3d curl intuition, part 1

Hello everyone. So, I'm going to start talking about three-dimensional curl, and to do that, I'm going to start off by taking the two-dimensional example that I very first used when I was introducing the intuition. You know, I talked about fluid flow, and I animated it here where with this particular vector field, you see a certain counterclockwise rotation on the right and a clockwise rotation on the top.

So, I'm going to take that vector field, which hopefully we have a little bit of an intuition for, and I'm going to plop it into three-dimensional space on the x-y plane. If I just take that whole vector field and I copy it onto the xy plane, here's how it looks. The vector spacing might be a little bit different; the choice of what sub-sample of points to use in displaying vectors might be a little bit different. But this is the same field, and it's actually worth writing out how it's defined in two dimensions.

This guy is a function of x and y, so it's a vector field function of x and y. What its components are? The first component is y cubed minus 9y, and then the next one is x cubed minus 9x. So, now if I look at this guy and I say, let's start thinking about the fluid rotation associated with it, because it's in three dimensions, it's natural to describe that rotation not just with a number at each point, you know, with a scalar value like the two-dimensional curl gives us, but instead to assign a vector to each one.

When you do that, when you associate a vector to each different point in space according to the fluid rotation that would be happening there, you get something that looks like this. Now, this is kind of complicated because there's two different vector fields going on. One of them, all the vectors are perpendicular to the xy plane, so let's just kind of take it piece by piece and see if we can understand it.

I've got four different circled regions here, and one of them is this one on the right where there's counterclockwise rotation happening. If you think to your right-hand rule—and I'll go ahead and bring in the picture of the right-hand rule here—where you imagine curling your fingers around that direction of rotation. When the fingers of your right hand curl, and you stick out your thumb, the direction your thumb is pointing will be the direction of vectors that describe that rotation.

So, if we do that here, and if we, you know, imagine curling your right fingers around there and sticking up your thumb, you're going to get vectors that point in the positive z direction. This is why in that region you have vectors pointing up positively in the z direction; they're telling you that as you view this xy plane from above, there's counterclockwise rotation.

But then what about at a different point? What about up here at the top, where you have clockwise rotation? Well, there, if you imagine taking the fingers of your right hand and curling them around that direction of rotation, your thumb is going to be pointing straight down; it'll be kind of in the negative z direction. We see that with this vector field here, where below that circle, below that point, you have vectors pointing straight down, indicating that that's the direction of rotation in that region.

So, if you do this at every single point, and you kind of get an understanding of what the rotation is at every point and assign a vector, this is the field that you're going to get. Let's go ahead and describe that with an actual function because we know how to compute the 2D curl at this point.

You say, if this whole thing, if we give names to the two different component functions as p and q, then the curl, the 2D curl of this guy, the 2D curl of the vector field v as a function of x and y, what it equals is the partial derivative of that second component with respect to x. So, the partial of q with respect to x minus the partial derivative of that first component with respect to y.

So, minus partial of p with respect to y. What we get when we do that, the partial of q with respect to x—so we take the partial derivative of this with respect to x—that just looks like a derivative since there's only x's in there, and you get 3x squared minus 9.

I did actually do this; there's another video where this is the example that I do. When you take this second derivative of p with respect to y, you're taking the derivative of this top part with respect to y, and that's three y squared minus 9. You can say this minus 9 cancels out with that, you know, minus minus 9. So, these guys cancel, and what you ultimately get is 3x squared plus 3y squared.

Now, what does this mean for the vector field that we see here? Because this scalar valued quantity, and yet the vector field that I'm showing with all these blue vectors indicating rotation, these are vectors. Because the rotation is happening purely in the x-y plane, which is perpendicular to the z axis, all of these vectors purely have a z component.

So, what you might say is that the curl—and I'm not going to say 2D curl, but actual curl of v of v as a function of x and y—is not a scalar value but a vector. It's going to be a vector that describes these blue ones in the pure z direction, and since they're in the pure z direction, the x and y components are 0. But that last component is the formula that we found that describes the magnitude of the curl: 3x squared minus 3y squared.

And this, let's see, kind of running out of room, this here, you could think of as kind of a prototype to three-dimensional curl. Because really this vector field v is not quite a three-dimensional vector field, is it? It only lives in the x-y plane; it only takes in x and y as input points.

So, what we want to do is start extending this to say, how can you make this look like a three-dimensional vector field and still kind of understand the rotation as a three-dimensional vector quantity? And that's what I'm going to continue doing in the next video.