Let's try to get our heads around the idea of divergence.
So first, like I did with gradients, I'll show you the
mechanics, which are actually pretty straightforward.
And then I'll try to give you the intuition.
And once you have the intuition, at first it
will seemed very, I don't know, unintuitive, maybe.
But it once you get it, you're like oh, that's it.
So let's see what divergence is.
Let's say I have a vector field.
And let's say this vector field, just for the purposes
of visualization it could be anything, but let's say it
represents the velocity of particles of fluid of any
point in two dimensions.
So it's going to be a two-dimensional vector field.
It's going to be a function of x and y, so the velocity at any
point-- it's a vector field -- let's say it is, and I'm just
going to make up something.
Let's say it's x squared, yi.
So at any point in the x-direction, at any point x
comma y, its velocity in the x-direction will
be x squared, y.
And then its velocity in the y-direction, I don't know
maybe it's just 3y, j.
That's its velocity in the x-direction.
So its velocity in the x-direction is actually
a function of x and y.
its velocity in the y-direction is just a function of y.
So what is the divergence?
So a couple of ways we can write it.
The correct way to write it is the divergence of
our vector field, v.
But a common mnemonic to remember the operation of
diverge and is to write the upside down triangle, which was
the same notation we used for gradient, but take the dot
product of that and the vector.
And if you remember from the gradient discussion, we said
that you can view, although it's kind of an abuse of
notation, but you could view this upside down triangle as
being equal to the partial derivative with respect to x in
the x-direction plus the partial derivative with respect
to y in the y-direction, which is the j-unit vector.
And then if we went to three dimensions, the partial
derivative with respect to z and the k-direction,
et cetera, et cetera.
But we're dealing with a two-dimensional vector here,
so let's just stick with two dimensions, x and y.
So what would this turn out to be?
If you took the dot product of this, which is this upside down
triangle, with this vector field, what would you get?
Well, you would just get the partial derivative of the x
dimension with respect to x, so you would get-- it's actually
pretty straight forward to memorize; you might not even
need this mnemonic right here, this abuse of notation; you
might just know it off hand --the x component, you take the
partial derivative with respect to x, and the y component, you
take the partial derivative with respect to y.
But I'll show you why it looks like the dot product.
So if you took the dot product of that and that, it would be
the partial derivative with respect to x of that
expression, of x squared, y and then plus the partial
derivative with respect to y of that second expression, the y
component of 3y, and then you would evaluate it.
What's the partial derivative of this with respect to x?
We just pretended y is a constant, just a number, so
the derivative of this with respect to x, would be
2x times the constant.
So it'll be 2xy plus-- what's the partial derivative
of 3y with respect to y?
Well, there's nothing else to hold constant, so it's just
like taking the derivative with respect to y
--so it's 2y plus 3.
So this is the divergence at a point x, y.
You could almost view it as a function of x and y.
So you could almost say you know, that the divergence of
v-- I'm going to make up some notation here --as long as you
get the point across, you can say that the divergence
of v, that this is a function of x and y.
That we just have an expression that if you give me a point
anywhere in this vector field, I can tell you the
divergence at that point.
So I think you'll find that the computation of divergence
isn't too difficult.
You just take the partial derivative of the x component
with respect to x, and you add that to the partial derivative
to the y component with respect to y.
And if you had the z, you would do the same thing,
so on and so forth.
Actually, let me do just do one more just hit the point home,
and then we'll work on intuition.
So if I said that I had, I don't know, let's say, my
vector field is cosine of yi plus-- so it's interesting; my
x-direction is dependent on my y-coordinate --plus, I
don't know, e to the xyj.
So then oh, that's difficult because I have these
e's and these cosines.
But we'll see; if you just keep your head straight on
what's constant and what's not, it's not too bad.
So the divergence of v is equal to the partial derivative
of this expression with respect to x.
Well, what's the derivative of this with respect to x?
If y is just a constant, cosine of y is just a number.
So the derivative of this with respect to x is just 0 plus--
what's the derivative of this with respect to y?
Well, you could just do x, since it's a constant,
as the coefficient on y.
So the derivative of x, y with respect to y is just x.
And then the derivative of e to anything is e to anything.
I just did the chain rule.
e to the x, y.
And so that is the divergence.
So you could just ignore this.
It's x, e to the x, y.
One thing to immediately realize, even before we work on
the intuition, is when we did gradient I gave you a surface
and it gave us a vector field.
Or I gave you a scalar field and you got a vector field.
When you take the divergence of something, you're going in the
opposite direction, in some ways.
You start with the vector field, right?
And what's a factor field?
It's something that if you give me any point x and y,
I'll give you a vector.
So if you wanted to graph it, in the x, y plane you'd have a
bunch of vectors, and I'll show you how that looks in a second
when we go over to intuition.
Well, when you take the divergence of it, you get a
value for any point x, y.
So even though a vector field has all these vectors on it,
the divergence tells you an actual scalar number at
any point in the field.
So let's get a little bit of intuition of what a
divergence actually is.
Let me do it in one dimension.
Or we can even, let's do it in two dimensions, but I'll
make it constant in the y.
So let's say that my-- let me erase this; I'll probably
need some space.
OK, oh, I didn't want to do that dot.
OK let's say the velocity of fluid, or the particles in
fluid, at any point in the x, y plane, let's say it is equal to
5xi plus, I don't know, 0y-- there's never any, sorry --0j,
right? j is the unit vector in the y-direction.
So there's never a y component to the velocity vector.
So what would that look like?
I don't need a computer to draw this.
I can handle this one myself I think.
So if that's the y-axis, that's my x-axis.
So when x is equal-- I'll just sample some points and draw
some vectors --when x is equal to 1-- let's say x is 1 there
--what's the magnitude of this vector?
It'll be 5, right?
Actually, let me make this a different number, because
it'll make it hard to do.
Let's make this 1/2 x.
So when x is 1, the magnitude of my vector is 1/2.
Only in the x-direction.
It has no y component; ignore this right here.
It's 1/2xi plus 0j.
Or you could just say 1/2xi.
And when x is equal to 2-- I could have picked any points,
but I'm just picking the numbers that's easy to
calculate --when x is equal to 2, what is the magnitude
of the vector?
It's 1/2 times 2, which is 1.
So it's going to be twice as big.
And remember, if I have a particle right here in my
fluid, if this is a particle, its velocity in the x-direction
is going to be 1 meter per second to the right.
If I have a particle here, it's velocity in the x-direction
is going to be 1/2 a meter per second to the right.
Let's just do one more point.
So let's say that x is equal to 3.
What's my velocity to the right?
I'll do it in a different color just so that you
don't get confused.
There's going to be 3/2; it's going to be even longer.
But the general idea here, and as we move up in x it
doesn't change much, right?
It doesn't change at all.
Our x value doesn't--
[COUGHS].
So for any y, the magnitude of the vector doesn't
change, right?
It's only dependent on x.
And then for example, here, it'll be even longer.
If we draw the vector here it'll be even longer, right?
If you do it here.
I think you get the point.
The further you go to the right, the faster the particles
are moving towards the right.
So now let's try to get a little bit of intuition.
Oh, I just realized that I ran out of time, so I will continue
this in the next video.