Okay, so what I'd like to do is just go through, I've got two of these things I'd like to do that are fairly simple, more or less like the book does where I just have some sort of function. And I've got this function here, I chose it for a reason, but you'll see what that is later on. This is a vector field with components in the x in the y direction and has nothing in the z direction. I want to find its divergence and its curl. Okay, so the divergence is a scalar, right, that it comes about from taking del dot v, right, so it's a scalar derivative. And it basically shows, you know, whether or not there are scalar sources at some point, that's one that's one way of thinking about it, shows how much the field separates. If you're going to be taking electricity and magnetism, that's actually exactly what it is. It is going to tell you about scalar sources. If you're taking something that has to do with, you know, plasma dynamics, it might have something to do with compressibility of the gas and things like that. So there are some other interpretations, depending on the actual physical situation. There are some other things that can happen with the divergence. So we've got this guy coming here, and now we just want to do this operator. So that product, remember, is just x component, x component, y component, y component, z component, z component. Right, so we take the extra derivative of the x component, minus 2 x y. And then we add in the y component, the y derivative of the y component. So ddy x squared minus y squared. And then I just, for completeness sake, could normally just ignore this. I can do ddz and multiply it by the z component, which is zero, which is always going to be zero. So I'm not really worried about it at the moment. Now I can go ahead and I can go through and hit this derivative. So I take the derivative of minus 2xy with respect to x. That just gets rid of the x. So we have minus 2y. And I do the same thing for ddy. And that's ddy x squared minus y squared. x squared ddy zero. y squared it's minus 2y again. So we've got minus 4y as our divergence. All right, so that's how much the field is spreading out. It spreads out, depending basically on how high it is. Right, how far up the it is in the y direction. B, we want to do the curl, right? The curl is the other of these vector products. The other derivatives on a vector function with respect to the vector components, which is again you take this Dell operator. And now you're going to find out how much it's swirling around or another way to think about it is where are the vector? Where are the vector sources? Right, vector sources have something to do with this cross product. x squared y squared zxyz, ddx, dy, dz cross this function that I wrote up there. It's going to be one of your favorites, right? You're going to write poems epic poetry to this wonderful function here. Now I can... what should I do with this? Why don't you tell me what I should do with this? Oh, well, I'm going to go ahead and I'll do with this cyclically. So what that means is I just take the, for the x component, I do things with the y and z components. So I start with the x component. And then I take the next one, which is y, and take the y for the first vector, and then z for the second vector. And that's the z component there, so the zero right there. So we have x, ddy of zero. And then we subtract the third from the second. So the z from the y. So we have ddz, or the product of the third in the second, ddz, x squared, y squared. So that has nothing to do with z. So this guy's going to end up being zero. And the y hat direction, I start with y. So I do z multiplied by x. So ddz minus 2xy. Right. Minus x multiplied by z. So that's d0dx. So again, we've got something that's going to be zero. So we're only going to have a z component. So getting this last one right is probably pretty important. So now for the z, the next one is, you come all the way back around in that one. So one, one times two, right. Well, that's going to be ddx of x squared minus y squared minus ddy, two times one of minus 2xy. All right. So that's all in the z direction. And let me see. How would I say something about this? Okay, ddx, that's 2x. Minus minus 2x. So that's plus 2x, which is equal to 4x in the z hat direction. That's it. All right. So that's pointing out. So whatever this is, it's circulating around in the xy plane, which makes a lot of sense because I've got these xy things here. And this is actually proportional to, I think sine 2 theta. And this is proportional to cosine 2 theta. So that makes some sense to me in general. And so this is what we get for these two operations. The dot product takes a vector, turns it into a scalar, a derivative of a dot product is a scalar. And for the vector v, that's for the cross product that turns it into another vector. There are some issues with that. I could talk about them, but I won't. But this only works really, really well in, this only works this way in three dimensions. So only think about the curl in three dimensions. If you want to go to multiple dimensions, talk to me. And we'll talk about all the issues that you'll have with it. All right. So anyways, this is where you get to start with all of these vector functions or vector derivatives. And I got to tell you, you're going to see this a lot in electricity and magnetism. So, you know, go ahead, practice, practice, practice, get this done, because this is something that you don't want to be behind on when you get into Ian. All right. Thank you very much for listening and I will see you soon. Bye.