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2d curl example

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

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  • male robot donald style avatar for user Nick Rainer
    Wouldn't it be much easier to simplify the formula to (3x^2)-(3y^2) first?
    (35 votes)
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  • blobby green style avatar for user Fatih Yaman
    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?
    (5 votes)
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  • blobby green style avatar for user Ahmed Sheta
    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?
    (5 votes)
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    • starky ultimate style avatar for user Chiarandini Pandetta
      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.
      (2 votes)
  • leafers ultimate style avatar for user jjakow
    At Grant 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?
    (4 votes)
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  • aqualine seed style avatar for user deankur7
    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..
    (4 votes)
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  • leaf green style avatar for user Soumil Sahu
    What does a higher or lower curl represent? For example, what is the difference between a positive curl of say, 20 and 50?
    (3 votes)
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    • piceratops ultimate style avatar for user Mark Eastwood
      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."
      (2 votes)
  • male robot hal style avatar for user Aaroh Gokhale
    So if the vectors surrounding a point form a perfect circle in the counterclockwise direction, would the curl be infinity?
    (2 votes)
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  • piceratops ultimate style avatar for user Ákos Vendrinszky
    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. ^^'
    (2 votes)
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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.