Before I actually show you the mechanics of what the curl of a
vector field really is, let's try to get a little
bit of intuition.
So here I've drawn, I'm going to just draw a two-dimensional
vector field.
You can extrapolate to 3, but when we're getting
the intuition, it's good to do it in 2.
And so, let's see.
I didn't even label the x and y axis.
This is x, this is y.
So when y is relatively low, our magnitude vector goes in
the x direction, when it increases a little bit, it
gets a little bit longer.
So as we can see, as our change in the y-direction, as we go
in the y-direction, the x-component of our vectors
get larger and larger.
And maybe in the x-direction they're constant, regardless
of your level of x, the magnitude stays.
So for given y, the magnitude of your x-component vector
might stay the same.
So I mean, this vector field might look something like this.
I'm just making up numbers.
Maybe it's just, I don't know, y squared i.
So the magnitude of the x-direction is just a
function of your y-value.
And as your y-values get bigger and bigger, the magnitude in
your x-direction will get bigger and bigger, proportional
to the square of the magnitude of the y direction.
But for any given x, it's always going to be the same.
It's only dependent on y.
So here, even if we make x larger, we still get
the same magnitude.
And remember, these are just sample points
on our vector field.
But anyway.
That's enough of just getting the intuition behind
that vector field.
But let me ask you a question.
If I were to, let's say that this vector field shows the
velocity of a fluid at various points.
And so you can view this, we're looking down on a river, maybe.
If I were to take a little twig or something, and I were to
place it in this fluid, so let me place the twig right here.
Let me draw my twig.
So let's say I place a twig, it's a funny-looking twig,
but that's good enough.
Let's say I place a twig right there.
What's going to happen to the twig?
Well, at this point on the twig, the water's moving to the
right, so it'll push this part of the twig to the right.
At the top of the twig, the water is also moving to the
right, maybe with a faster velocity, but it's also going
to push the top of the twig to the right.
But the top of the twig is going to be being pushed to
the right faster than the bottom of the twig, right?
So what's going to happen?
The twig's going to rotate, right?
After, I don't know, some period of time, the
twig's going to look something like this.
The bottom will move a little bit to the right, but the
top will move a lot more to the right.
Right?
And the whole thing would have been shifted to the right.
But it's going to rotate a little bit.
And maybe after a little bit further, maybe it looks
something like this.
So you can see that because the vectors increasing in a
direction that is perpendicular to our direction
of motion, right?
This fairly simple example, all of the vectors point
in the x-direction.
But the magnitude of the vectors increase, they increase
perpendicular, they increase in the y-dimension, right?
And when this happens, when the flow is going in the same
direction, but it's going at a different magnitude, you see
that any object in it will rotate, right?
So let's think about that.
So if the derivative, the partial derivative, of this
vector field with respect to y is increasing or decreasing, if
it's just changing, that means as we increase in y, or as we
decrease in y, the magnitude of the x-component of our vectors,
right, the x-direction of our vectors changes.
And so if you have a different speed for different levels of
y, as something moves in the x-direction, it's going
to be rotated, right?
You could almost view it as if there's a net torque on an
object that sits in the water here.
And the ultimate would be, let me draw another vector field,
the ultimate would be if I had this situation.
Let me draw another vector field.
If I had this situation, where maybe down here it's like this,
then maybe it's like this, and then maybe it gets really
small, then maybe it switches directions, up here, and then
the vector field goes like this.
So you could imagine up here that's going to the left, with
a fairly large magnitude.
So if you put a twig here, you would definitely hopefully see
that the twig, not only will it not be shifted to the right,
this side is going to be moved to the left, this side is going
to be the right, it's going to be rotated.
And you'll see that there's a net torque on the twig.
So what's the intuition there?
All of a sudden, we care about how much is the magnitude of a
vector changing, not in its direction of motion, like in
the divergence example, but we care how much is the magnitude
of a vector changing as we go perpendicular to its
direction of motion.
So when we learned about dot and cross product,
what did we learn?
We learned that the dot product of 2 vectors tells you how much
2 vectors move together, and the cross product tells you how
much the perpendicular, it's kind of the multiplication
of the perpendicular components of a vector.
So this might give you a little intuition of what is the curl.
Because the curl essentially measures what is the rotational
effect, or I guess you could say, what is the curl of a
vector field at a given point?
And you can you can visualize it.
You put a twig there, what would happen to the twig?
If the twig rotates and there's some curl, if the magnitude
of the rotation is larger, then the curl is larger.
If it rotates in the other direction, you'll have the
negative direction of curl.
And so just like what we did with torque, we now care
about the direction.
Because we care whether it's going counterclockwise or
clockwise, so we're going to end up with a vector
quantity, right?
So, and all of this should hopefully start fitting
together at this point.
We've been dealing with this Dell
vector or this, you know, we could call this abusive
notation, but it kind of is intuitive, although it really
doesn't have any meaning when I describe it like this.
You can kind of write it as a vector operator, and then it
has a little bit more meeting.
But this Dell
operator, we use it a bunch of times.
You know, if the partial derivative of something in the
i-direction, plus the partial derivative, something with
respect to y in the j-direction, plus the partial
derivative, well, this is if we do it in three dimensions
with respect to z in the k-direction.
When we applied it to just a scalar or vector field, you
know, like a three-dimensional function, we just multiplied
this times that scalar function, we got the gradient.
When we took the dot product of this with a vector field, we
got the divergence of the vector field.
And this should be a little bit intuitive
to you, at this point.
Because when we, you might want to review our original videos
where we compared the dot product to the cross product.
Because the dot product was, how much do two
vectors move together?
So when you're taking this Dell operator and dotting it with a
vector field, you're saying, how much is the vector
field changing, right?
All a derivative is, a partial derivative or a normal
derivative, it's just a rate of change.
Partial derivative with respect to x is rate of change
in the x-direction.
So all you're saying is, when you're taking a dot product,
how much is my rate of change increasing in my
direction of movement?
How much is my rate of change in the y-direction increasing
in the y-direction?
And so it makes sense that it helps us with divergence.
Because remember, if this is a vector, and then as we increase
this in the x-direction, the vectors increase, we took a
little point, and we said, oh, at this point we're going to
have more leaving than entering, so we have a
positive divergence.
But that makes sense, also, because as you go in the
x-direction, the magnitudes of the vectors increase.
Anyway, I don't want to confuse you too much.
So now, the intuition, because now we don't care about the
rate of change along with the direction of the vector.
We care about the rate of change of the magnitudes of
the vectors perpendicular the direction of the vector.
So the curl, you might guess, is equal to the cross product
of our Dell operator and the vector field.
And if that was where your intuition led you, and that
is what your guess is, you would be correct.
That is the curl of the vector field.
And it is a measure of how much is that field rotating, or
maybe if you imagine an object in the field, how much is the
field causing something to rotate because it's
exerting a net torque?
Because at different points in the object, you have a
different magnitude of a field in the same direction.
Anyway, I don't want to confuse you too much.
Hopefully that example I just showed you will make
a little bit of sense.
Anyway, I realize I've already pushed 9 minutes.
In the next video, I'll actually compute curl, and
maybe we'll try to draw a couple more to hit
the intuition home.
See you in the next video.