The velocity of a rigid body with angular velocity \(\boldsymbol\omega=\langle 1,2,3\rangle\) is \(\mathbf v=\boldsymbol\omega\times\langle x,y,z\rangle\). Its curl is:
Expanding, \(\mathbf v=\langle 2z-3y,\,3x-z,\,y-2x\rangle\), and its cross differences give \(\langle 2,4,6\rangle\). In general \(\nabla\times(\boldsymbol\omega\times\mathbf r)=2\boldsymbol\omega\): the curl is twice the angular velocity.
The curl of a vector field \(\mathbf F=\langle P,Q,R\rangle\) is the vector field that measures, at each point, the field's tendency to rotate, with axis and rate of rotation given by its direction and length.