I started reading Spivak's Calculus about a month ago and I'm at the end of chapter two, so this is not really calculus yet. However, I find the problems really difficult and the answer keys are not very helpful. I can answer some problems but a lot of them, I've no idea what I should even start with. Also, it takes me a very long time to do each problem. (10-15 minutes on average).

I've no prior experience to writing proofs, but I've watched a few youtube videos on it. The problem is generally not the proofs problems but the one where they say "Find a formula" or "derive this equation". Those, i can never do.

Is there some prior knowledge that I should know before reading this besides knowing how to do proofs? Or, are these problems generally difficult to begin with? Or, am I not paying attention to what's written in the chapter?

Also, should I look at/try every single problem are is doing like the first page of them sufficient? I had plan to do every problem in the first chapter but called it quits after they started putting epsilons everywhere.


  • $\begingroup$ Most of the first page problems in Spivak's book are computation problems for practicing some technique, so you should try to do more than just those if you want to really understand the ideas, because understanding will mostly come from the problems: my experience with reading Spivak is you finish a chapter, think understand the ideas well, and then after 5 proof problems you realize you know nothing. But you've also learned a lot from these 5 problems, and each new one teaches you even more. So Don't do all of them, which would take forever, but try to get at least 75%. $\endgroup$ – user71641 Oct 6 '13 at 23:57
  • $\begingroup$ Chrystal's Algebra seems to me to be the only preparation for Spivak, I'd say Spivak himself used Chrystal or an old book like that before even learning calculus. A good idea would be to use Chrystal & something like Piskunov for computational calculus, then come back & try either Spivak, Apostol or Courant for the theory, or mix them all up & use them at the same time, whatever works. $\endgroup$ – bolbteppa Oct 7 '13 at 0:00
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    $\begingroup$ You might want to read a book like amazon.com/How-Think-Like-Mathematician-Undergraduate/dp/… to get used to logic and proofs. $\endgroup$ – Ted Shifrin Oct 7 '13 at 0:24
  • $\begingroup$ I only read spivak just before starting grad school... you are definitely ahead. $\endgroup$ – Squirtle Dec 26 '13 at 22:33
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    $\begingroup$ 10 to 15 minutes is not "a very long time" at all! $\endgroup$ – Pedro Tamaroff May 18 '14 at 22:54

Given that you mention that you have "no prior experience [in] writing proofs", you may find Velleman's How to Prove It: A Structured Approach to be a helpful preparation for Spivak's wonderful book. And it is a wonderful book, so please keep at it, or come back to it after Velleman. Note that I haven't read Velleman, but it seems to get universal praise for "preparing students to make the transition from solving problems to proving theorems".

I'd recommend that you do not obsess about finishing all, or even most, of the problems on first reading. I think it's worthwhile to give it a first read without worrying that you are getting every ounce out of it. If you feel like you're getting most of it, keep going. Do plan to give it a second read, though.

Also, there's some good discussion on how to read mathematics books in how-to-read-a-book-in-mathematics.


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