Divergence 3

Analyzing a vector field using its divergence.

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

analyze the vector field.

And hopefully this will
make everything a little

bit more tangible.

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

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

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

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

2 in the y-direction.

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

So let's just do
the math first.

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

divergence of our field.

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

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

my not-so-accurate drawings.

So what's the divergence?

We take the partial derivative
of the x-component

with respect to x.

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

really have to worry about
keeping y or z constant.

It's really just a derivative
of this expression

with respect to x.

So it's 2x minus 3.

And then we add that to the
partial derivative of the

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

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

take the derivative
with respect to y.

So it's plus 2y minus 3.

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

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

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

function a little bit.

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

and think about when does that
vector field have some

interesting points?

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

or the y-components
are equal to 0.

So when is the
x-component equal to 0?

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

same thing as, we could
rewrite our vector field.

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

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

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

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

the roots of this polynomial,
when x is equal to

1 or 2, right?

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

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

of these points.

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

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

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

So these are the points where
the magnitude of the velocity

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

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

And let me ask
another question.

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

the divergence equal to 0?

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

Let me click clear
up some space.

I think I can delete this.

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

magnitude of the
vector field 0.

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

equal to 0?

So this is divergence.

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

plus 2y, oh, sorry.

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

So this is minus 6, right?

Minus 3 minus 3,
that's minus 6.

That's my major flaw,
adding and subtracting.

Anyway, so the divergence.

2x plus 2y, minus 6.

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

So let's set it equal to 0.

And we can simplify
this a little bit.

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

plus y minus 3 is equal to 0.

You get x plus y is equal to 3.

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

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

easier to visualize a line.

We could say y is
equal to 3 minus x.

So along this line, the
divergence of the vector fields

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

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

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

sign, that would carry over.

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

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

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

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

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

divergence is negative.

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

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

consistent with our intuition
of what a divergence is,

and the numbers we found.

I hope you can see this.

So this is the vector field.

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

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

This is the definition
of our vector field.

Have it graphed here.

And we figured out, we just
figured out when the

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

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

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

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

the vectors are 0
at that point.

And actually, I could
zoom in a little bit.

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

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

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

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

get very, very, very small.

We could zoom in again.

And we see, the magnitude
gets very small.

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

get small, and then 2, 2.

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

field gets very small
at this point.

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

the divergence equal 0?

Well, the divergence equaled
0 along the line y is

equal to 3 minus x.

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

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

Right?

So anything along, at any
point along that line,

the divergence is 0.

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

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

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

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

time, just as many particles
are entering through the top

right as leaving through
the bottom left, right?

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

through the bottom left.

And the vectors, though,
are about the same.

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

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

also less leaving.

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

you see just as much
entering as leaving.

And that's why the
divergence is 0.

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

Up here, we figure the
diverge is positive.

And does that make
sense?

Let's pick an arbitrary point.

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

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

because I can't draw
on this graph.

But actually, let's
just say the square.

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

This one right here.

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

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

the vectors entering it, right?

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

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

that the particles
are diverging.

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

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

pick an arbitrary spot.

Let's say this
square right here.

We see that the vectors
entering it, the magnitude of

the vectors entering it, are
larger than the magnitude of

the vectors exiting it.

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

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

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

or it's actually converging.

Actually, something
interesting is happening

at these two points.

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

y-direction it's actually
converging, right?

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

and below it the arrows
are pointing up, right?

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

have a negative divergence.

Things are entering
any given spot.

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

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

have particles entering above
and below a certain space, but

you have just as many particles
exiting to the left

and the right.

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

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

increase or decrease in density
along that line y is

equal to 3 minus x.

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

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

that means that things are
flowing out, when the rate

of change is positive.

So we said the
divergence is positive.

Let's say at the spot, right?

So it makes sense, if our
partial derivatives are

positive, that means that the
magnitude of our vector is

getting larger and larger for
larger values of our

x's and y's, right?

So if the magnitude of our
vectors are getting larger and

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

right are going to have
a larger magnitude than

the factors on the left.

They're increasing
in magnitude.

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

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

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

getting less dense.

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

but I have run out
of time once again.

I'll see you in the next video.