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# 2d curl intuition

A description of how vector fields relate to fluid rotation, laying the intuition for what the operation of curl represents. Created by Grant Sanderson.

## Want to join the conversation?

• Why is positive curl anti-clockwise while negative curl's clockwise?
Is it more usefull in other fields of study that the positive curl is anti-clockwise, or is it just contrived convention? • Counterclockwise is defined as positive curl for the same reason the cross product is defined as it is (the right hand rule -- the cross product of i and j is k). For example, torque is the cross product of the arm and force (I'll just use x for cross product). So tau = r x F. If r points in the x direction and F points in the y direction, then tau is in the positive z direction, by the definition of cross product. But as we know, the torque is counterclockwise. So positive torque means counterclockwise.

So if one vector points forward, and another vector drags it to the left, the cross product is pointing up. Up means positive. Yet another way to say the cross product points up is to say the curl is counterclockwise. So counterclockwise is positive curl.

The convention of counterclockwise being positive and cross product using the right hand rule are dependent on each other. If you want to change one of them, you have to change the other.
• Please could you tell me the formula for this particular vector field so I can simulate on mathematica? • Why do I feel like Grant made a mistake with clockwise/counterclockwise rotations? The one that's clockwise rotation was called "counterclockwise", and vice versa! Could I be wrong? • What I don't understand about this concept is that a vector is defined as having a length. So why are some of the molecules travelling half way along a vector then changing direction supposedly following another vector that is in the field but not in the image? Why doesn't the molecule follow the vector it is currently on for the entirety of its length and then change direction?
(1 vote) • It can also depend on what the vectors are representing. In this case, unless I am wrong, the vectors represent the velocity of every particle, so each particle doesn't have to follow each vector from tail to point, as it just represent the direction to which particles travel on that instant, and the magnitude of the vector, aka the length, is just the speed that particles have exactly when passing by that point in space.
So actually when a particle changes its location by an infinitesimal amount in some direction, it's already subject to a new velocity represented by another vector.  