What is advanced calculus?

Answer 1

Advanced Calculus means different things in different context. But in general there are only two different meanings.

i) Multivariable and Vector Calculus, where instead of looking at just the boring and trivial real line, we move up to look into functions over real vector spaces Rn, where functions are not as nice, but much more practical in Physics, Engineering etc. (Space is R3, Space-Times os R4 with a weirdly defined dot product)

ii) "Real Math" Calculus, or more accurately, rigorous Calculus or Modern Calculus where we don't just take what the professor tells us to be the absolute truth, but rather using logic, axioms and tools such as mathematical induction, symbolic logic, and other type of mathematical proofs to come to our own conclusion of what is true (and more importantly why) and what is false. It is "advanced" as there is no direct reason of doing so in Engineering, Physic, or any field of Science. It is a way for people going into research in Mathematics to be prepared as it is how modern Math is really done.

I am in Advanced Calculus myself where it is the second meaning. Clearly, the second case involves discussions and theorem proving for the first case.