I have that book and read the first four chapters. It’s good stuff, but the Lisp took a while to parse through, for me at least.
I took a physics class that used Jose and Saletans book, and that helped my understanding a lot. After carefully reading the first few chapters of that I’m kind-of able to understand Betancourt and Girolami’s papers. I also like Elementary Differential Geometry, but they don’t talk about differential forms.