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I started reading Hubbard and Hubbard's Vector Calculus, Linear Algebra, and Differential Forms a few months ago on the recommendation of a former professor, but I found myself confused both reading the proofs provided and writing the proofs from the exercises. I finally gave up recently, feeling that I simply wasn't getting what I should have out of the book, as I was often skipping the most important parts because I didn't understand them! Thus, I started looking for "Intro to Proofs"-type books. So far, the two I have looked at are The Art of Proof: Basic Training for Deeper Mathematics and Mathematical Proofs: A Transition to Advanced Mathematics. I found the former one to be too focused on justifying every single step with an axiom in proofs for which I feel this was not necessary. The latter I found better, but I simply wasn't interested in the proofs - so many even and odd number problems!

What I'm looking for is something that

  • (a) will prepare me for the difficult-to-understand proofs in Hubbard and Hubbard

  • (b) will intrigue me and expose me to new math (new to me, not necessarily new in the world) and interesting problems

N.B. Obviously I would prefer the books be available online, but I understand that this is not always possible.

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This started as a comment, but got away from me. :\

I rather like Richard Hammock's Book of Proof as a good stepping stone to higher math. That being said, I am not sure that there is any way of knowing whether or not it will satisfy your point (b). I imagine that most (if not all) books written at this level are going to be somewhat pedantic and cover similar topics. In a way, I feel that this really is necessary for learning—you complain about "justifying every single step" and claim that it is "not necessary," but learning how to do this is part of the process. In Tao's taxonomy, you need to get through the rigorous stage in your development; you can't skip straight into the post-rigorous stage.

Some non-free alternatives if you want more topical books (i.e. books that deal with a particular topic, rather than just "intro to proof"):

  • You might try to pick up an "advanced calculus" text. Personally, I like Spivak's book. If you can get through Spivak, you should have the basic grounding necessary to start tackling more advanced topics like differential forms. I might similarly recommend Apostol's book. I would, however, recommend that you avoid Rudin—Rudin's Principles of Analysis is one of the great books, but is daunting for a beginner.

  • If you are more interested in the algebra side of things, a couple of non-free options include Dummit and Foote (which is very approachable) and Hoffman and Kunze (which is not quite as approachable, but still an excellent book on linear algebra from an abstract point of view).

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    $\begingroup$ "you can skip straight into the post-rigorous stage"—typo "can" for "can't"? $\endgroup$ – Brian Tung Mar 28 '18 at 4:13
  • $\begingroup$ @BrianTung Yes, thank you. $\endgroup$ – Xander Henderson Mar 28 '18 at 14:54
  • $\begingroup$ @XanderHenderson Thank you for your response! Right now, I'm leaning towards either Spivak or Apostol. My main concern is that by reading them, I will be essentially re-learning things I already know. Thus, my main question is to what extent these texts incorporate rigorous proofs of the concepts of calculus (which I have not learned, except for what little I gleaned from Hubbard and Hubbard)? $\endgroup$ – Calico Mar 28 '18 at 15:04
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    $\begingroup$ If you haven't proved the results in calculus, then you don't really understand how calculus works, and cannot claim to "already know" it. If you are an engineer or biologist, it might not matter, but if you want to study higher mathematics (particularly analysis, which seems to be what you are interested in), then you need to learn the topic inside and out. That means going back and making the calculus you learned in high school or in a lower division class rigorous. This might seem redundant, but it really isn't. $\endgroup$ – Xander Henderson Mar 28 '18 at 15:08

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