We have our way to take derivatives now, Del d, dx I hat + d, dy j hat + dz k hat and we applied it to a scalar field now let's apply it to a vector field, which All right, just big V. And a vector field we've dealt with many fields, vector fields by now, electric fields and magnetic fields. If you're going to write one really general, you write it like this x I hat + Vy J hat + Vz k hat. And these are not constant Vx Vy Vz they aren't constants they're not even just functions of X, Y and Z. There any function so Vx Vy Vz are some function of x, y, z. And see there could be constant there could be anything the point is these are really functions here even though I'm calling them Vx Vy Vz. So we want to have our operator, operate on a vector field so how is it going to do that? We said they're not really multiplying their really operating on, but they operate on things in a way that looks like multiplication. So if this thing kind of looks like a vector and this looks like a vector, how are you going to combine them? Well, one way is the dot product so what we can write is no dotted with V Okay? It looks kind of the same V was an unfortunate choice so that's a Dell and this is a V the vector sign on top of it okay, if you do that, d, dx operates on Vx. So that's d, dx and d, dy operates on d,Vy and d, Vy d, Vz and d, dx operates on Vz so that's your dot product. And again, keep in mind these are functions Vx V,y, Vz can be functions of X Y or Z? So if we do this, it's called, it gives you a scalar field first of dot products always make scalars and these make scalar fields. So this is some function X, Y, Z, function X, Y, Z, function X, Y Z add them together it's a great big function of X, Y and Z. It's a scalar field and it's called the divergence is the name of this derivative take vector field, take it stop product or take Dell dot with that medical field. And you get the divergence, let's Look at one and see why it's called divergence. Let's actually just make a field V let's call it X, I hat +y j hat, okay no, I said you can have a function of X, Y and Z here. And you can be very complicated, but I'm actually going to draw this so I would rather not make it too complicated. Let's see if I were to draw X and Y axis like this, then at X zero, Y equals zero you'll have nothing, zero, zero, but if you start to move away so you move away in X. You move to X equals one, Y equals zero here well you have a vector with a value of one pointing in the I hat direction. Kind of like that and if you moved up one and y, it would just point up and y If you move down one and y would be negative, moved down 1 -1 and X. And kind of look like that and he went to one, you would be what you'd be kind of like that and one, one here we like that and like that and like that kind of looks like the field around a point charge but it is not okay so field around a point charge is spherical symmetry. This is actually sort of a Cartesian thing we're making if you move out to two here now, it's twice as long and this one would be twice as long and this one would be much longer. So this field kind of looks like the vectors are just exploding out of the origin if how would you describe it, kind of like that? And if you go further away, they get even bigger like this and if we kept doing these really, we need to fill in all the Cartesian points if we wanted to completely drawn. Okay so you get the idea the field is blowing out of the board okay, so let's think what is the divergence of this field? Well, Dell, dot V is d, dx of X is one all right? All right d dy of Y is one I just took the derivative of Y and I got one and we're not doing Z. So the divergence of this field is two, It has a divergence of two. So the way to think of it is diverging is just to drop a little cube on it, say like this Llttle three dimensional cube and just think about how much field is going in and how much field is coming out. So right here you have small vectors going in, you have big vectors coming out, you get a positive number. If you do the number coming out minus the number coming in, that's because the field has divergence there. Okay, if you put the box way out here now anywhere you go, smaller fields are going in that are coming out. That's because the field has a constant divergence it's divergence everywhere is two if you go on an axis right here, a little bit of field goes in, a larger field comes out the field is diverging. If you go right on the origin, you can go on the origin all it is is field coming out right? Because all the vectors are pointing out so divergence, the property divergence is like creation of field. We could contrast it with a uniform field so if I were to over here, just let me draw a uniform field equals one. I'll be more interesting two, I hat okay, plus zero j plus zero k, I X, so I may just vectors like this. This is the generic uniform field that we've drawn many times, electricity and magnetism. It's divergence is what Dell dot v is zero this thing has no divergence because a derivative of that with respect to X, it's just a constant to zero,00. And if you draw these cubes, you see uniform field, the amount of field going in, eagles amount of field coming out. So, also there's no field created inside the cube, so that also matches the idea that the divergence is zero.
