What Is Curl in Vector Fields?

[Nikitin]

define curl "rotation per area"

When they define curl, they say it is a measure of "infinitesimal rotation", or "rotation per area".

What does that mean? Does it mean they measure how much something goes around in an infinitesimal point (which makes no sense), kind of like a whirlwind shrunk down? Or do they mean its the measure of how much something changes its direction, or "bends", at any point?

Second, can you guys give me an intuitive explanation for the following formula?

"The k-component of the curl of a vector field F = M*i + N*j at the point (x,y) is the scalar:
(Curl F)*k = ∂N/∂x - ∂M/∂y"

I sense that this formula represents how much the M and N components change direction, ie how much they bend, but I cannot get it down on paper..

Thanks for any help

 

[DrZoidberg]

I can give you an example from physics.
Curl can be defined for an arbitary number of dimensions. I'm going to do this in 2 dimensions to make it easier to describe. So imagine you are in a 2 dimensional universe.
Let's say you have a liquid and there are turbulences in the flow of that liquid. That flow can be represented as a vector field where each vector represents the velocity of the liquid at that point. Now draw a circle anywhere in the liquid. That circle is not going to affect the flow. It's just an imaginary circle. At the surface of that circle the liquid may flow in different directions. Take the component of the liquids velocity that's tangential to the circle and integrate it over the circumference. What you got now is the integral of the curl over the entire area of the circle. If you want the curl at a specific point you need to make the circle infinitesimally small.
The curl is also important in electrical engineering. In a transfomer you have a circular electric field going around the iron core. A loop of wire that's wound around the core will therefore have a voltage induced that's equivalent to the integral of the curl of the electric field inside the loop.

 

[Nikitin]

Thanks! could you offer insight on why the formula Curl(F)=∇×F/formula in my OP makes sense?

 

[HallsofIvy]

Any mention of "infinitesmal" in basic Caluculus is shorthand for a limit. It is essentially saying that if you calculated the "amount of motion" in a circle of radius r, then took the limit as r goes to 0, you would get the curl.