2d curl example

So let's compute the two-dimensional curl of a vector field. The one I have in mind will have an x-component of, let's see, not nine, but y cubed minus nine times y. Then the y-component will be x cubed minus nine times x. You can kind of see I'm just a sucker for symmetry when I choose examples.

I showed in the last video how the two-dimensional curl, the 2D curl of a vector field, which is a function of x and y, is equal to the partial derivative of q, that second component, with respect to x minus the partial derivative of p, that first component, with respect to y. I went through the reasoning for why this is true, but just real quick, kind of in a nutshell here: this partial q partial x is because, as you move from left to right, vectors tend to go from having a small or even negative y component to a positive y component. That corresponds to counterclockwise rotation.

Similarly, this dpdy is because, if vectors, as you move up and down, as you kind of increase the y value, go from being positive to zero to negative, or if they're decreasing, that also corresponds to counterclockwise rotation. So, taking the negative of that will tell you whether or not changes in the y direction around your point correspond with counterclockwise rotation.

In this particular case, when we start evaluating that, we start by looking at the partial of q with respect to x. So we're looking at the second component and taking its partial derivative with respect to x, and in this case, nothing but x’s show up. So, it's just like taking its derivative, and you'll get 3x squared minus minus 9. There, 3x squared minus 9.

And that's the first part. Then we subtract off whatever the partial derivative of p with respect to y is. So we go up here, and it's entirely in terms of y. We can undo the symmetry; we're just taking the same calculation: three y squared, that derivative of y cubed minus nine. So this right here is our two-dimensional curl.

Let's go ahead and interpret what this means. In fact, this vector field that I showed you is exactly the one that I used when I was kind of animating the intuition behind curl to start off with, where I had these specific parts where there's positive curl here and here, but negative curl up in these clockwise rotating areas. So we can actually see why that's the case here and why I chose this specific function for something that'll have lots of good curl examples.

Because if we look over in that region where there should be positive curl, that's where x is equal to 3 and y is equal to 0. So I go over here and say, if x is equal to 3 and y is equal to 0, this whole formula becomes, let's see, 3 times 3 squared, so 3 times 3 squared minus 9 minus 9, and then minus the quantity. Now we're plugging in y here, so that's 3 times y squared is just 0 because y is equal to 0, minus 9, minus 9.

And so this part is 27. Now, that's 3 times 9 is 27, minus 9 gives us 18. Then we're subtracting off a negative 9, so that's actually plus 9. So this whole thing is 27. It's actually quite positive. So this is a positive number, and that's why when we go over here and we're looking at the fluid flow, you have a counterclockwise rotation in that region.

Whereas, let's say that we did all of this, but instead of x equals 3 and y equals 0, we looked at x is equal to 0 and y is equal to 3. Okay, so in that case, we would instead—so x equals 0, y equals 3—let's take a look at where that is. x is 0, and then y, the tick marks here are each one-half, so y equals 3 is right here. It's in that clockwise rotation area.

So if I kind of play this, we've got the clockwise rotation. We're expecting a negative value, and let's see if that's what we get. We go over here, and I'm going to evaluate this whole function again by plugging in 0 for x. So this is 3 times 0 times 0 minus 9, and then we're subtracting off 3 times y squared. So that's 3 times 3 squared, 3 squared minus 9.

And this whole part is 0 minus 9, so that becomes negative 9. Over here, we're subtracting off 9 minus 27 minus 9, which is 18. So we're subtracting off 18, so the whole thing equals negative 27. So maybe I should say equals—that equals negative 27.

So because this is negative, that's what corresponds to the clockwise rotation that we have going on in that region. If you went and you plugged in a bunch of different points, you can perhaps see how if you plug in 0 for x and 0 for y, those 9s cancel out, which is why over here there's no general rotation around the origin when x and y are both equal to 0.

You can understand that every single point and the general rotation around every single point just by taking this formula that we found for 2D curl and plugging in the corresponding values of x and y. So it's actually a very powerful tool because you would think that's a very complicated thing to figure out, right? That if I give you this pretty complicated fluid flow and say, "Hey, I want you to figure out a number that will tell me the general direction and strength of rotation around each point," that's a lot of information.

So it's nice just to have a small, compact formula.