Divergence example

So I've got a vector field here V of XY where the first component of the output is just ( x \cdot y ) and the second component is ( y^2 - x^2 ). The picture of this vector field is here; this is what that vector field looks like. What I'd like to do is compute and interpret the divergence of V.

The divergence of V as a function of X and Y. In the last couple videos, I explained that the formula for this, and hopefully, it's more than just a formula, but something you have an intuition for, is the partial derivative of P with respect to X, and by P, I mean that first component. So if you're thinking about this as being ( P(x, y) ) and ( Q(x, y) ), it could use any letters, right? P and Q are common, but the upshot is it's the partial derivative of the first component with respect to the first variable plus the partial derivative of that second component with respect to that second variable Y.

As we actually plug this in and start computing, the partial derivative of P with respect to X of this guy with respect to ( x ) looks like a variable, and ( y ) looks like a constant. The derivative is ( y ), that constant. Then, the partial derivative of Q, that second component, with respect to Y; we look here, ( y^2 ) looks like a variable, and its derivative is ( 2 \cdot Y ). ( 2 \cdot Y ) and then ( X ) just looks like a constant, so nothing happens there.

So on the whole, the divergence evidently just depends on the Y value; it's ( 3 \cdot y ). So what that should mean is if we look, for example, let's say we look at all points where ( y = 0 ), we'd expect the divergence to be zero. That fluid neither goes towards nor away from each point.

So ( y = 0 ) corresponds with this x-axis of points. At a given point here, evidently, it's the case that the fluid kind of flowing in from above is balanced out by how much fluid flows away from it here. And wherever you look, I mean here, it's kind of only flowing in by a little, and I guess it's flowing out just by that same amount, and that all cancels out.

Whereas, you know, let's say we take a look at ( y = 3 ). So in this case, the divergence should equal ( 3 ), so we'd expect there to be positive divergence when ( Y ) is positive.

If we go up and ( Y ) is equal to 1, 2, 3, and if we look at a point around here and we kind of consider the region around it, you can kind of see how the vectors leaving it seem to be bigger. So the fluid kind of flowing out of this region is pretty rapid, whereas the fluid flowing into it is less rapid.

So on the whole, in a region around this point, the fluid, I guess, is going away. You know, you look anywhere where ( Y ) is positive, and if you kind of look around here, the same is true where fluid does flow into it, it seems, but the vectors kind of going out of this region are larger. So you'd expect on the whole for things to diverge away from that point.

In contrast, you know, if you look at something where ( Y ) is negative, let's say ( Y = -4 ). It doesn't have to be three there, so that would be a divergence of -12. So you'd expect things to definitely be converging towards your specific points.

So you go down to, you know, I guess I said ( Y = -4 ), but really I'm thinking of anything where ( Y ) is negative. Let's say we take a look at like this point here. Fluid flowing into it seems to be according to large vectors, so it's flowing into it pretty quickly here. But when it's flowing out of it, it's less large; it's flowing out of it just kind of in a lack of physical way.

So it kind of makes sense just looking at the picture that the divergence tends to be negative when ( Y ) is negative. What's surprising, what I wouldn't have been able to tell just looking at the picture, is that the divergence only depends on the Y value. That once you compute everything, it's only dependent on the Y value here.

As you go kind of left and right on the diagram, you know if I go up as we look left and right, the value of that divergence doesn't change. That's kind of surprising! It makes a little bit of sense; you know you don't see any notable reason that the divergence here should be any different than here. But I wouldn't have known that they're exactly the same.