If you don't mind me asking, what are you studying? I'm always down to learn about new topics, especially if they might lead to an interesting article.
I'm definitely doing a Differential Geometry approach with a hint of Topology. This article is part of a series talking about Physics, and I've found that Differential Geometry is the natural language of Physics. As for the resources, I've either looked through or know of
- Michael Penn's unfinished Differential Forms playlist, which has a ton of worked examples (in Cartesian/Minkowski coordinates only).
- I found this PDF while I was finishing up the Generalized Stokes' Theorem article (the upcoming sequel to this article unless I decide to make an article for k-chains) that goes into some more detail in parts.
https://www.diva-portal.org/smash/get/diva2:1680707/FULLTEXT01.pdf
- V.I. Arnold's Mathematical Methods of Classical Mechanics includes a chapter on Differential Forms similar to the recent articles.
- Michael Spivak's Calculus on Manifolds is a standard in the field that also covers k-chains and there's a good chance that's what Michael Penn learned from. I haven't read it directly, but a lot of answers in various forums mention it by name in answering questions I have.
- Keenan Crane's Discrete Differential Geometry lecture series and his course notes talk about a lot of these topics, but go in a different direction to generalize these ideas to discrete meshes like the ones used in 3D rendering.
- These online notes from Vincent Bouchard.
https://sites.ualberta.ca/~vbouchar/MATH215/preface.html
- The textbook Analysis on Manifolds by J.R. Munkres to be helpful, but it is definitely written for mathematicians somewhat familiar with Topology.
- Mark E. Fels's book An Introduction to Differential Geometry through Computation. I can't remember what this one did for me.
- Tristan Needham's Visual Differential Geometry and Forms has a lot of diagrams and drawings. I felt a little disappointed by the proof of the Generalized Stokes' Theorem in the book.
- C.H. Edwards, Jr.'s Advanced Calculus of Several Variables is the only resource that I could find with a proof of the Generalized Stokes' Theorem using k-chains. Every other proof works with the upper half-plane for some reason.
- There's a few YouTubers like Xy|yXy|yX and eigenchris whose series I found useful, but didn't really focus on Differential Forms.
With that being said, I've had to fill in a lot of gaps. It's quite rare to find a resource that works through any problems in coordinate systems other than Cartesian/Minkowski coordinates. Maybe 1 out of every 10 resources mentions the Hodge dual, and those that do leave out a lot. I've only found people implying ways you could visualize the wedge product in forums across the internet. Many resources just give a definition of things like the exterior derivative and the Hodge dual without motivation.
tl;dr: I took everything I could find that made sense to me and derived everything else to fill the gaps.
