Curl 1 | Partial derivatives, gradient, divergence, curl | Multivariable Calculus | Khan Academy

Curl 1 | Partial derivatives, gradient, divergence, curl | Multivariable Calculus | Khan Academy

January 1, 2020 28 By Kody Olson


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.