Jumpstart - Questions and Answers Week 5

Question 1 (Michael Schirle, incoming 2018)

I'm having a really hard time grasping the intuition behind differential forms. What do they look like visually? And also, I know the lectures say they're supposed to be coordinate independent, but we're defining them in terms of $\mathbb{R}^n$. How does this preclude us from being dependent on $\mathbb{R}$? Are there any other resources where I could get a more in depth treatment of differential forms?

Although this is my first time learning differential forms, too, I know of two resources. "Real Mathematical Analysis" by Pugh has a short section dedicated to Differential Forms. The section is understandable and most of notations should be familiar. If you're on campus, you should be able to download it from Springer for free, I think. Otherwise, there is a pdf copy available just by google searching. There is also this note. It mainly talks about 1-forms and 2-forms as they correspond to line integrals and surface integrals, respectively.

(Patrick Guidotti) These videos about differential forms could be useful. In particular, the first provides some motivation and the second some visualization.