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Comment by senderista | Hacker News Reader

Every vector calculus instructor should teach their students the intuitive (by which I mean visual/physical) meaning of grad, div, and curl (and the intuition behind results like Stokes's and Gauss's theorems). Even engineering students uninterested in proofs should be able to grok the intuition.

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Have you ever seen a book that doesn't? It's in every book I've seen.

The textbook my Emag professor wrote himself made sure to avoid anything intuitive or visual, and was just a dense tome of text and equations with nothing else. He had a lot of trouble getting it published, but made sure to teach from it for decades. If you asked nicely, he'd give you a copy of the errata that he never fixed in the book. That class was essentially "vector calc for EEs" so it was my introduction to all these concepts, and I never intuitively understood them until much later.

University education in a nutshell.

We need to burn it down, better sooner than later.

> Have you ever seen a book that doesn't?

The brown Rudin.

Do you have other good resources for this? (I'm sure I understood this better back doing my degree but it was 40 years ago)

Just search for videos, stuff like this:

https://www.youtube.com/watch?v=eEwZeY51mT0

You can make similar kinds of videos for all 3 of them. That video shows a divergence free field since number of particles aren't changing, I easily see that since I know the intuitive explanation for divergence, it is useful to have intuition for those things.

Gradient is just the equivalent of slope but for higher than 1 dimension.

Edit: Or no, that field has divergence, I'm dumb I didn't watch the start, many particles accumulate at a few points, that is due to divergence. Divergence is essentially areas that attracts or repels particles in that simulation.

Found the divergence video, in case it is hard to understand what I said above: https://www.youtube.com/watch?v=c0MR-vWiUPU

Note: This series (Multivariate calc on Kahn Academy) is done by Grant Sanderson of 3Bl1Brown. It's outstanding, and goes over the concepts related to how the article here describes the fields.

Wikipedia explains the basics

https://en.m.wikipedia.org/wiki/Del and the related articles on gradient (slope), divergence (flow across a boundary), and curl (circulation)

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