Water Analogies for Divergence, Curl and Gradient

Comment: A really nice water analogy for the field properties Divergence, Curl and Gradient from the Blog Starts With a Bang

....it's pretty mathematically intensive, but what's missing from most textbooks and E&M courses are physical explanations of what the mathematics means. For instance, I've started teaching about fields, and pretty much every textbook out there goes on and on about the properties of fields. They say you can do three things to fields, take the gradient, divergence, orcurl of them.

(Are you asleep yet? I'm sorry!)

What do these things mean? An easy way to picture it is in terms of water. If you placed a drop of water anywhere on, say,Earth, the magnitude and direction of how it rolls down is the gradient of the Earth's elevation.

If you let that drop of water flow, as it goes downhill, it can either spread out or converge to a narrower stream. When we quantify that, that's what the divergence of the field is.

And finally, when that water is flowing, sometimes it gets an internal rotational motion, like an eddy. A measure of that rotational motion is called thecurl of the field.

Well, one math geek statement is as follows: the curl of the gradient of a scalar field is always zero. What does this mean, in terms of our water? It means that I can take any topography I can find, invent, or even dream up.

I can drop a tiny droplet of water on it anywhere I like, and while the water may roll downhill (depending on the gradient), and while the water may spread out or narrow (depending on the divergence of the gradient), it will not start to rotate. For rotation to happen, you need something more than just a drop starting out on a hill, no matter how your hill is shaped! That's what it means when someone says, "The curl of the gradient is zero."