If anyone is seeking a more formal treatment of differential forms than the (admittedly informal!) description given in class, a good reference is
Abraham, Marsden, Ratiu, “Manifolds, Tensor Analysis, and Applications”
Note that an electronic version of this book is available for free for CMU students through the library webpage.
A big difference from the treatment we’ve seen in class is that this book first spends several chapters defining and studying manifolds before introducing differential forms. We instead started with differential forms in \(\mathbb{R}^n\), and will later talk about how to work with them on curves and surfaces. Interestingly enough, however, differential forms in \(\mathbb{R}^n\) is essentially all we need to define discrete differential forms, which in turn are sufficiet to work with “curved” polyhedral surfaces. (The joys of being piecewise Euclidean…)