= Solution
Let the <angular velocity> vector be the spatially constant $\boldsymbol\omega$. Choose an origin on the fixed rotation axis. The <velocity field> of a <rigid body> is $v=\boldsymbol\omega\times r$. In components,
$$
v=(\omega_yz-\omega_zy,\ \omega_zx-\omega_xz,\ \omega_xy-\omega_yx).
$$
Taking the <curl> gives
$$
\boxed{\nabla\times v=(2\omega_x,2\omega_y,2\omega_z)=2\boldsymbol\omega.}
$$
An added spatially uniform translational <velocity> has zero <curl>, so it does not affect this relation. Equivalently, the <vorticity> of rigid rotation is twice the <angular velocity>; the local rotational part of a general smooth flow has <angular velocity> $\tfrac12\nabla\times v$.
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