What is “advanced calculus”?

Calculus is taught in America (and maybe in other parts of the world) in several "passes," with increasing rigor and scope. The exact order of the approaches varies from place to place, but these different trips through the theory of calculus all have a few things in common.

1) Naive calculus: This is calculus which is highly computation and application based. Students see limits in terms of tables of values, and the idea of "go close but don't touch." You compute so many derivatives and integrals your eyes bleed. You see things like related rates, applications to physics, arc length, volumes of revolution, etc. These are the "easy" applications of calculus that you can do without much theory.

2) Multivariable calculus: This is calculus where all the stuff from the "naive" section is generalized to several variables. You do more complicated integrals, learn about partial derivatives and a chain rule for several variables. There is some amount of emphasis on geometry in the sense of vectors, surfaces, and maybe some introductory linear algebra, but the bulk of the theory is left out.

Both these classes are the kind of things new students take in their early years in mathematics. They are likely aimed at engineers or physicists rather than aspiring mathematicians.

What follows is material that is more likely to be "advanced calculus."

3) Elementary analysis - limits of sequences, series, topology of the reals, functions, continuity, etc. The bulk of the course is in these "foundational" topics in analysis. The idea is to figure out what makes the theorems tick, rather than proving the theorems, as by now, students have seen the intermediate value theorem before, but since they've probably never heard of a "supremum" they are not in a position to prove it.

4) Multivariable analysis - A rigorous generalization of the material from (3). Certainly includes a substantial amount of linear algebra and topology of metric spaces, and things like a rigorous treatment of the Jacobian and its significance, Taylor's theorem in several variables, the inverse and implicit function theorems, Stoke's theorem, etc. This is what we call "advanced calculus" in my school.

5) Measure theory(?) - This could be included in that scope. It typically refers to the study of Lebesgue measure and integral, as a generalization of Riemann integration that people are familiar with.

Depending on context, it's not unreasonable to think that "advanced calculus" refers to calculus on manifolds, which refers to taking the ideas of (4) and transferring them to more abstract settings where more powerful geometric tools can be developed.

In my studies, the path I took was (1) -> (3) -> (2) -> (5) -> (4), which is a little non-standard I think, but I do a lot of self-studying, so I picked things up kind of hodge-podge, and have been assembling them in my brain ever since. I think the order I listed them is probably a little more standard, although I think people probably study measure theory before worrying about some of the things I listed under (4). It all comes down to what the exact content of the courses are.

Here is my undergraduate bulletin. The courses correspond roughly to

1: MAT 131/132. 2: MAT 203 3: MAT 319/320 4: MAT 322 5: MAT 324

http://sb.cc.stonybrook.edu/bulletin/current/courses/mat/