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.