Let's say I have the vector field v.
The vector at any given point is equal to minus-- or the
magnitude of my x direction is actually dependent on y.
So where we are in the y coordinate and the xy plane.
Plus-- and then my magnitude in the y direction
is dependent on x.
Fair enough.
So first, let's just chuck through this, and
figure out it's curl.
So the curl of v.
It is equal to our del vector operator, cross v.
Which is nothing but this.
And even though this looks like a two dimensional vector field,
we actually have to take the cross product in 3 dimensions.
Because a curl is just like torque, and when you-- like we
did the right hand rule when we studied-- well, I hope you
watched some of the videos on magnetism and torque-- but the
torque actually goes in a direction that is perpendicular
to both of the vectors in your cross product.
If both of these only have x and y components, your actual
result is going to be in the z direction.
It's going to be perpendicular to both of these vectors.
So when you take the cross product, you still have to
do it in 3 dimensions.
So i j k, partial with respect to x, partial with respect to
y, partial with respect to z.
The x component is minus y.
The j component is x.
And we have no k component.
This one should be a little bit cleaner to calculate
than the last example.
So this is going to be equal to-- well, see the i
component-- let's cross out its column and its row.
So it's going to be the partial with respect to y of 0
minus the partial with respect to z of x.
Minus that.
And all of that times i minus-- and now it's going
to be the j component.
Remember, you do plus minus plus.
So minus and this is the j component.
Cross out its row and column, the partial derivative
of x with respect to 0.
Or partial derivative of 0 with respect to x, actually.
Minus the partial derivative of z.
Or the partial derivative of the [? vector ?]
z of minus y.
That's its j component.
And finally, plus its k component.
Row and column.
So the partial derivative with respect to x minus the partial
derivative of y with respect to minus y.
And I know you can't read it, but that's that minus y there,
and that's the k component.
And now let's simplify it.
I'll simplify it above it.
So this term-- let's see, partial derivative of 0.
Well, that's 0.
Partial derivative of x with respect to z.
Well, as far as z's concerned, x is a constant, so that's 0.
Partial derivative of 0 with respect to x, 0.
Partial derivative of minus y with respect to z.
As far as z's concerned, y is also constant.
So that's 0.
So all we're left is with this last term.
That was pretty straightforward.
Why don't we just cross all of that out?
And that makes intuitive sense, too, right?
Because at least from x's point of view, the rotation is--
although, the rotation, if you think about it, is going to be
in the direction perpendicular to both the x direction and
perpendicular to the direction in which it is changing.
You could kind of view it orthogonally, to its direction
of motion, which is y.
So it would be the z direction.
If that confuses you, don't worry about it.
But if it doesn't, then you could apply the same
argument to the j vector.
But anyway, lets simplify this.
So this is equal to the partial derivative of
x with respect to x.
Well, that's just 1.
Minus the partial derivative of y-- of minus y
with respect to y.
Well, that's just minus 1.
So it equals 2.
So the curl, at any point of this vector field, is 2.
Let's see what this vector field looks like, and let's
see if that gives us-- if our intuition holds
in this example.
And let me try to make it a little bit bigger.
Make the window bigger.
There you go.
Well, I think it's clear, you know, right when you look at
it, that this vector field looks like it's spinning.
If you were to stick something, especially in the middle, it's
very clear that it would spin.
But what might be a little unintuitive-- you might think,
wow, well, wouldn't something spin faster near the center
than it would here?
Why is the-- you know, the curl we got is 2.
It's a constant.
The curl is the same throughout this entire vector field.
So you'll be like, whoa, that's kind of implying that the field
is making something spin equally, no matter where
you are in the field.
Let's see if that makes sense.
Well, in the middle, it definitely makes sense that
something is spinning.
If I had a little stick here, I'd be pushing in this
direction, with not that much of a magnitude in
that direction.
And then I'd be pushing down to the right in this direction,
so it would cause it to spin.
But what if I had that same stick here?
You'd say, well, on the top right I'm pushing
up, up and to the left.
And the bottom left, I'm also pushing up and to the left.
You know, it wouldn't spin as much.
But it would.
Because the difference in magnitudes of these 2-- you
could almost view them as the torque producing forces-- the
difference in magnitude is enough that you'd still have
the same counterclockwise rotation here as you
would have here.
So because the curl is a constant positive number, when
we look at the xy plane like this, if you put a twig-- that
same twig anywhere where you put it on this plane--
it'll have the same counterclockwise rotation.
I think that's pretty neat.
Now let's do a little experiment.
What would have happened if this was plus y.
So let's just do that experiment.
If this was plus y then this would have been plus y, then
this final term-- we would have taken the partial derivative
with respect to y of plus y-- and so this would have
been 1 minus plus 1.
And then our curl would have been 0.
Which would have meant that we would have had no rotation.
And what's the intuition of that?
Well, if we just look at it mathematically, if we have 0
curl, somehow, the rotation in our x direction must be being
offset by a rotation in the y direction.
That the torques must be just perfectly
offsetting each other.
Let's see what happens if I were to change.
So this was our old graph.
Let me actually change it to my new vector field.
So that's our new vector field.
This is our vector field plus yi plus xj.
And now the curl is 0 everywhere.
Which implies, or which means, that I could put a twig
anywhere here and I'm not going to get any kind of rotation.
Let's see if that makes sense.
If I put a twig in the center, or some kind of stick in
the center, let's see.
I would have the forces or the fluids pushing in in this
direction, but they're not helping to rotate, and pushing
out in that direction.
Well, that's not going to help me either.
So I'm definitely not going to rotate there.
And actually, you could put a twig anywhere.
And maybe a twig might be pushed in a direction.
For example, if I put a twig here, it's going to be pushed
outward by the flow of the water, by the velocity
of the water.
But it's not going to rotate.
So the [? ink ?]
kind of holds.
That's even though I kind of have curl in the x direction,
or I have curl in the y direction, they're
offsetting each other.
So that in 2 dimensions, I actually end up
having no rotation.
And actually, this is called an irrotational-- I think that's
the word-- vector field.
Where you're not going to have any rotation here.
All of the-- if you think of it as force or velocity of the
vector field-- is going to be applying translation to
objects in that field.
And actually, just for fun, let's think about the
divergence of that field.
Just to, I don't know.
Just because I have 2 minutes.
So if I were to think about the divergence of that field, it's
fairly easy to calculate.
Let me erase this real fast.
It's always fun to just interpret a vector
field to death.
Let's do the divergence of this one.
So the divergence of that vector field, is just a
partial derivative of this with respect to x.
So div of v is the same thing as our del operator dot
our vector field, v.
And that's the partial derivative of this
with respect to x.
Well, the partial derivative of y with respect to x is just 0.
Plus the partial derivative of x with respect to y.
So this is 0.
So the divergence of this field is 0.
And even in the other case, when this was a minus, it
still would have been 0.
And does that make sense?
Well, if we take a circle anywhere here-- let's take it
in the center, because the center's the most interesting.
Let's take a circle.
So we do have some fluid or particles coming in at a
certain velocity from the bottom right and the top left.
But just as much is coming out through the top right
and the bottom left.
Whatever's coming in through here and here is leaving
through there and there.
So if I had an infinitesimally small circle, or sphere, in
this vector field, I would have no net density increasing.
Or nothing would be entering into that circle, or in any
given amount of time the concentration of that
circle wouldn't change.
And that's true pretty much anywhere.
If I were to draw a circle here, because the divergence
is 0, it's telling us that whatever's coming in in 1
direction is coming out in the other directions.
So I'm not getting any denser, or any less dense.
I have no divergence or convergence.
So that's just interesting.
Well, now I've run of time.
So we're done analyzing this vector field.
See you in the next video.