Learning vector calculus and I'm still confused over what the curl represents for a vector field. It is stated that the curl represents the magnitude of rotation of surrounding vectors to a given point. But which direction does it point?
Let's say we were to use the field $\vec{F}(x,y,z)=\langle-y,x,0\rangle$, which gives us the velocity vector of a particle moving on the path $x^2+y^2=r^2$ with speed $r$ counter-clockwise on the x-y plane with the plane translated upwards for any $z$.
Intuitively by the argument of rotation, we have circular rotation over any point on the z-axis with the same axis of rotation, while for any other point there doesn't appear to be the same circular behavior surrounding that point, so I would expect the curl to return a different quantity.
Yet we obtain
$$\nabla\times\vec{F}=\langle0,0,2\rangle$$
a constant vector for any point. If the behavior of the vector field around a point is different for points on the z-axis and any other point, yet the curl yields the same vector, what exactly does this mean?
If you put an object in this field (just for simplicity, suppose it's a thin bar), then it would experience rotation at the same rate, no matter where it is placed. It's this local rotation-causing torque we're measuring with the curl.
The field may also cause the object to travel in a circle, but that isn't itself what the curl measures, even though you can use an integral to connect the two phenomena.