Books to read after AOPS Calculus?

Question

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?

Answer 1

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.