2d curl intuition

Hello everyone! So I'm going to start talking about curl. Curl is one of those very cool vector calculus concepts, and you'll be pretty happy that you've learned it once you have it, for no other reason than because it's kind of artistically pleasing.

There are two different versions: there's a two-dimensional curl and a three-dimensional curl. Naturally enough, I'll start talking about the two-dimensional version and kind of build our way up to the 3D one. In this particular video, I just want to lay down the intuition for what's visually going on. Curl has to do with the fluid flow interpretation of vector fields.

Now, this is something that I've talked about in other videos, especially the ones on divergence, if you watch that. But just as a reminder, you kind of imagine that each point in space is a particle, like an air molecule or a water molecule. Since what a vector field does is associate each point in space with some kind of vector—and remember, we don't always draw every single vector; we just draw a small subsample—but in principle, every single point in space has a vector attached to it.

You can think of each particle, each one of these water molecules or air molecules, as moving over time in such a way that the velocity vector of its movement, at any given point in time, is the vector that it's attached to. So as it moves to a different location in space and that velocity vector changes, it might be turning or it might be accelerating, and that velocity might change. You end up getting kind of a trajectory for your point.

Since every single point is moving in this way, you can start thinking about a flow, kind of a global view of the vector field. For this particular example, this particular vector field that I have pictured, I'm going to go ahead and put a blue dot at various points in space, and each one of these you can think of as representing a water molecule or something. I'm just going to let it play.

At any given moment, if you look at the movement of one of these blue dots, it's moving along the vector that it's attached to at that point—or if that vector's not pictured, you know, the vector that would be attached to it at that point. As we get kind of a feel for what's going on in this entire flow, I want you to notice a couple particular regions.

First, let's take a look at this region over here on the right, kind of around here. Just kind of concentrate on what's going on there, and I'll go ahead and start playing the animation over here. What's most notable about this region is that there's counterclockwise rotation, and this corresponds to an idea that the vector field has a curl here.

I'll go very specifically into what curl means, but just right now I should have the idea that in a region where there's counterclockwise rotation, we want to say the curl is positive. Whereas if you look at a region that also has rotation but clockwise, going the other way, we think of that as being negative curl.

Here, I'll start it over here. In contrast, if you look at a place where there's no rotation, like at the center here, you have some points coming in from the top right and from the bottom left, and then going out from the other corners, but there's no net rotation. If you were to just put a twig somewhere in this water, it wouldn't really be rotating.

These are regions where you think of them as having zero curl. So with that as a general idea, you know, clockwise rotation regions correspond to positive curl, counterclockwise rotation regions correspond to negative curl, and then no rotation corresponds to zero curl.

In the next video, I'm going to start going through what this means in terms of the underlying function defining the vector field and how we can start looking at the partial differential information of that function to quantify this intuition of fluid rotation.

What's neat is that it's not just about fluid rotation, right? If you have vector fields in other contexts and you just imagine that they represent a fluid, even though they don't, this idea of rotation and curling actually has certain importance in ways that you totally wouldn't expect. The gradient turns out to relate to the curl, even though you wouldn't necessarily think the gradient has something to do with fluid rotation.

In electromagnetism, this idea of fluid rotation has a certain importance even though fluids aren't actually involved. So it's more general than just the representation that we have here, but it's a very strong visual to have in your mind as you study vector fields.