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

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.

Newtonian pooh sticks.

Good explanation of the physical significance of curl.

I love you, Sal =)

I was struggling with the wording of my lecturer's notes but you've set that straight.

thank you, you give us intuition that explains all the fundamentals and roots of the concepts.

he loves the word intuitionÂ

And suddenly an entire semester of aerodynamics makes sense.

I just understood curl in one shot. Thanks.

I am trying to download this video, unfortunately half – the best part is corrupt.

Any help…

all three videoes on curl were superb..your eg under consideration simplifies the topics by great deal, Sir

best explanation ever

thank you Khan

oh man!

it just saved me from losing the entire semester.

OH MY GOD THANKS

LITERALLY A LIFESAVER

i don't understand why every time he uses the del operator, he calls it an "abusive notation". can anyone tell me why that's so. i really hope someone tell me why.

Only college students can relate…

rotation is happening in the xy plane?

thanks a ton… was struggling with the meaning behind divergence… understood completely from previous videos

I can tell from the ugly handrawn graph in the thumbnail that we're back with Khan. đ

Man I just don't know how to thank you enough for making me feel that mathematics are extremely logical and very beautiful to comprehend, cheers man and merry christmas

im not religious but …God bless youâ€đ

such a good explanation. The mathematical definition makes SO much sense now with this intuitive understanding.

Thank you so mvh sir….helped a lot.â

What level maths is this?because I'm in 9th grade and I totally understand this..

Good job! â€

first thought needs more jpeg, but then realized its from 2008

God of Teaching

Sal likes the words intuition and twig!

Khan Academy has been my life line against all the professors in this bloody country who are afflicted but acute 'tenure-itis'….

Thank you for your generous and free knowledge !

Intuitive academy

'lemme draw my twig…'