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Studying for a test? Prepare with these 13 lessons on Derivatives of multivariable functions.
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Video transskription
- [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.