# How do I join the dots between the formal definition of exterior derivative and the intuition

I am self-studying differential forms, and I'd like to understand the proof of Stoke's theorem.

I have read through the document here by Dan Piponi and really liked his explanation of the exterior derivative as finding the boundary in the picture (section 4, page 6), but I have a hard time to connect this with the formal definition of the exterior derivative.

Reading through the comments of his answer here, I am not sure if the concept of 'finding the boundary in the picture' applies to just xdy or any differential forms?

• two earlier MO questions are similar: one and two --- is there something left to answer here? Jun 11 '20 at 11:17
• well, I think joining the dots is essentially proving, at least some local version of Stoke's theorem. I understand that there is a gap between the definition, via essentially a formula, of the exterior differential, and the intuition, which is essentially the local version of Stoke's theorem. Historically, it came from generalizing several lower dimensional special cases, and is thus non-trivial. Many books discuss it. I would guess you could find it in some volume of Spivak, one of Lee's books, possibly in Boothby's book and many other places. Jun 11 '20 at 13:10
• Perhaps my recent answer in Carlo's reference link "one" may join your dots. Aug 9 '20 at 15:26