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I've been going through the AOPS Calculus textbook, and I genuinely really enjoy reading this textbook. I enjoy the format, where if I had to break it down:

1 - Introduce a problem, or a goal, basically give something that the section of the book is working towards (for example, when talking about limits towards/involving infinity, presenting the need to more generally be able to rigorously take the limits of quotients of functions)

2 - Begin working to the solution, developing "suboptimal" or incomplete methods (for example, taking the limits of quotients of rational functions going to infinity by dividing both functions by some power of x)

3 - Work towards a more general solution, with a good amount of rigour and providing good intuition but without getting too lost in the weeds (introducing the reader to use tangent line approximations to try to approximate the limit, then showing how the error of the approximation of f cancels with that of g, but leaving some of the more complex proofs involving the generalization of L'Hopital's rule to a special section)

4 - Have creative, hard, clever, and interesting exercises to drive the learned material home

I really enjoy this format, I feel as though I have a good amount of rigor here, a strong ability to prove the things that I use when solving, good intuition for what I'm doing, and I'm really having fun solving these creative difficult problems. I'm looking for books that achieve all/most of the above well, while not losing the intuition/beauty of the mathematics in the weeds of the specific proofs (I like rigor, but I think it would be best if someone of the more complex proofs, (to give an example from calc, the proof for IVT) were left to their own section/subsection or an exercise). I would like to do something that leverages calculus in some way, but something which does not is also fine. I've been looking into Hubbard & Hubbard Vector Calc and Linalg, as well as Strogatz Nonlinear Dynamics and Chaos. Thoughts or recommendations?

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I think at this stage, if you haven't studied any solid analytic geometry (at the level of the AOPS Precalculus book), you should become familiar with that, for example with Schuster's Elementary Vector Geometry. If you haven't acquired some familarity with complex numbers, you might want to do that too.

After that or alongside it, the next step for most people interested in math would be to take a stricter theoretical look at single-variable calculus. Burkill's concise A First Course in Mathematical Analysis is good for that. It's at a level comparable to Spivak's Calculus, but it doesn't belabor the mechanical aspects of calculus, which it assumes readers have already mastered.

After that, I think the next steps could be:

  • Algebra (linear and abstract), in a book like Algebra by Artin.

  • Analysis, in a book like Apostol's Mathematical Analysis.

If you have an interest in physics, learning multivariable calculus will be more of a priority initially than algebra or analysis (like what's in Apostol's book), so after Burkill, you could read something like C.H. Edwards' Advanced Calculus of Several Variables. It has enough linear algebra and topology for multivariable calculus. I would recommend you use a fully rigorous book like that one for multivariable calculus, which will require you to first study single-variable calculus more theoretically (as in Burkill). I think Hubbard & Hubbard is below this level.

If you enjoy number theory, you'll be able to study it in a more sophisticated way after learning some abstract algebra. (And Artin discusses some of this anyway.) But if you're eager to get started before then, Stark's Introduction to Number Theory has very few prerequsites.

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  • $\begingroup$ I'm realizing that what I wrote was quite a generic answer to "What next after AOPS Calculus?" Reading proofs of theorems contributes a lot to your ability to solve problems. So if your real interest is math as opposed to applications, it's not likely that a book postponing proofs of moderate difficulty, like the one for L'Hôpital's rule, is going to help you much. There can be exceptions where the flow of the book would really be broken, but they shouldn't be anywhere near the frequency they are in AOPS Calculus at higher levels. With that being said, I feel a book that comes close to $\endgroup$
    – Mike
    Oct 2, 2021 at 18:32
  • $\begingroup$ your requirements might be the second volume of Apostol's Calculus. It's mostly focused on problem-solving, not on theory. But the theory and the proofs are there. You need to be aware that some of the material on vector calculus will need to be relearned in a different way later, but that's not too bad. $\endgroup$
    – Mike
    Oct 2, 2021 at 18:35
  • $\begingroup$ Honestly, I'm not too interested in doing analysis, I never really enjoyed many of the more "obvious" delta-epsilon proofs, and I don't want to retread calculus ground with more proofs of that style. I'd like to move on to something else. Looking at linear algebra (don't mind something a bit harder than AOPS calculus), how is LADR (Axler)? Any other reccomendations? Thank you! $\endgroup$
    – Tuatarian
    Oct 6, 2021 at 4:38
  • $\begingroup$ @Tuatarian Sorry, I've heard of the linear algebra book by Axler, but I don't really know it, so I can't comment. Generally for people interested in math, I recommend studying abstract and linear algebra together in a book like Artin's Algebra, but this takes some preparation on vector geometry and complex numbers. I believe these subjects are addressed sufficiently in books earlier in the AoPS series. If it's for linear algebra alone, you could consider Lang's Introduction to Linear Algebra or Linear Algebra. $\endgroup$
    – Mike
    Oct 17, 2021 at 11:49

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