If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# 2d curl example

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

## Want to join the conversation?

• Wouldn't it be much easier to simplify the formula to (3x^2)-(3y^2) first? •  Absolutely! To be honest, I'm not really sure why I didn't do that. Just got caught up focussing on the relevant curl interpretation I suppose.
• 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? • 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? • 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.
• 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? • 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.. • What does a higher or lower curl represent? For example, what is the difference between a positive curl of say, 20 and 50? • 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."  