To get started, we need a derivative. Maybe we'll be able to take a derivative of a field. So let's consider a scalar field. A scalar field, and I'll call it S. Of X and Y. Lower case s because I don't want to get to cookie. Okay, so let's think then, if we had sort of an xy and a plane of the board, and we're going to plot the scalar field s is sort of a height. We could have kind of a lump kind of like that. And I'll kind of put a line like this. You've got to kind of feel that's sort of a lump spreading out in space, in x and y. And we're plotting s okay, there's your scalar field plotted as a height. So the question is, what is the derivative? And well derivatives we know, at least at this point we know the derivative is the slope. If we just had a line, a single 1D plot, the derivative at eight point is just a slope. So you can kind of imagine this thing is flat here and it goes up. So you can say, well the derivative here, the slope here must be small, and the slope here must be big. So the derivative, it depends on position, we have to do something. But that's not too surprising, because we have a function. We have a scalar that depends on position, its value depends on position. And even in a 1D derivative, it depends on where you are. That's not so bad. But there's something else that it depends on. Let's look here. So this is what's coming down is the slope, it depends on direction, it depends on which way you go. If I go downhill here, then I have a big slope. Let's see, I have a negative slope. Right, if I'm coming down the hill. What if I go up the hill? I could go either way, right. I have a positive slope. What if I go around the hill? What if I go around like that, around this hump and at 0? So not only depend on position, it depends on direction. So you actually have the case that your derivative of this 2D scalar field is a vector field. Anywhere you go, it has a vector. But you have to think in terms of a vector, the value depends on the direction you're looking. So, we need a way to calculate this thing ,we want to know, I don't care about direction. What is the total differential? What is ds, how much does ds change? And we got to think about when I make a move, when I move this way or that way, or whatever way you want to go. How do we figure it out? What you do is you say, it's equal to the partial derivative, which you remember we talked about partial derivatives. You take the derivative of the function, just considering with respect to X. And you hold Y and Z if there's a constant. So it's like the partial (ds/dx) times the move you made in dx. Okay, so if I want to know what's the derivative this way, I move a little bit in x and move a little bit in y. And say that's the way I'm going. Well, to get it, It's the partial (ds/dx) times that differential dx, plus the partial (ds/dy) times the differential dy. So if you take a look at it, you could imagine what if you made this dx be a motion of just a unit of one in x. And this be emotional unit of y and y. You can think of that as sort of like the unit factors. So let's look at a way that we could write this. We could say is to find this. We need to use something that looks like this. d/dx i hat + d/dy j hat + d/dz k hat, if we had a three dimensional function. Right now, we just have to so you can ignore this. But I'm trying to be general for Cartesian coordinates. This thing is what we call Del. This is the differential operator for fields. Okay, it doesn't really multiply fields, it just operates on fields. It's like taking a derivative of a function. You don't multiply the function times the derivative, you just take a derivative. So when you apply Del to a field, you're not multiplying by just taking the Del of the field. So for example, if we have the scalar field s, then Del s is ds/dx. I hat + ds/dy j hat + ds. In this case we're doing all three dimensions dz k hat, like that. And this is called is the gradient of the field. because like we just showed, if you imagine you're going to some unit direction, these different directions multiplying by ds/dx gets you the changed a little bit of the change in one direction, a little bit of change and the other, a little bit of the change in the other. If instead of thinking of a differential in a specific direction, if you just do this to the field, what it gives you is a vector pointing along the. The maximum change. And its magnitude of the vector is equal to the maximum change. So if we came back down to our drawing like this, it's true that in all these different directions with different ds as you move in different directions. But if you take the gradient, you just get the maximum change straight up, and give you the direction and the magnitude of the gradient does. So the gradient is one of the ways take a derivative of field. It's the way applying it this way, taking the gradient, having applying it directly to a scalar field gives you basically the three dimensional version of the derivative you're used to. It gives you the three dimensional gradient of the field.
