= Solution
The <curl> gives the <vorticity>:
$$
\boldsymbol\nabla\times\mathbf v
=\bigl(-\partial_z(\alpha xz),\ \partial_z(-\alpha yz),\
\partial_x(\alpha xz)-\partial_y(-\alpha yz)\bigr)
=(-\alpha x,-\alpha y,2\alpha z).
$$
Since $x\mathbf i+y\mathbf j=\rho\widehat{\boldsymbol\rho}$, the requested form is
$$
\boxed{\boldsymbol\nabla\times\mathbf v
=-\alpha\rho\widehat{\boldsymbol\rho}+2\alpha z\mathbf k.}
$$
At the axis the radial <unit vector> is undefined, but its coefficient vanishes and the Cartesian expression remains smooth. The radial <vorticity> reflects the change in rotation rate with height.
Back to article page
