Differential Forms and Integration (2008) [pdf]
I took a Differential Geometry course in university. The course covered basic theory on smooth manifolds and Riemannian Geometry. While I did appreciate the beauty of the subject from a theoretical perspective, I have to admit that I somewhat lacked intuition for the material, especially with regards to differential forms. I understood that their properties make sense to define the notion of integration on a smooth manifold, but never really saw whether they had any purpose besides using them for integration. Even in their use in integration they were kind of a mystery to me, if I recall correctly we just defined integration on subset of R^n, where differential forms played the same role as d_{x_1}d_{x_2}...d_{x_n}, which is known from the riemann integral, but then to define integration on manifolds, we just used the pullback of the charts, which didn&#x27;t alleviate any of the mystery and didn&#x27;t add anything about why differential forms themselves are supposed to be important. I wish I had taken some physics courses, where vector calculus is used heavily and that might have helped some, but in the end I was somewhat disappointed, because even though taking the course certainly did make me a better mathematician, I still didn&#x27;t have the feeling of really completely grasping the concepts, even though I could prove statements.