To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts. Few realize that the world of modern mathematics is rich with vivid images and provocative ideas. -Ivars Peterson
Those of you who know me well know that I have little love for math for its own sake, but that I prize it as an incredible tool to help us understand and quantify physical systems. This has incredible applications, from figuring out what trajectory will succeed in a game of basketball,
to being able to predict what happens to matter — both dark matter and normal matter — when galaxy clusters collide.
Well, I’m teaching my very first advanced undergraduate course this semester, on electricity and magnetism.
To nobody’s surprise, 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, or curl 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 the curl 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.”
So, you know, the next time you’re at a bar, and some friendly math geek comes up to you (they’re easy to spot by their T-shirts),
you’ll actually be able to talk vector calculus with them. Err… maybe that isn’t the best idea after all. But you will know what divergence, curl, and gradient mean, and how many people will know that at bar trivia?