2d curl nuance

In the last couple of videos, I've been talking about curl, where if we have a two-dimensional vector field v defined with component functions p and q. I've said that the 2D curl of that function v gives you a new function that also takes in x and y as inputs. Its formula is the partial derivative of q with respect to x minus the partial derivative of p with respect to y.

My hope is that this is more than just a formula, and you can understand how this represents fluid rotation in two dimensions. But what I want to do here is show how the original intuition I gave for this formula might be a little oversimplified. For example, if we look at this, the partial q partial x component, I said that you can imagine that q at some point starting off a little bit negative, so the y component of the output is a little negative.

Then as you move positively in the x direction, it goes to being zero, and then it goes to being a little bit positive. With this particular picture, it's hopefully a little bit clear why this can correspond to counterclockwise rotation in the fluid. But this is only a very specific circumstance for what partial q partial x being positive could look like. You know, it might also look like q starting off a little bit positive, and then as you move in the x direction, it becomes even more positive and then even more positive.

According to the formula, this should contribute as much to positive curl as this very clear-cut kind of whirlpool example. To illustrate what this might look like, if we take a look at this vector field here, if we look in the center, this is kind of the clear-cut whirlpool counter-clockwise rotation example. If we play the fluid flow, the fluid does indeed rotate counter-clockwise in the region.

But contrast that with what goes on over here on the right; this doesn't look like rotation in that sense at all. Instead, the fluid particles are just kind of rushing up through it. But in fact, the curl in this region is going to be just as strong as it is over here. I'll show that with the formula and kind of computing it through in just a moment.

But the image that you might have in your mind is to imagine a paddle wheel of sorts, where let's say it's got arms kind of like that, and then you hold down with your thumb that middle portion. Even though the paddle wheel, left to its own devices, would just kind of fly up, I want to say, let's say you're holding that down with your thumb, but it's free to rotate. Then the vectors on its left are pointing up, but less strongly than the vectors on its right, which are even greater.

So if you imagine that setup and you kind of have your paddle wheel there, then when you play the fluid rotation, holding your thumb down but letting the paddle wheel rotate freely, it's also going to rotate just as it would over here in the easier-to-see whirlpool example. In terms of the formula, this is because a situation like this one here, where q goes from being negative to zero to positive, should be treated just the same as a situation like this as far as 2D curl is concerned.

This term in the 2D curl formula is going to come out the same for either one of these. It's worth pointing out, by the way, that curl isn't something that mathematicians and physicists came across trying to understand fluid flow. Instead, they found this term as being significant in various other formulas and circumstances. I think electromagnetism might be where it originally came about, but then in trying to understand this formula, they realize that you can give a fluid flow interpretation that gives a very deep understanding of what's going on beyond just the symbols themselves.

So let me go ahead and walk through this example. In terms of the formula representing the vector field, it's a relatively straightforward formula, actually. So p and q: that x component is going to be negative y, and the y component q is equal to x. When we apply our 2D curl formula and play the partial of q with respect to x, the partial of this second component with respect to x is just one.

Then we subtract off the partial of p with respect to y, which up here is negative 1 because p is just equal to negative y. So the 2D curl is equal to 2. In particular, it's a constant 2 that doesn't depend on x and y, which is pretty unusual. Most times that you apply 2D curl to a vector field, you're going to get some kind of function of x and y.

The fact that this is constant tells us that when we look over at this fluid flow, the sense in which curl—the formula for curl—wants to say that rotation happens around the center is just as strong as it's supposed to happen over here on the right or anywhere on the plane for that matter. So, for playing this and if you imagine, you know, the paddle wheel in the center, evidently, it would be rotating just as quickly as the paddle on the right.

Even though it might, I don't know, to me that feels a little unintuitive because the one on the right, I'm thinking okay, you know, there’s maybe there's a little bit more torque on the right side than there is on the left, and that's kind of a counterbalancing act. But the idea that that's actually the same as the very clear-cut, I see the counter-clockwise rotation with my eyes example in the center does seem a little unusual.

But I think it's important to understand what else two-dimensional curl can look like and what else this formula might be representing. So with that, I'll see you next video.