Fixed solid bondan o a b nx=g D Figure 2: Sketch of local parametrization of a patch of surface, Dl,2, D=l=2 and Drri denote parametric domains with respect to coordinates x, 22),a, 22) and (z, r) respectively 4 Conclusions Basically, some primary relations in general theory of vorticity dynamics have been extended to surface tensor fields based on Levi-Civita connection operator. Subsequently, the governing equation of vorticity, Lagrange theorem on vorticity, Caswell formula on strain tensor are at tained. Furthermore the stream function vorticity algorithm with pressure Possion equation has been set up for incompressible Hows. It has been revealed that Gaussian and mean cur vatures are taking part directly in some governing equations but all of the effects due to the geometry of the surface will disappear as flows on general fixed surfaces degenerates to ones or the plane. All of the theoretical results presented in this paper are exactly deduced without any approximation and constitute a theoretical framework for two dimensional flows on general fixed smooth surfaces Acknowledgements This work is supported by National Nature Science Foundation of China(Grant No. 11172069 and some key projects of education reforms issued by the Shanghai Municipal Education Com- mission(2011) Appendix 1. Construction of local curvilinear coordinates 3,J As sketched in Figure 2, one can suppose without lost of generality that the preimage of a segment of boundary on the surface in the parametric space [ x, 2) can be represented locally (a,b) v(x2) Then the following coordinates can be constructed locall (x2,x2)= The segment of the boundary is corresponding to il E(a, b)and 12=0. The local co-variant basis with respect to [_ is denoted by 9:12_. Thirdly, one introduces another coordinatesFigure 2: Sketch of local parametrization of a patch of surface, Dx1x2 , Dx˜ 1x˜ 2 and Dxqx⊥ denote parametric domains with respect to coordinates {x 1 , x2}, {x˜ 1 , x˜ 2} and {x q , x⊥} respectively. 4 Conclusions Basically, some primary relations in general theory of vorticity dynamics have been extended to surface tensor fields based on Levi-Civita connection operator. Subsequently, the governing equation of vorticity, Lagrange theorem on vorticity, Caswell formula on strain tensor are at￾tained. Furthermore, the stream function & vorticity algorithm with pressure Possion equation has been set up for incompressible flows. It has been revealed that Gaussian and mean cur￾vatures are taking part directly in some governing equations but all of the effects due to the geometry of the surface will disappear as flows on general fixed surfaces degenerates to ones on the plane. All of the theoretical results presented in this paper are exactly deduced without any approximation and constitute a theoretical framework for two dimensional flows on general fixed smooth surfaces. Acknowledgements This work is supported by National Nature Science Foundation of China (Grant No.11172069) and some key projects of education reforms issued by the Shanghai Municipal Education Com￾mission (2011). Appendix 1. Construction of local curvilinear coordinates {x q , x⊥} As sketched in Figure 2, one can suppose without lost of generality that the preimage of a segment of boundary on the surface in the parametric space {x 1 , x2} can be represented locally as (a, b) ∋ x 1 7→ [ x 1 x 2 ] (x 1 ) = [ x 1 ψ(x 1 ) ] Then the following coordinates can be constructed locally [ x 1 x 2 ] 7→ [ x˜ 1 x˜ 2 ] (x 1 , x2 ) = [ x 1 x 2 − ψ(x 1 ) ] The segment of the boundary is corresponding to ˜x 1 ∈ (a, b) and ˜x 2 = 0. The local co-variant basis with respect to {x˜ i} 2 i=1 is denoted by {g˜i} 2 i=1. Thirdly, one introduces another coordinates 10