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 nuance

The meaning of positive curl in a fluid flow can sometimes look a bit different from the clear cut rotation-around-a-point examples discussed in previous videos. Created by Grant Sanderson.

## Want to join the conversation?

• The video's title is nuance but Grant never says what Nuance is. Can I get an exact definition of 2D Curl Nuance please?
• Nuance is just a noun. The title of the video is "2D curl nuance", which means the essence of 2D curl. Nuance isn't a mathematical term.
• in the video, where does the forumla [-y x] come from?
• It's the example equation he used for the fluid flow diagram in the video.
• Although the paddle wheel idea is a great educational tool for curl, I can't help but feel that it doesn't really hold true in a real-world setting (hence, Grant's comment of counterintuitiveness). Just curious. Maybe I'm wrong. Any thoughts?

(Thanks for the videos Grant!)
• Introduction of the paddle wheel would change the local flow significantly in a real life example. (Thus changing the problem definition) And that change to the flow would be different for different points in the flow field. However, assuming that the flow is exactly as given by the formula in the example, and unaffected by the introduction of the paddle wheel, the paddle in the center would rotate with the same angular velocity anywhere in the flow field.
• How does one find the "center" of curl (the point the fluid is rotating about)? For example, the fluid in the video rotates about the origin, but the curl is the same regardless of what point it's evaluated at. Is there another way to determine that the origin is the "center"? Also, what if there are multiple "centers"?
• My best guess would be to take that function which we can use to find the rotation:

2d-curl(v) = ∂Q/∂x - ∂P/∂y

And that is but a function of x,y itself. And the defining property of the 'centre' you describe (If I am not mistaken) is the local maximum of that multivariable function, the place where we have that x,y function most fitting to the perfect example!

This must mean that you just have to find the point on that x,y function [ let's call it U(x,y) ] where

∂U/∂x = 0,
∂U/∂y = 0.

Which implies
∂U/∂x + ∂U/∂y = 0.

The expression on the right is again a multivariable function, find where it is equal to zero, and you have all your centres.

P.S. this was all on the spot typing and thinking, so if I am wrong, please correct me.
• Up till now the 2D curl has given us an intuitive concept about the direction of rotation. But I want to know how curl might give us a quantitative approach in measuring the magnitude of rotation?
• I don't understand why the rotations of the two regions are the same visually.
(1 vote)
• got a bit of a question regarding how the curl would be shaped if both dq/dx and dp/dy were positive but dp/dy was lesser in magnitude than dq/dx. Would this not imply a positive curl but result in a shape different from that typical counterclockwise rotation? As x goes up, the flow of the field will push things up because of the dq/dx being positive, but then would that increase in vertical distance push things to the right and result in just an infinitely long, not very curly part of the vector field?

I'm mostly asking cuz I have a pretty hard time visualizing these things.

Also kinda curious how zero dp/dy would work, as it should just mean the function does not move to the right or left at all with a change in height which kinda just... means no curling at all and just straight lines. How would you still have positive curl with that?

This could just be a massive misinterpretation of what curl really is, but it doesn't feel right to call it a positive curl if it doesn't curl. Does the formula for curl only apply as a means to measure that curl and not identify if there is any curling?
(1 vote)
• Maybe a dumb question but wanted to know what is the meaning of the blue dots moving in the animation?
Does it mean coordinates increasing?
(1 vote)