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I’m not a complete beginner. I’ve got a PhD in Mathematics Education which gave me a pretty strong Master’s degree level of pure math knowledge.

That was 30 years ago, and I’ve forgotten tons. Seeing Michael Spivak’s Calculus book mentioned all over, I thought I’d give it a try. The problems seem from pretty simple to impossible.

I’d like to get my analysis chops back in order. I have my copy of Taylor & Mann for Advanced Calculus, Churchill Downs for Complex Analysis, and a few others I’ve collected here and there.

So I’d like your informed opinion: should I try to regain my math chops using Spivak, or use a more traditional Analysis text—maybe not as traditional as Taylor and Mann, but maybe Understanding Analysis by Abbott.

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    $\begingroup$ Go with Spivak. I learnt my calculus from that book 30 years ago and in all that time I have not seen any other calculus textbook that comes even close to it. Its only fault is that it doesn't have anything on multivariable calculus. $\endgroup$
    – Sam
    Commented Aug 15, 2022 at 0:26
  • $\begingroup$ I have gone with Spivak and am not regretting it at all. I just have to realize that my hour-long session may mean I get to one problem (I just did Chapter 2 question 3 part (e) part (iii) and it took forever. I finally looked at the solution and spent the rest of the hour shouting at the ceiling because the answer was essentially one line long, but requiring quite a sophisticated amount of cleverness). $\endgroup$ Commented Aug 29, 2022 at 17:41

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Spivak is good, also so are Tao's books. I mostly recommend reading a bunch of books and using the ideas that make the most sense to you. I usually try to have a primary book that I like the most, and a few others that I read a little more quickly just to see how other authors handle the same topic.

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  • $\begingroup$ I have Tao’s books but have only glanced at them. They were far easier to read than I would have guessed. $\endgroup$ Commented Aug 29, 2022 at 17:42
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One reason people use Spivak's book for self-study is that it has a very extensive solution manual containing the answers to the exercises, so if you are studying by yourself you can look up all the answers. Now there is stackexchange so the solutions manual is less relevant, although some people might be embarrassed to admit when they got stuck on question 1.1 or something. Spivak also grades the difficulty of the problems with stars, which are quite accurate about the relative amount of time the problem will take.

It was published in 1966 so it's already "traditional". Also Spivak was the course book for first-year analysis at my university, so I really don't think it's some kind of unconventionial book (or perhaps I went to a weirdo university). Anyway I just went and looked at the university's page and they are still using Spivak. Spivak was ahead of his time because he also invented "gender-neutral pronouns".

If you do decide to go with Spivak it would be smart to start from the beginning and do the "easy" problems and then work through to the end. People who start in the middle of the book end up like those weightlifters who only work on one part of the body, so they have giant biceps and tiny legs or something. Then they can do all kinds of complicated-looking proofs but they cannot do the simple parts of problems.

Spivak also does not cover set theory. Sometimes you need to understand at least the basics of that.

Another opinion: When I mentioned that Spivak was the course book at my university, a person who attended another place told me that he thought that "Spivak has no soul". I have absolutely no idea what he meant by that comment.

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    $\begingroup$ There were very substantial revisions done by the third and fourth editions of Spivak, adding hundreds of exercises and some improvements to the exposition. I recommend not using the first or second editions at this point. $\endgroup$ Commented Aug 15, 2022 at 1:03
  • $\begingroup$ I agree 100%. I’ve seen editions 1, 3, and 4 and the difference between 1 and 3/4 are noticeable. I found a copy of the 4th edition and a copy of the solution manual, so it couldn’t be better for self study except that it is quite humbling. So many of the problems take me forever, and a full sheet of paper. I look at his solution and it’s literally something like “Subtract part (ii) from part (i).” His answers are so elegant and mine are like a high schooler cranked them out. I had the opposite experience with real analysis in school, finding it easy—but finding Algebra impossible. $\endgroup$ Commented Aug 29, 2022 at 17:48
  • $\begingroup$ Any chance you still have your syllabus, or a link to the current one? I’m curious to see how a real classroom would use this book, as opposed to my method of doing every section and every non-starred problem. $\endgroup$ Commented Aug 29, 2022 at 19:12
  • $\begingroup$ Sorry but I can't share that. $\endgroup$ Commented Aug 29, 2022 at 20:56

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