that introduces students to the fundamentals of writing proofs. He tackles all the background topics like basic logic, truth tables, elementary number theory, set theory, etc. I found the exercise sets to be very helpful. They are chosen to be progressively more difficult, and he occasionally introduces an interesting side topic as well. Also, he has numerous solution sets in the back so it's a good book for self-study.


Spivak is a fine book. Of course the student should be proficient at algebra before beginning. If that algebra experience was a few years ago, brushing-up will be essential.

When I teach from Spivak, of course I spend time working on "how to write a proof".

Neither Spivak nor Apostol is best suited for self-study, however. Both books will benefit when studied with an experienced instructor!

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    $\begingroup$ What would you recommend for self study? I've tried many times unsucessfullly with Tomas/Finney's - Calculus. I've been using khan academy lately. $\endgroup$ – gideon Aug 17 '14 at 15:46

If you're still looking for video lectures on real analysis, Dr. Francis Su at Harvey Mudd College has some very excellent ones on Youtube linked in their entirety at his course blog here: http://analysisyawp.blogspot.com. He does not use Spivak; he is using Rudin's real analysis book.

However, if you are looking to learn the math you need for computer science, I think that real analysis might not be the best choice to start. Discrete math/structures are more relevant and applicable to the courses you'll be taking at first. For discrete math, I think Susanna Epps' discrete math book is good and has a lot of computer science application sections. Grimaldi's discrete math book is also excellent as well.

Also, to make your studying productive and ensure that you'll be covering material you need to know, I think you should look into what books they're using in discrete math and calculus at your school.


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