Main content

## Multivariable calculus

- 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 example

A worked example of computing and interpreting two-dimensional curl. Created by Grant Sanderson.

## Want to join the conversation?

- Wouldn't it be much easier to simplify the formula to (3x^2)-(3y^2) first?(32 votes)
- Absolutely! To be honest, I'm not really sure why I didn't do that. Just got caught up focussing on the relevant curl interpretation I suppose.(41 votes)

- Shouldn't the divergence of this vector field at origin be negative. But using the divergence formula, I get a 0. Can you explain the logic?(4 votes)
- Nope, I know that this video was about curl. But for fun I calculated divergence at origin. And the divergence of this function is 0. But when watching the animation it seems like the divergence should be negative. I couldn't understand the reason.(3 votes)

- According to the formula..if for example the y-component of the vector field changed from being slightly positive to more positive as we move in the x-direction, that would still give us positive curl. However, in practice, that would not give us counterclockwise rotation around that point. So, how is this problem solved?(4 votes)
- This is explained in the 2d curl intution video Grant posted just after the video watched for this question. I am assuming for your question, the same goes for the y direction, that the x is somewhat positive then 'more positive'. To summerize the 2d-curl nuance video : if you put a paddle wheel in a region that you described earlier, if there is a positive curl, that means the force of the vector along the x axis will push harder on the right than on the left, and same principle on the y axis (the upper part will be pushed more than the lower). The left and right(as well as the up and down force) forces cancel out each others extra energy, and where left with the simplified 2d-curl!

If this is not clear, Grant does a much better job in thee earlier quoted viedos in 5 minutes.(1 vote)

- At3:20Grant mentions that a curl of 27 is "quite positive". What is this "quite positive" value relative to? I mean, I know 0 corresponds to no curl and the more positive the value, the greater the counterclockwise curl what is the range of how high the curl scale goes?(3 votes)
- At around5:30Grant interprets curl values as the "general direction and strength of rotation around a point." I'm guessing higher (absolute) values may mean larger magnitude vectors overall going around in circles, and over time at a faster rate.(1 vote)

- Okay , this is just out of curiosity .. curl of a vector field gives a result , thats a vector ... as it can be described by cross product of gradient and vector ... so Is in this case the curl is directed along z axis ?? and if it so then is it appropriate to relate its geometrical representation of counter-clockwise or clockwise rotation by right hand thumb rule to assign positive and negative sign respectively..(3 votes)
- What does a higher or lower curl represent? For example, what is the difference between a positive curl of say, 20 and 50?(2 votes)
- This is from betterexplained.com

"Curl is simply the circulation per unit area, circulation density, or rate of rotation (amount of twisting at a single point). Imagine shrinking your whirlpool down smaller and smaller while keeping the force the same: you'll have a lot of power in a small area, so will have a large curl. If you widen the whirlpool while keeping the force the same as before, then you'll have a smaller curl. And of course, zero circulation means zero curl."(1 vote)

- So if the vectors surrounding a point form a perfect circle in the counterclockwise direction, would the curl be infinity?(1 vote)
- So, let's say I choose an arbitrary point like (3,0) as in your example, and I see that it corresponds to a positive curl. Stuff rotates counterclockwise around (3,0). But how do I know for how far from my point (3,0) is this true? (The curl is also positive for the value (4,0), and still, only (3,0) seems to be a center of rotation.) If I got it right, the point I choose to evaluate the curl at, will be considered as a center, around which the curl function tells me how much particles rotate. How do I know the radius of the circle in which the particles rotate around my specific point, and not around another? Or diverge away? Or is that not even relevant? Sorry if I'm way off, I'm fairly new to multidimensional calculus. ^^'(1 vote)

## Video transcript

- [Voiceover] So let's compute
the two dimensional curl of a vector field. The one I have in mind will
have an x component of, let's see not nine. But y cubed minus nine times y and then the y component will be x cubed minus nine times x. You can kind see I'm just
a sucker for symmetry when I choose examples. When I showed in the last video how the two dimensional curl, the 2D curl of a vector field, of a vector field v which
is a function of x and y, is equal to the partial derivative of q, that second component, with respect to x minus the partial derivative
of p that first component, with respect to y. And I went through the
reasoning for why this is true but just real quick kinda
the in the nutshell here, this partial q, partial x
is because of as you move from left to right vectors
tend to go from having a small or even negative y component
to a positive y component, that corresponds to
counter clockwise rotation. And similarly this dp,
dy is because if vectors as you move up and down as you
kind of increase the y value go from being positive to zero to negative or if they're decreasing
that also corresponds to counter clockwise rotation. So taking the negative of that
will tell you whether or not changes in the y direction
around your point correspond with counter
clockwise rotation. So in this particular case
when we start evaluating that, we start by looking at partial
of q with respect to x. So we're looking at the second
component and taking its partial derivative with
respect to x and in this case, nothing but x's show up so
it's just like taking its derivative and you get
three x squared minus nine. Three x squared minus nine,
and that's the first part. Then we subtract off whatever
the partial derivative of p with respect to y is, so we
go up here and it's entirely in terms of y and trying to
do the symmetry we're just taking the same calculation,
three y squared that derivative of y cubed minus nine. So this right here is
our two dimensional curl. And lets go ahead and
interpret what this means. And in fact this vector
field that I showed you is exactly the one that
I used when I was kind of animating the intuition
behind curl to start off with, where I had these specific
parts where there is positive curl here and
here but negative curl up in these clockwise rotating areas. So we can actually see
why that's the case here and why I chose this specific
function for something that'll have lots of good curl examples. Cause if we look over in that
region where there should be positive curl, that's
where x is equal to three and y is equal to zero. So I go over here and say
if x is equal to three, and y is equal to zero, this whole formula becomes let's see, three
times three squared so, three times three squared minus nine, minus nine and then minus the
quantity now we're plugging in y here so that's three
times y squared is just zero cause y is equal to zero, minus nine. Minus nine and so this part is 27, that's three times nine is
27 minus nine gives us 18. And then we're subtracting
off a negative nine so that's actually plus nine
so this whole thing is 27, it's actually quite positive,
so this is a positive number and that's why when we go
over here and we're looking at the fluid flow, you have
a counter clockwise rotation in that region. Whereas, let's say we did
all of this but instead of x equals three and y
equals zero, we looked at x is equal to zero and
y is equal to three. So in that case, we would instead, so x
equals zero, y equals three, let's take a look at where that is. X is zero, and then y the tick
mark's here are each one half so y equals three is
right here, it's in that clockwise rotation area
so if I kind of play this, we got the clockwise
rotation, we're expecting a negative value. Now let's see if that's what we get. We go over here and I'm gonna evaluate this whole function again. How about plugging in
zero for x so this is three times zero times zero minus nine. And then we're subtracting off three times y squared so that's
three times three squared. Three squared minus nine. And this whole part is zero
minus nine so that becomes negative nine and over
here we're subtracting off 27 minus nine which is 18
so we're subtracting off 18, so the whole thing equals negative 27. So maybe I should say
that equals negative 27. So because this is negative
that's what corresponds to the clockwise rotation that we have going on in that region. And if you went and you plugged in a bunch of different points like
you could perhaps see how if you plug in zero
for x and zero for y, those nines cancel out
which is why over here there's no general
rotation around the origin when x and y are both equal to zero. And you can understand
that every single point and the general rotation
around every single point just by taking this formula
that we found for 2D curl, and plugging in the
corresponding values of x and y. So it's actually a very powerful
tool cause you would think that's a very complicated
thing to figure out right, that if I give you this
pretty complicated fluid flow and say hey I want you to
figure out a number that'll tell me the general direction
and strength of rotation around each point, that's a
lot of information so it's nice just to have a small compact formula.