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I'm trying to intuitively understand curl of a vector field, and I was watching this video. In it, they present the following scenario:

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I'm curious as to why the curl is zero in the four squares that are zero. They are two arrows pointing in the same direction, shouldn't that contribute twice the curl to a particle at a given point, and not cancel the other arrow out? I just don't see how the two arrows in the same direction cancel each other out, unless maybe it has to do something similar to how two torques applied in the same direction at opposite ends will produce no net torque.

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    Try the right hand rule to compute the cross product. That will give you meaningful results for all but the squares with parallel vectors. For those, is there any "rotation" for the cross product to pick up? – 2017-02-12
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    Don't forget the curl represents a very small rotation of the vector field. There is a rotation at the four corners and at the centre (indicated by the changing arrow directions), but not along the edges of the box (indicated by the parallel arrows) in this example. So the curl is zero where there is no rotation in the field. – 2017-02-12
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    @ChrisC that does make sense, you're right. And Steve, is that because it is almost "not providing a centripetal force" in a way when looking at the grid with zero curl? – 2017-02-12
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    M.Boas, Mathematical Methods: "Consider an under-current of water passing through a sub-merged water-mill. The net rotation of that mill is also called the Curl of the vector field at that point." – 2017-02-12

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