May 22, 2020

The point of differential forms is that they give a way to express geometric theorems in a coordinate free way. Coordinates are seen as obscuring the pure geometric content of theorems. They are sometimes necessary artifacts of doing concrete calculations, but the idea is that geometry shouldn't depend on a choice of coordinates.

The important ideas can be found in pages 9-10 in this link:

https://www.math.ucla.edu/~tao/preprints/forms.pdf

Note halfway down page 9 we get a really clean equation for how to change variables (change coordinates) in an abstract way.

Also note the simple form that the general n-dimensional stokes theorem takes in terms of differential forms at the top of this page:

https://en.wikipedia.org/wiki/Stokes%27_theorem

That they allow the expression of substantial theorems in concise form is a clue that they are the "right" way to do differential geometry.