The math is fun, but what does the curl really mean? We had a feeling for divergence, that means that creation of field. What does the curl mean? Let's look at some examples. The curl, if I had to put it in words, I would say, it is how much the field makes something twist. You'll see what I mean as we draw some fields. Let's start with a uniform field. Let's say that V is 5 i-hat plus 0, j hat plus 0, k hat. That's just a uniform field pointing to the right with magnitude 5. We could draw a field like that these vectors say are five units long and they're all the same size. They're all pointing in the x hat direction. Let's do another layer. To decide visually, if something has curl, I like to just imagine drop a disk in it like this, put it in a disk, a little solid disk. Of course we're thinking in two-dimensions. It's really a three-dimensional field, but let's just do it and think about it in 2D and ask yourself, will the disk start to twist? If we think of the field vectors is frictional forces pushing on the sides of the disk will it feel a network or will it twist? That's what I mean. How much does the field makes something twist? Here you would say this doesn't look like it's going to twist. It has some vectors on this side making it want to go around that way and has some vectors on this side making it want to go around that way, and since it's symmetric, they're equal and opposite and they're just going to cancel out. Here I would say, it doesn't appear to have any curl and mathematically you're right, it's not going to have any curl. Because all those terms in the mathematical determinant all were partial derivatives. If you take partials of zeros or fives, so you're going to get zero. This, the curl del cross V is definitely zero in this case. Let's look at one, that's not uniform. Let's look at a growing field. Let's see. Let's have V equal, let's let it be some x on the i hat plus 0, on the j hat plus 0, k hat. Instead of five, it's going to increase. We're going to have short vectors here. All small vectors where x is small. Then they get bigger as you move to the right. They get even bigger as you move further. They're just increasing with x, and they're doing it linear. I don't know if I'm drawing them perfectly linear and my voice is getting louder when they get longer. I'm not sure why. That's just for dramatic effect, I guess. There it is, a growing field to the right, the field x, i hat plus 0 plus 0. Let's do our thing. Let's drop in a solid disk and ask ourselves is the disk going to feel a net twist? Again, this one is not, because it feels some vectors here pushing this way. If the vectors are growing, maybe they're pushing a little harder here than here to make it want to twist. But the exact same thing is happening over here. Again, when I look at that and say, is that giving me any twist? I would say no. It really isn't. If you were to evaluate the curl of this, mathematically you would also get zero. Because remember every one of those partial derivatives, it was taking the partial of one of the components, say the x component with respect to z, or the x component with respect to y, or the y component with respect to z, they were always crossed. If we just have something in the x-component that was going to have a partial. But it's with x that's not in the curl. If you go through and do this curl, you'll still get del cross v. It's still zero. In our little story here, it also appears to be zero. Let's do another one. It would seem to make sense to do one that does have a curl. Let's do a field. I don't know what to call it, but I've given the other two names. Let's do one. It's growing laterally. What I mean by that is that it's v equals y, i hat plus 0, j hat plus 0, k hat. This is probably the simplest field you can make that has some curl. Let's think, what would that look like? What we're saying is that the fields still points in the x-direction. Here's some little field vectors here. But as you go up in the y direction, they get bigger. Like that. There's another big one, but then they get smaller when you go down here. Another big one, they get smaller when you go down here. You could also draw this with field lines. You could have a low density of lines down here, as the density is getting higher and higher up there. I'm drawing it a little bit awkwardly here with just a little quiver plot. But now if I drop in my disk, you can see what's going to happen. Drop in my disk like this and ask ourselves, is that thing going to feel a twist? Now maybe you can see that it is. We have big vectors here pushing it around like that, we have little vectors here pushing back. The net effect is it does feel a net twist on this object. Sure enough, if you were to do the determinant to get the curl, you would actually have a value, because on the term when you take dvx/dy, here it is, it's one. That shows up on the i-hat, or its shows up second in a list, so it's minus 1 is the answer in the k-hat direction. Curls are vectors, it's a cross-product. In this case, Del cross V, the curl of V is minus 1 in the k-hat direction. That's telling you even something further is that the curl of the field is what makes it something want to twist. Then as we often have in physics, the vector that you use to describe that is perpendicular to the plane of the motion. In this case, it points in on the z-axis, it happens to be in. There's even a right-hand rule, either it makes it curve this way, take our right hand, follow it around, z has to be this direction in this case, because we have x cross y, z is out of the board, it's negative, it's into the board, everything makes sense. Curl is not, mean that the field is curvy. Here, I've given you all three fields and all three fields, the field was never curvy, it was always just straight lines, it was just simple linear functions or constants. But this one actually had a curl, even though the field always only points to the right, you can still get a curl. You can also get a curvy field that has no curl. Just when you see a field with the vectors making smooth curves, don't assume there's going to be a curl. Let me give you a fourth field, which I have no pre-planned a name for. I'm just going to call it the curvy field, but it's fairly simple. It's that V is xi-hat plus yj-hat plus 0k-hat. I certainly don't want to draw it in 3D. Let's think about that field for a second. If we were to draw an x-y thing going on like that. Well, it's always just x, y, so if you go along this line, it just grows. If you go negative x, negative y, it points down like that, and then if you go on these lines, it points in towards the origin like that, and it points in when you come here like that. That doesn't look curvy. But if you go and add other vectors around it, you actually get that it makes a very curvy looking thing. If you do a quiver plot of it, it does stuff like this, and then they get bigger like that. That one goes that way, the top goes like this. You get little vectors making paths like that, and the right would go up. Yeah, it's going to follow this pattern, so you get a little one like that, and then it gets bigger when you get further away, of course, like that, and then this one comes around like that. If you ever want to make this, I've seen some of you making very cool vector plots on MATLAB, it's a very curvy looking field. It's got curving around like this, curving around like that, like that, and like that. But mathematically is zero, and you can see why. We only have x on the vx function, we only have y on the vy function. All those partial derivatives are going to be zero, you're not going to get any partial derivatives. Even though this is curvy, it's hard to really see it exactly, but if you were to drop a little puck in here, you might say, well, the field looks curvy. Surely it's going to spin, but actually no, it doesn't. Curvy is not the same as a twist. If you look, this one's pushing up and would make it go this way, so it might go that way. But this vector is making it go this way, but maybe you can see these weaker vectors that might make it twist less, they follow the contour better. There's more of these small vectors making it want to push this way, there's less of these big vectors making it want to push that way, and in the end, they'll cancel out if you work it all out mathematically because this field has no curl, it's just curvy. Curvy and curl are not the same thing, for a long explanation of that fact.
