shinobi20 said:
I have already taken Calculus in a "Stewart" style, that is why I am re-learning calculus from a rigorous way, but my question is HOW rigorous should I be knowing I want to pursue theoretical physics (probably in the field of quantum field theory). I know that Apostol is for the pure mathe majors but I'm confused why a lot of posters here recommend apostol for physics majors.
I would argue, it's not how rigorous but rather how thoroughly or how quickly you should learn it. Being able to answer any question you could see on an exam is one measure of knowledge in a subject, but that is not enough. What you need is to be able to answer any question at some time in the future, and being able to answer those questions now is not a guarantee that you'll be able to answer them in the future.
That is the difference between the "Stewart" style recipe-book learning that one is likely to forget, and a more rigorous understanding that is likely to stay there forever. So your goal should be to gain a knowledge of calculus that is permanent because it all fits together and makes sense, and because it is constructed from simple parts and is justifiable, in the sense that one can justify why something is like it is. Why does integration by parts work, etc.
So that is my answer to how rigorous it should be: rigorous enough that you won't forget it. That said, rigorous books can be difficult to learn from so part of the challenge is to learn how best to learn from such books.
And I mentioned that how quickly you should learn it is an important question. I'm sure theoretical physics has a ton of stuff you would have to learn and in a hurry. So speed is paramount. My question is this, how quickly can you reach a level of knowledge that is permanent and makes calculus a tool that you can wield when needed?
To be more specific, I would aim to be as quick as possible, take as many shortcuts as possible, and get yourself to the level needed, being able to answer any calculus question that you would find in a typical book like Stewart and understanding what calculus is about. And if you have questions like what ##dy \over dx## really means, find the answers.
And once you get there, move on to the multivariable stuff and race through it as well. Try to keep in mind that you want to learn physics, not calculus. Calculus is going to be a tool and using it to answer questions is the point.