Given a flow field \(\textbf v(\textbf x,t)\), the vorticity \(\boldsymbol{\omega}\) of \(\textbf v\) is defined by taking its curl \(\boldsymbol{\omega}:=\frac{\partial}{\partial\textbf x}\times\textbf v\). For a flow field rotating rigidly with angular velocity vector \(\boldsymbol{\omega}_0\) so that \(\textbf v=\boldsymbol{\omega}_0\times\textbf x\). The vorticity associated with this purely rotational flow is:
\[\boldsymbol{\omega}=\frac{\partial}{\partial\textbf x}\times(\boldsymbol{\omega}_0\times\textbf x)=\boldsymbol{\omega}_0\left(\frac{\partial}{\partial\textbf x}\cdot\textbf x\right)+\left(\textbf x\cdot\frac{\partial}{\partial\textbf x}\right)\boldsymbol{\omega}_0-\textbf x\left(\frac{\partial}{\partial\textbf x}\cdot\boldsymbol{\omega}_0\right)-\left(\boldsymbol{\omega}_0\cdot\frac{\partial}{\partial\textbf x}\right)\textbf x=2\boldsymbol{\omega}_0\]
Thus, \(\boldsymbol{\omega}=2\boldsymbol{\omega}_0\). In other words, it is possible to rewrite the original flow field as \(\textbf v=\boldsymbol{\omega}_0\times\textbf x=\frac{1}{2}\boldsymbol{\omega}\times\textbf x=\frac{1}{2}\left(\frac{\partial}{\partial\textbf x}\times\textbf v\right)\times\textbf x\) (cf. Lamb vector?). Can this also be gotten from the vorticity equation?