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## Multivariable calculus

### Course: Multivariable calculus > Unit 2

Lesson 10: Curl- 2d curl intuition
- Visual curl
- 2d curl formula
- 2d curl example
- Finding curl in 2D
- 2d curl nuance
- Describing rotation in 3d with a vector
- 3d curl intuition, part 1
- 3d curl intuition, part 2
- 3d curl formula, part 1
- 3d curl formula, part 2
- 3d curl computation example
- Finding curl in 3D
- Symbols practice: The gradient

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# 2d curl formula

Here we build up to the formula for computing the two-dimensional curl of a vector field, reasoning through what partial derivative information corresponds to fluid rotation. Created by Grant Sanderson.

## Want to join the conversation?

- Wouldn't curl be a vector despite the dimensions as it's defined by cross product?(17 votes)
- Excellent question. Yes, curl indeed is a vector. In the x,y plane, the curl is a vector in the z direction. When you think of curl, think of the right hand rule. It should remind you of angular momentum, who's vector direction is in the direction perpendicular to the plane of rotation.(14 votes)

- Around4:00, How do you know which vector to start from like P>0 to P<0 , because if we start from top and go counter clockwise we get positive dP/dy(7 votes)
- No Akshat even if we reverse the order of the vectors the partial derivative remains the same. Grant has taken the convention of taking the derivative of P along increasing y. In the case of anticlockwise rotation, when you go from the bottom to top P decreases and change in P (i.e. the final value minus the initial value) is negative. But the change in y is positive and therefore the ratio (Delta P / Delta y) is negative. If you do the same in your way, Delta P is positive and Delta y is negative because you climb down the ladder of y. And the resulting derivative is still negative.

Some numbers would help you imagine better.

Let's say P goes from 8 at y = 2 to -4 at y = 4.

Going from y = 2 to 4 : Delta y = (4 - 2) = +2

and Delta P = (-4 - 8) = - 12

and the derivative is -12/+2 = -6.

In the opposite way i.e. going from y = 4 to y = 2

Delta y = (2 - 4) = -2

and Delta X = (8 - (-4)) = +12

and the derivative remains equal to +12/-2 = -6.(7 votes)

- Is the number for curl have any physical meaning beside its sign- i.e, can we assing units to it?(5 votes)
- Yes, Stoke's theorem says that the Flux of the Curl is equal to the work done by going around the boundary; usually the units on work are Newtons.(5 votes)

- at5:06i couldn't understand why partial(P)/partial(y) is subtracted from partial(Q)/partial(x)(4 votes)
- We want our formula for curl to give us a
**positive**value when there is**counterclockwise**rotation around a point. One of the conditions that Grant described in the video as giving counterclockwise rotation is when**Partial(P)/Partial(y) is less than 0**.

To get positive values for curl when we have counterclockwise rotation we take the negative of (subtract) Partial(P)/Partial(y), which is negative if we have clockwise rotation. We end up subtracting a negative giving us a positive value which is what we want by definition when we have counterclockwise rotation.(4 votes)

- If curl of a vector field is 0, how is it conservative in nature? is there any proof?(2 votes)
- The idea is that when the curl is 0 everywhere, the line integral of the vector field is equal to 0 around any closed loop. Thus, if the vector field is a field of force (gravitational or electric, usually), then the line integral along any path gives us the total work done by the force. For conservative forces, the total work done should be independent of the path, and only depend on the endpoints (for example, it takes the same amount of work to lift 1 kg 1 meter, regardless of whether you lift it straight up or use ramps). Also, if you move an object in a closed path, the total work done by the force is zero (mechanical energy is conserved). The mathematical proof that curl = 0 at every point implies path independence of line integral (and thus line integral of 0 for all closed loops) is called Stokes' Theorem, and it is one of the great accomplishments of all mathematics. You could try to look at these two Khan articles for more info:

https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/line-integrals-in-vector-fields-articles/a/conservative-fields

https://www.khanacademy.org/math/multivariable-calculus/greens-theorem-and-stokes-theorem/stokes-theorem-articles/a/stokes-theorem(2 votes)

- at1:11he says that the curl of the field will be a scalar value in this case because at every point we gonna get a value(positive or negative) telling about the curl but will not the curl be a vector in z direction? i understand that the final value of curl will tell weather its in +ve Z direction or -ve and will give the magnitude too but will not it still be a vector? then why its said that it will be a scalar value? please reply.(2 votes)
- Yes, curl is a 3-D concept, and this 2-D formula is a simplification of the 3-D formula. In this case, it would be 0i + 0j + (∂Q/∂x - ∂P/∂y)k. Imagine a vector pointing straight up or down, parallel to the z-axis. That vector is describing the curl. Or, again, in the 2-D case, you can think of curl as a scalar value.(2 votes)

- Doesn't the formula of the Curl look like the Cross Product of nabla and the vector that the function corresponds to?(2 votes)
- Exactly! That is a "trick" for remembering how to compute the curl. In this video, the result is not a vector, but the components of the vector would be 0i + 0j + (∂Q/∂x - ∂P/∂y)k. You can imagine that for 2-dimensional curl, the vector describing the rotation is pointing straight up, parallel to the z-axis. Grant describes 3-dimensional curl in another video, which I would recommend watching.(1 vote)

- I'm kinda confused how he brings up partial derivatives at4:01- if p < 0, doesn't that already indicate that the curl is positive?(1 vote)
- The vector being negative doesn't imply the curl being positive. For example, if the vector field is defined in a way where it is negative everywhere (for example, F = <-1 , 0>), the curl is 0.

Hence, we involve partial derivatives. The vector's sign at a point doesn't tell us about how it is curling. However, how the vector changes as you move along an axis does (as Grant showed in the video)(2 votes)

- Why does he take the partial derivative of p with respect to y instead of x and the partial derivative of q with respect to x instead of y?(1 vote)
- Around
, I can't seem to understand why Dp / Dy should be < 0. From my perspective,4:16`If P > 0 , it has no y component, it's vector could be`

(+ve x, 0 )

If P < 0, it's vector could be (-ve x, 0 )

By that logic, if an increase in y-direction to the p-vector , causes x component of the vector to turn negative, shouldn't Dp / Dy be > 0 ?(1 vote)

## Video transcript

- [Voiceover] So after
introducing the idea of fluid rotation in a
vector field like this, let's start tightening up
our grasp on this intuition to get something that we can
actually apply formulas to. So a vector field like
the one that I had there, that's two-dimensional,
is given by a function that has a two-dimensional input and a two-dimensional output. And it's common to write the
components of that output as the functions p and q. So, each one of those p and q, takes in two different
variables as it's input. P and q. And what I want to do here, is
talk about this idea of curl and you might write it down as just curl, curl of v, the vector field. Which takes in the same inputs
that the vector field does. And because this is the
two-dimensional example, I might write, just to distinguish it from
three-dimensional curl, which is something we'll get
later on, two d curl of v. So you're kind of thinking of
this as a differential thing, in the same way that you have, you know, a derivative, dx is gonna take
in some kind of a function. And you give it a function and
it gives you a new function, the derivative. Here, you think of this 2d
curl, as like an operator, you give it a function,
a vector field function, and it gives you another function, which in this case will be scalar valued. And the reason it's scalar valued, is because at every given point, you want it to give you a number. So if I look back at the vector field, that I have here, we want, that at a point like this, where there's a lot of
counter-clockwise rotation happening around it, for the curl function to return a positive number. But at a point like
this, where there's some, where there's clockwise
rotation happening around it, you want the curl to
return a negative number. So, let's start thinking
about what that should mean. And a good way to understand this two-dimensional curl function and start to get a feel for it, is to imagine the
quintessential 2d curl scenario. Well let's say you have a point and this here's going to be our point, xy, sitting of somewhere in space. And let's say there's no
vector attached to it, as in the values, p and
q, and x and y, are zero. And then let's say that
to the right of it, you have a vector pointing straight up. Above it, in the vector field, you have a vector pointing
straight to the left, to it's left, you have one
pointing straight down, and below it, you have one
pointing straight to the right. So in terms of the
functions, what that means, is this vector, to it's right, whatever point it's evaluated at, that's gonna be q is greater than zero. So this function q, that
corresponds to the y component, the up and down component of each vector, when you evaluate it at this point, to the right of our xy point, q's gonna be greater than zero. Where as if you evaluate
it to the left over here, q would be less than zero, less than zero, in our kind of, perfect curl
will be positive example. And then these bottom guys, if you start thinking
about what this means for, you'd have a rightward vector below, and a leftward vector above, the one below it, whatever
point you're evaluating that at, p, which gives us the kind of, left right component of these vectors, since it's the first
component of the output, would have to be positive. And then above it, above it here, when you evaluate p at that point, would have to be negative. Where as p, if you did it on
the left and right points, would be equal to zero because
there's no x component. And similarly q, if you did it
on the top and bottom points, since there's no up and down
component of those vectors, would also be zero. So this is just the, the very specific, almost contrived scenario
that I'm looking at. And I want to say, hey if this
should have positive curl, maybe if we look at the information, the partial derivative
information to be specific, about p and q, in a scenario like this, it'll give us a way to
quantify the idea of curl. And first let's look at p. So p starts positive, and as y increases, as the y value of our input increases, it goes from being positive
to zero, to negative. So we would expect, that
the partial derivative of p, with respect to y, so as we change that y component,
moving up in the plane, and look at the x
component of the vectors, that should be negative. That should be negative in circumstances where we want positive curl. So all of this we're looking at cases, you know the quintessential
case where curl is positive. So evidently, this is a fact, that corresponds to positive curl. Where as q, let's take a look at q. It starts negative,
when you're at the left. And then becomes zero,
then it becomes positive. So here, as x increases, q increases. So we're expecting that a
partial derivative of q, with respect to x, should be positive. Or at the very least,
the situations where, the partial derivative of q
with respect to x is positive, corresponds to positive
two-dimensional curl. And in fact, it turns out, these guys tell us all you need to know. We can say as a formula, that the 2d curl, 2d curl,
of our vector field v, as a function of x and y, is equal to the partial
derivative of q with respect to x. Partial derivative of
q, with respect to x, and then I'm gonna subtract
off the partial of p, with respect to y. Because I want, when this is negative, for that to correspond
with more positive 2d curl. So I'm gonna subtract off, partial of p, with respect to y. And this right here, is the formula for two-dimensional curl. Which basically, you can
think of it as a measure, at any given point you're asking, how much does the surrounding
information to that point, look like this set-up, like this perfect
counter-clockwise rotation set-up? And the more it looks like this set-up, the more this value will be positive. And if it was the opposite of this, if each of the vectors was turned around and you have clockwise rotation, each of these values
will become the negative of what it had been before. So 2d curl would end up being negative. And in the next video,
I'll show some examples of what it looks like to use this formula.