Analysis on manifolds after course on Lebesgue integration

Question

I am an junior currently taking a course on measure theory and Lebesgue integration using Royden's text. Before this, I took a standard intro to analysis course covering the first seven chapters of baby Rudin. My institution doesn't offer a multivariable analysis course. I have room in my schedule next semester for two independent studies on topics of my choice. I want to go to graduate school to pursue a PhD in math and do not want to have any holes in my undergraduate preparation. I'm not yet sure what I direction I want to go in during grad school.

Would it be worthwhile for someone in my situation to do an independent study using a book such as Munkres' Analysis on Manifolds or similar such as Spivak's?

Answer 1

M.Spivak has written a few popular books in math. One of which is, Calculus on Manifolds. This small book will surely whet your appetite when you see it. Check it out.

Answer 2

Yes, if you intend to do graduate work, you should know the inverse and implicit function theorems and experience with manifolds and differential forms is all for the better. Munkres is a more readable rewrite of Spivak's Calculus on Manifolds.

Answer 3

Maybe, it would be better for you to choose a slightly more advanced topic (like introduction to topological manifolds using Lee's book, or Boothy's book) instead of multi-variable analysis for next semester. You should be able to cover multi-variable mathematical analysis yourself by using a suitable textbook (like Zorich's Mathematical Analysis II, which covered some more material).

I think the point for independent study is to have a faculty guide you through the learning process. So if the material is not too difficult you can cover it yourself instead of following the instructor. And you can always ask questions at here.