To me differential forms are an alternate notational tool used to concisely write out high dimensional integral expressions. I think they have important uses for differential geometers, but I know nothing about that business haha
Recently my adviser wanted to use the differential form notation to do some calculations and I found myself needing two important identities.
For a map T
- T* ( e_1 ^ e_2 ) = T e_1 ^ T e_2 (where T* is the map on differential forms induced by T)
- T( e_1 ^ e_2 ) = ( det T ) e_1 ^ e_2 (Change of basis)
These identities hold for higher dimensional forms other than 2-forms and I wanted a reference for these identities and I found the following notes online to be helpful:
First the 2016 Differential Topology Lecture notes by Lorenzo Sadun at UT. In particular, I liked the part 1 notes for the basics on differential forms and the section 3 on Pullbacks (and exercise 4) showed the two identities above. I would also suggest looking at the part 4 notes to see Stokes Theorem phrased using differential forms.
Additionally, I also liked these notes on differential forms from the University of Utah. I felt like these notes were written more for someone coming from an analysis background and page 3 clearly demonstrates the second identity which I stated above.
I hope you find these references as enlightening as I did!