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Current time:0:00Total duration:4:11

- [Voiceover] So I've got a
vector field here, v of x y. Where the first component of
the output is just x times y, and the second component is
y squared, minus x squared. And the picture of this
vector field is here. This is what that vector field looks like. And what I'd like to do
is compute and interpret the divergence of v. So, the divergence of v,
as a function of x and y. And in the last couple of
videos I explained that the formula for this, and
hopefully it's more than just a formula, but something
I have an intuition for, is the partial derivative
of p with respect to x. By p, I mean that first component. So if you're thinking about
this as being p of x y and q of x y. So I could use any letters
right, and 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. And 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. X looks like a variable,
y looks like a constant. The derivative is y, that constant. And then the partial derivative
of q, that second component, with respect to y. We look here. Y squared looks like a
variable and it's derivative is two times y. And then x just looks like a constant so nothing happens there. So in the whole, the divergence
evidently just depends on the y value. It's three times y. So what that should mean is
if we look at, for example, let's say we look our points
for y equals zero, we'd expect the divergence to be zero. The fluid neither goes towards
nor away from each point. So y equals zero corresponds
with this x axis of points. So to give it a point here,
evidently it's the case that the fluid kind of flowing
in from above is bounced out by how much fluid flows away from it here and wherever you look. I mean, here its only
flowing in by a little bit, and I guess it's flowing
out just by that same amount and that all cancels out. Whereas, let's say we take
a look at y equals three. So in this case the
divergence should equal nine. So we'd expect there to be
positive divergence when y is positive. So if we go up, and y is
equal to one, two, three. And if we look at a point
around here, I'm gonna kinda consider the region around it. You can kinda see how the
vectors leaving it seem to be bigger. So the fluid 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. And 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, if you look at
something where y is negative, let's say it was y is equal
to negative four, it doesn't have to be three there. So there would be a
divergence of negative 12. So you'd expect things to
definitely converging towards your specific points. So you go down to, I guess I
said y equals negative four. But really, I'm thinking of
anything where y is negative. Let's say we take a
look at 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 in a
kind of a lackadaisical way. So, it kinda makes sense,
just looking at the picture the divergence tends to be
negative when y is negative. And 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. But once you compute everything,
it's only dependent on the y value here. And that as you go kind
of left and right on the diagram there. As we look left and right, the value of the divergence doesn't change. That's kind of surprising. It makes a little bit of sense. 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 were exactly the same.