This is very interesting, I did not know that the theory of differential geometry can be discretised so nicely. I admire how they build from nothing but elementary linear algebra and calculus, and end up at quite advanced topics like De Rham cohomology, of which I never imagined it had any application in computer science at all. That said, definitions are mathematically very informal, and it skips handwavily over topics like topology. But these details are arguably less important for doing computations.