The Curl of a Vector Field: Local Rotation

The first animation shows the vector field \(\mathbf F=\binom{1-y^2}{0}\) with curl $\nabla\times\mathbf F=2y\,\mathbf k$. The ducks move in straight lines, but rotate at an angular velocity $\pmb\omega=y\,\mathbf k$ depending on their location in the stream.
The remaining animations illustrate local versus global rotation. In all three vector fields the ducks race in concentric circles. The curls are different resulting in different local rotation.
$\mathbf F=\binom{-y}{x},\quad \nabla\times\mathbf F=2\,\mathbf k$ $\mathbf F=\frac 1{x^2+y^2}\binom{-y}{x},\quad \nabla\times\mathbf F=\mathbf 0$ $\mathbf F=\frac 1{(x^2+y^2)^2}\binom{-y}{x},\quad \nabla\times\mathbf F=-\frac 2{(x^2+y^2)^2}\mathbf k$
Counter-clockwise local and global rotation 1 rad/s Counter-clockwise global rotation. No local rotation Counter-clockwise global rotation. Clockwise local rotation