Past exam of the mathematical methods II course of the University of Oxford / 2019 / 8 / a

Let the angular velocity vector be the spatially constant $\mathbf{\omega }$. Choose an origin on the fixed rotation axis. The velocity field of a rigid body is $v=\mathbf{\omega }\times r$. In components,Taking the curl givesAn 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 $\frac{1}{2}\nabla \times v$.