system
Given the following post, respond as if you're commenting on Reddit:
human
But what ARE divergence and curl?

When I took Calc III, we mostly talked about divergence and curl in the context of how they can help you solve surface integrals and line integrals with the divergence theorem and with Stokes' theorem. That was the extent of it. We also discussed how curl explains how 'spiraly' a vector field is at a certain point, but nothing behind the intuition. From further investigation I've learned that divergence is a measure of 'source vs. sink' and that both are kind of derivatives.

My question is: Sure, as mathematicians we can define whatever we want, but why? What are divergence and curl really? Are they only useful in the context described above or are there more? What was the motivation for defining them? How can I use them to solve problems? Why are they an interesting topic of study? Are there extensions to higher/lower dimensions? Etc.
gpt
Drop some puffed rice onto a point in a gently flowing stream.  If the rice flows away from that point or flows toward it, the water's flow (which can be described by a vector at every point on the surface) has a positive or negative divergence there.  It the rice swirls around counter-clockwise around that point as you watch it, then the water has a curl vector that points up away from that point.  If the rice swirls around clockwise, then the water has a curl vector that points down into the water at that point.  This is just an intuitive picture, of course.  In physics, the geometry of the electric and magnetic fields are specified entirely by the divergence and curl of their field vectors, and these are related to the electric charges (both stationary and moving) and time-varying fields that produce the fields in the first place.  The divergence and curl were developed to describe electric and magnetic fields by Oliver Heaviside in the 1890s.