- [Voiceover] Hello everyone.

So I'm gonna start talking about curl.

Curl is one of those very cool vector calculus concepts,

and you'll be pretty happy that you've learned it

once you have, if for no other reason

because it's kind of artistically pleasing.

And, there's two different versions,

there's a two-dimensional curl and a three-dimensional curl.

And naturally enough, I'll start talking

about the two-dimensional version

and kind of build our way up to the 3D one.

And in this particular video,

I just want to lay down the intuition

for what's visually going on.

And, curl has to do with the fluid flow interpretation

of vector fields.

Now this is something that I've talked about

in other videos,

especially the ones on divergents if you watch that,

but just as a reminder, you kind of imagine

that each point in space is a particle,

like an air molecule or a water molecule.

And since what a vector field does

is associate each point in space with some kind of vector,

now remember we don't always draw every single vector,

we just draw a small sub-sample,

but in principle, every single point in space

has a vector attached to it.

You can think of each particle,

each one of these water molecules or air molecules,

as moving over time in such a way that the velocity vector

of its movement at any given point in time

is the vector that it's attached to.

So as it moves to a different location in space

and that velocity vector changes,

it might be turning or it might be accelerating,

and that velocity might change.

And you end up kind of a trajectory for your point.

And since every single point is moving in this way,

you can start thinking about a flow,

kind of a global view of the vector field.

And for this particular example,

this particular vector field that I have pictured,

I'm gonna go ahead and put a blue dot

at various points in space,

and, each one of these you can think of

as representing a water molecule or something,

and I'm just gonna let it play.

And at any given moment, if you look at the movement

of one of these blue dots,

it's moving along the vector that it's attached to

at that point, or if that vector's not pictured,

you know the vector that would be attached to it

at that point.

And as we get kind of a feel for what's going on

in this entire flow,

I want you to notice a couple of particular regions.

First, let's take a look at this region

over here on the right.

Kind of around here.

And just kind of concentrate on what's going on there.

And I'll go ahead

and start playing the animation over here.

And what's most noticeable about this region

is that there's counterclockwise rotation.

And this corresponds to an idea that the vector field

has a curl here, and I'll go very specifically

into what curl means, but just right now

you should have the idea that in a region

where there's counterclockwise rotation,

we want to say the curl is positive.

Whereas, if you look at a region that also has rotation,

but clockwise, going the other way,

we think of that as being negative curl.

Here I'll start it over here.

And in contrast, if you look at a place

where there's no rotation, where like at the center here,

you have some points coming in from the top right

and from the bottom left, and then going out

from the other corners.

But there's no net rotation.

If you were to just put like a twig somewhere in this water,

it wouldn't really be rotating.

These are regions where you think of them

as having zero curl.

So with that as a general idea, clockwise rotation regions

correspond to positive curl.

Counterclockwise rotation regions

correspond to negative curl,

and then no rotation corresponds to zero curl.

In the next video I'm gonna start going through

what this means in terms of the underlying function

defining the vector field and how we can start looking

at the partial differential information of that function

to quantify this intuition of fluid rotation.

And what's neat is that it's not just about fluid rotation.

If you have vector fields in other context

and you just imagine that they represent a fluid,

even though they don't,

this idea rotation and curling

actually has certain importance

in ways that you totally wouldn't expect.

The gradient turns out to relate to the curl,

even though you wouldn't necessarily think the grading

has something to do with fluid rotation.

In electromagnetism, this idea of fluid rotation

has a certain importance,

even though fluids aren't actually involved.

So, it's more general than just the representation

that we have here, but it's a very strong visual

to have in your mind as you study vector fields.