The Visualization of Differential Forms (2021)

I’m working on a PhD adjacent to computational differential geometry, and while I’ve made a lot of progress on the computation, I still don’t have much intuition for k-vectors and k-forms. I love coming across articles like this that help me build intuition. The article for which this is the second part was really helpful, but I’m going to have to come back to this second part a few times to fully grasp it. I also loved this quote from one of the articles listed as a source (but with a broken link, I found it at <a href="http:&#x2F;&#x2F;yaroslavvb.com&#x2F;papers&#x2F;notes&#x2F;piponi-on.pdf" rel="nofollow">http:&#x2F;&#x2F;yaroslavvb.com&#x2F;papers&#x2F;notes&#x2F;piponi-on.pdf</a>): “Think of a vector as a pin, and a one-form as an onion. You evaluate a one-form on a vector by counting how many onion layers it goes through.”<p>Edit: This one also looks good: <a href="https:&#x2F;&#x2F;math.uchicago.edu&#x2F;~may&#x2F;REU2018&#x2F;REUPapers&#x2F;Bixler.pdf" rel="nofollow">https:&#x2F;&#x2F;math.uchicago.edu&#x2F;~may&#x2F;REU2018&#x2F;REUPapers&#x2F;Bixler.pdf</a>

In some domains differential forms are mind-mindbogglingly expressive. E.g. Maxwell&#x27;s equations boil down to [1]:<p>dF = 0, d*F = J<p>You literally could not use fewer letters. But the corresponding visualizations are not particularly easy or illuminating.<p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Mathematical_descriptions_of_the_electromagnetic_field" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Mathematical_descriptions_of_t...</a>

Web Archive link: <a href="https:&#x2F;&#x2F;web.archive.org&#x2F;web&#x2F;20240921132410&#x2F;https:&#x2F;&#x2F;web.tecnico.ulisboa.pt&#x2F;~ist189623&#x2F;wordpress&#x2F;2021&#x2F;09&#x2F;08&#x2F;on-the-visualization-of-differential-forms&#x2F;" rel="nofollow">https:&#x2F;&#x2F;web.archive.org&#x2F;web&#x2F;20240921132410&#x2F;https:&#x2F;&#x2F;web.tecni...</a>

I’d like to recommend Prof Chern’s course on Discrete Differential Geometry <a href="https:&#x2F;&#x2F;cseweb.ucsd.edu&#x2F;~alchern&#x2F;teaching&#x2F;cse270_wi24&#x2F;" rel="nofollow">https:&#x2F;&#x2F;cseweb.ucsd.edu&#x2F;~alchern&#x2F;teaching&#x2F;cse270_wi24&#x2F;</a>. The lecture slide on Exterior Calculus includes many good illustrations of differential forms, and there are more cool figures on related topics. There is also a pdf textbook neatly formatted if you want to dive deeper.

Backup <a href="https:&#x2F;&#x2F;web.archive.org&#x2F;web&#x2F;20240213003201&#x2F;https:&#x2F;&#x2F;web.tecnico.ulisboa.pt&#x2F;~ist189623&#x2F;wordpress&#x2F;2021&#x2F;09&#x2F;08&#x2F;on-the-visualization-of-differential-forms&#x2F;" rel="nofollow">https:&#x2F;&#x2F;web.archive.org&#x2F;web&#x2F;20240213003201&#x2F;https:&#x2F;&#x2F;web.tecni...</a>

I think this is a fancy way of saying, &quot;the visualization of a vector field&quot;. Differential forms are a generalization of a vector field, most of the images in the post that are visualizations of differential forms are images of vector fields. Using vector field instead of differential form, makes the title accessible to people who&#x27;ve taken multivariable calculus, which is a much larger group of people than people that know what a differential form is.<p>I&#x27;m sure there&#x27;s some interesting stuff in the difference between a differential form and a vector field that the author is trying to get at, it&#x27;s just interesting that all the images are of vector fields.