How does one represent differential geometry on computers?

Question

So something that I've wondered is that how does one represent differential geometry concepts on a computer, since a lot of the concepts seem "continuous" and "derivative" (such as having $\frac{\partial}{\partial x^i}$s as bases, rather than unit vectors).

Answer 1

Keenan Crane at CMU has written an excellent introduction to the subject of discrete differential geometry, from an applications perspective.

The scope is far too vast to cover in an answer, but the high-order bit is that there are various ways to discretize different notions from differential geometry, and the name of the game is to choose the "right" set of discretizations, in order to make the whole theory coherent.