Joseph Mellor
2 min readMar 11, 2023


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.

- 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.

- 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.



Joseph Mellor

BS in Physics, Math, and CS with a minor in High-Performance Computing. You can find all my articles at