(* Line integrals *)
L[v_,g_,a_,b_]:=Module[{},u[t_]:=D[g[s],s] /. s->t; NIntegrate[v[g[t]].u[t],{t,a,b}]];

(* Example: v0 is vector field w. curl 1, g0 is unit circle path. Lineintegral is 1 *)

v0[{x_,y_}]:={-y,0}; g0[t_]:={Cos[t],Sin[t]}; L[v0,g0,0,2Pi]