Discrete Calculus

[TimothyOBrien]

This is fascinating, I had no idea these concepts and theorems could be explained so simply with pretty much just arithmetic. Playing with this a bit... if I start by specifying a scalar field and then calculate gradients, the curls will always be zero (including curls based on the gradient of the 0-form Laplacian). It's easy to prove, but I'm not sure what it means. The diagrams in the examples are illustrative but I'm curious about the process for constructing the diagrams (and generally how to apply these diagrams to solve problems). Also, as you hinted at, I'm curious about the dynamics when the diagrams are embedded on different manifolds (connections to triangulations of manifolds in combinatorial topology) and how it related to physics. Is there a natural way to make these diagrams 3D? (maybe space-filling cubes and measure curl on faces, but not sure how to quantify the volumes). Lastly, and perhaps related the previous question, does this extend easily to complex numbers? That seems interesting because then each node is sort of 2D. Sorry that's a lot, but any insight you have or terms you can provide for me to Google would be great. Very interesting video, nice and concise, and your use of color in the diagrams is really helpful.