Defining a vector field f and a parametrized curve r in the plane:

M1[x_,y_]:= 2x
N1[x_,y_]:=-3y
f1[x_,y_]:= {M1[x,y],N1[x,y]}

g[t_]:=2Cos[t]
h[t_]:=2Sin[t]
r[t_]:= {g[t],h[t]}
v[t_]:=r'[t]

Plotting the vector field and the curve in the plane:

Show[VectorPlot[f1[x,y],{x,-4,4},{y,-4,4}],
ParametricPlot[r[t],{t,0,2Pi}]]

Computing Circulation

circulationintegrand1=f1[g[t],h[t]].v[t]//FullSimplify
Integrate[circulationintegrand1,{t,0,2Pi}]

Computing Flux

fluxintegrand1=M1[g[t],h[t]]*h'[t]-N1[g[t],h[t]]*g'[t]//FullSimplify
Integrate[fluxintegrand1,{t,0,2Pi}]

A second example: The shear vector field with the same curve

M2[x_,y_]:= y
N2[x_,y_]:=0
f2[x_,y_]:= {M2[x,y],N2[x,y]}

Show[VectorPlot[f2[x,y],{x,-4,4},{y,-4,4}],
ParametricPlot[r[t],{t,0,2Pi}]]

circulationintegrand=f2[g[t],h[t]].v[t]//FullSimplify

Integrate[circulationintegrand,{t,0,2Pi}]

fluxintegrand=M2[g[t],h[t]]h'[t]-N2[g[t],h[t]]g'[t]//FullSimplify

Integrate[fluxintegrand,{t,0,2Pi}]