I am currently taking an undergraduate course in diff geo (see below for a description). I don't know if I'll take the second course (see below for a description of this as well)--I might just take the grad course on Riemannian geometry, differential topology, or Lie groups. I like diff geo, but I like generality (to an obscene degree) and so I find learning things in such specificity to be annoying--although this can give you good intuition for the more general notions.What is your guys' experience in undergraduate differential geometry?
Not really, depending on how you do it. I know here (at UMD) we offer a two-sequence course on undergraduate differential geometry. The first covers what you have quoted (a lot more actually, but presumably this is just because its a course description) and the second is a course on the generalization of this to real manifolds of finite type, not thought of as immersed in --differential forms--Lie groups over such manifolds. So, I think if you want to you can do it as an undergraduate course. If you're interested I've attached an example of a midterm and final in this course from a past year.Is the generalization of Stoke's Theorem and the (introductory) study of differential forms and manifolds really a graduate level only topic?
Anything can be done independently (that's the only way I like it!). But I think just basic multivariate calculus and linear algebra might be a little sparse. If you instead meant mutlivariable analysis (and what you call "vector analysis") and more advanced linear algebra (things like tensor algebras) you should be fine.Can these topics be studied independently (at least at the basic undergraduate level) or is one area of study (beyond the basics of multivariate calculus/linear algebra) requisite to the other?