  Two dimensional flows on fixed smooth surfaces have been studied in the point
of view of vorticity dynamics. Firstly, the related deformation theory
including kinematics and kinetics is developed. Secondly, some primary
relations in vorticity dynamics have been extended to two dimensional flows on
fixed smooth surface through which a theoretical framework of vorticity
dynamics have been set up, mainly including governing equation of vorticity,
Lagrange theorem on vorticity, Caswell formula on strain tensor and stream
function & vorticity algorithm with pressure Possion equation for
incompressible flows. The newly developed theory is characterized by the
appearances of surface curvatures in some primary relations and governing
equations.
