(* 2D vector field *)
F={x^2+y^3,x^4+y^5};
(* a circle around (x,y) with radiu r *)
q={x+r Cos[t],y+r Sin[t]};
Foq=F/.{x->q[[1]],y->q[[2]]}
qt=D[q,t]
qtperp={qt[[2]],-qt[[1]]}
area=r^2 Pi;

(* flux density *)
flux=Integrate[Dot[Foq,qtperp],{t,0,2 Pi}] ;
fluxdens=Limit[Expand[flux/area],r->0]

(* circulation density *)
circ=Integrate[Dot[Foq,qt],{t,0,2 Pi}] ;
circdens=Limit[Expand[circ/area],r->0]

VectorPlot[F,{x,-1,1},{y,-1,1},VectorScaling->True]
ContourPlot[fluxdens,{x,-1,1},{y,-1,1}]
ContourPlot[circdens,{x,-1,1},{y,-1,1}]