Rudin's PMA Chapter 10

Question

I am having a hard time trying to learn Rudin's Principles of Mathematica Analysis chapter 10 on differential forms. Please suggest a book for reference on this chapter. I would most prefer a book where there is a lot of geometric intuition. Also I don't know Vector calculus. Let me know if I need to learn that before learning this chapter. In that case suggest a book for that as well. Please take note that this is for self study. Thanks.

Answer 1

The approach in Chapter 10 of Rudin is somewhat convoluted in large part because he seems to want to avoid discussion of manifolds (or submanifolds of $\mathbf{R}^n$). That is, he considers parametrized surfaces within $\mathbf{R}^n$ rather than considering the surfaces as subsets of $\mathbf{R}^n$.

Vector analysis is the special case of this theory where the dimension is 3. If you need this material for physics, it's probably best to study the special case in depth first, for example in Volume 2 of Apostol's Calculus. If your interest is pure math, that may not be necessary.

Good sources for differential forms include Volume 2 of Zorich's Mathematical Analysis, and any number of introductory books on differentiable manifolds.

I remember when I first learned about differential forms, I was able to follow all the proofs and do the calculations, but I had very little intuition for what I was doing. Reading the chapter on differential forms in Arnold's Mathematical Methods of Classical Mechanics helped a lot in this regard.

Answer 2

I highly recommend Hubbard’s ‘Vector Calculus, Linear Algebra and Differential forms’. The chapters that deal with differential forms are crystal clear and should open the door to understanding Rudin’s chapter 10 which would otherwise be impervious.