Prove. If v is the velocity at a point P(x, y, z) in a moving fluid, the vector angular velocity of an infinitesimal portion of the fluid about P is given by ω = ½ curl v.
Proof. A rigid body rotates about a fixed axis. Set up a Cartesian coordinate system with the x axis coinciding with the axis of rotation, origin at any selected point and y axis in any selected direction. Let r be the position vector of any infinitesimal particle P. Then the velocity v of P is given by v = ω r where ω is the instantaneous vector angular velocity (a vector quantity directed along the axis of rotation). Thus
ω = ½ curl v