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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.

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• Wouldn't curl be a vector despite the dimensions as it's defined by cross product?
• 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.
• Around , 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
• 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.
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.
• Is the number for curl have any physical meaning beside its sign- i.e, can we assing units to it?
• 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.
• at i couldn't understand why partial(P)/partial(y) is subtracted from partial(Q)/partial(x)
• 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.
• If curl of a vector field is 0, how is it conservative in nature? is there any proof?
• at he 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.
• 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.
• Doesn't the formula of the Curl look like the Cross Product of nabla and the vector that the function corresponds to?
• 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 at - 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)
``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 )``