WEBVTT

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Now let's do a slightly fancier
example, then we'll try to

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analyze the vector field.

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And hopefully this will
make everything a little

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bit more tangible.

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So let's say that the velocity
of the fluid, or the particles

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in the fluid, at any given
point in the x-y plane, let's

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say in the x-direction, it is x
squared, x squared minus 3x

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plus 2 in the x-direction, plus
y squared minus 3y plus

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2 in the y-direction.

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Make it simple so that we only
have one thing to factor.

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So let's just do
the math first.

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Let's figure out the divergence
of our vector field, the

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divergence of our field.

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And going to show you a graph
of this field soon, so we'll

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get an intuition of what it
actually looks like, instead of

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my not-so-accurate drawings.

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So what's the divergence?

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We take the partial derivative
of the x-component

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with respect to x.

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So that's just, there's only an
x-variable here, so we don't

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really have to worry about
keeping y or z constant.

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It's really just a derivative
of this expression

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with respect to x.

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So it's 2x minus 3.

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And then we add that to the
partial derivative of the

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y-component, or the y-function,
with respect to y.

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There's only y's in the
y-component, so we just

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take the derivative
with respect to y.

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So it's plus 2y minus 3.

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Or we could just say that the
divergence of v, at any point

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xy, so this is a function of x
and y, is 2x plus 2y minus 3.

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Now before I show you the
graph, let's analyze this

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function a little bit.

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First of all, let's just look
at the original vector field,

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and think about when does that
vector field have some

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interesting points?

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Well, I think some interesting
points are when either the x-

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or the y-components
are equal to 0.

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So when is the
x-component equal to 0?

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Well, if we factor the
x-component, that's the

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same thing as, we could
rewrite our vector field.

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If we just factored that,
that's x minus 1 times x minus

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2i plus, and it's the same
polynomial, just with y, for

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the y-component, so y minus
1 times y minus 2 times j.

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So the x-component is 0 when x
is equal to 1, these are just

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the roots of this polynomial,
when x is equal to

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1 or 2, right?

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And the y-component is 0
when y is equal to 1 or 2.

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And they're both equal to 0
if we have any combination

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of these points.

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So the points where they're
both equal to 0 are 1, 1, x's

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is 1, y is 2, right, because
then both components

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are 0, 2, 1, or 2, 2.

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So these are the points where
the magnitude of the velocity

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of our fluid, or the particles
of the fluid, are 0.

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And we'll see that on
our graph in a second.

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And let me ask
another question.

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At what points are, well, let's
first decide, what point is

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the divergence equal to 0?

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Let's say, at what point is
a divergence equal to 0.

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Let me click clear
up some space.

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I think I can delete this.

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We got our points, we figured
out what coordinates are the

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magnitude of the
vector field 0.

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So let's try to figure out,
when is the divergence

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equal to 0?

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So this is divergence.

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So if we set that equal
to 0, 2x plus 2y, 2x

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plus 2y, oh, sorry.

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You know what, this is 2x
minus 3 plus 2y minus 3.

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So this is minus 6, right?

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Minus 3 minus 3,
that's minus 6.

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That's my major flaw,
adding and subtracting.

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Anyway, so the divergence.

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2x plus 2y, minus 6.

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And we want to know,
when does that equal 0?

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So let's set it equal to 0.

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And we can simplify
this a little bit.

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We can divide both sides of the
equation by 2, and you get x

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plus y minus 3 is equal to 0.

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You get x plus y is equal to 3.

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We could be finished there, or
we could just put it in our

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traditional mx plus b form,
that's the way I find it

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easier to visualize a line.

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We could say y is
equal to 3 minus x.

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So along this line, the
divergence of the vector fields

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b is equal to 0, along the line
y is equal to 3 minus x.

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And if we're above that line,
the divergence is going to be

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positive, right, because if you
just made this a greater than

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sign, that would carry over.

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You'd have y is
greater than 3-x.

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So y greater than 3-x, the
divergence is positive.

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And y is less than 3 minus x,
the divergence is negative.

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And you could just make this a
less than sign then solve, and

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you'll get y is less than 3-x,
you want to know when the

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divergence is negative.

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So I think we've done all of
the analyzing we can do.

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So let's take a look at the
graph and see if it if it's

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consistent with our intuition
of what a divergence is,

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and the numbers we found.

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I hope you can see this.

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So this is the vector field.

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I don't have space to show it,
but I think you remember, this

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is, you know, x squared
minus 3x plus 2.

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This is the definition
of our vector field.

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Have it graphed here.

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And we figured out, we just
figured out when the

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x-components and the
y-components are equal to 0,

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and then we said, when are
both of them equal to 0?

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And we said, oh, well
at the point 1, 1.

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Well, this is the point 1, 1,
and we that the magnitudes of

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the vectors are 0
at that point.

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And actually, I could
zoom in a little bit.

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Right around here, they're all
pointing inwards,, but they get

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smaller and smaller as you
approach the point 1, 1, right?

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We also said at the point
1, 2. x is 1, y is 2.

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And here, too, we see that
the magnitude of the vectors

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get very, very, very small.

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We could zoom in again.

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And we see, the magnitude
gets very small.

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The other point, 2, 1, once
again, we see the magnitude

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get small, and then 2, 2.

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So that's consistent with what
we found out, that the vector

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field gets very small
at this point.

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And the other interesting
thing we said, OK, when does

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the divergence equal 0?

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Well, the divergence equaled
0 along the line y is

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equal to 3 minus x.

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So the line y is equal to 3
minus x starts, the y-intercept

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is going to be 3, and it's
going to come down like this.

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Right?

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So anything along, at any
point along that line,

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the divergence is 0.

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And if we actually look at
the graph, it makes sense.

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Because I can't draw on this
graph, but if we drew a circle

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right there, let's assume that
that's on that, the line y is

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equal to 3 minus x, we would
see that in a given amount of

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time, just as many particles
are entering through the top

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right as leaving through
the bottom left, right?

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A lot of entering through the
top right and a lot of leaving

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through the bottom left.

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And the vectors, though,
are about the same.

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And if we go to the bottom, if
we go here on the line, it

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looks like maybe there are less
entering, but there's

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also less leaving.

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I know it's hard to see, but
anywhere along the line,

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you see just as much
entering as leaving.

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And that's why the
divergence is 0.

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Now, let's look at some
of the other points.

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Up here, we figure the
diverge is positive.

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And does that make
sense?

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Let's pick an arbitrary point.

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If we were to draw a circle
around here, we see the vectors

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on the left-hand side of that
circle that you can't see,

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because I can't draw
on this graph.

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But actually, let's
just say the square.

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Let's say the square
is my region, right?

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This one right here.

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If that square is my region,
we see the vectors on the

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left-hand side are larger, than
factors leaving are larger than

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the vectors entering it, right?

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So if, in a given amount of
time, more is leaving than

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entering, then I'm becoming
less dense, or you could say

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that the particles
are diverging.

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And that makes sense, because
I have a positive divergence.

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And if we go here, where the
divergence is negative, let's

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pick an arbitrary spot.

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Let's say this
square right here.

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We see that the vectors
entering it, the magnitude of

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the vectors entering it, are
larger than the magnitude of

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the vectors exiting it.

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So in any given amount of time,
more is coming in than leaving.

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So it's getting denser,
or it's converging.

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So negative divergence, you can
view it as getting denser,

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or it's actually converging.

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Actually, something
interesting is happening

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at these two points.

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So we said that at the point
2, 1, we see that in the

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y-direction it's actually
converging, right?

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Above y is equal to 1, the
arrows are pointing down,

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and below it the arrows
are pointing up, right?

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So in the y-direction, we're
actually converging, or we

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have a negative divergence.

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Things are entering
any given spot.

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But in the x-direction, things
are getting pushed out, right?

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So the reason why the
divergence is 0 here, you might

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have particles entering above
and below a certain space, but

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you have just as many particles
exiting to the left

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and the right.

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So it's kind of like particles
are getting deflected out.

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So on a net basis, between both
dimensions, you have no

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increase or decrease in density
along that line y is

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equal to 3 minus x.

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And before I run out of time, I
just want to give you that core

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intuition again, of why the
divergence is positive, and why

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that means that things are
flowing out, when the rate

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of change is positive.

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So we said the
divergence is positive.

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Let's say at the spot, right?

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So it makes sense, if our
partial derivatives are

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positive, that means that the
magnitude of our vector is

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getting larger and larger for
larger values of our

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x's and y's, right?

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So if the magnitude of our
vectors are getting larger and

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larger for larger values of x
and y, the vectors on the

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right are going to have
a larger magnitude than

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the factors on the left.

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They're increasing
in magnitude.

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And so if I were to draw a
boundary, more is going to

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be coming out of the right
than entering to the left.

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And so you have a positive
diverge, or you're

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getting less dense.

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Anyway, I hope I haven't
confused you too much,

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but I have run out
of time once again.

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I'll see you in the next video.