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Suppose you are looking at a field [m[x,y],n[x,y]] which has one singualrity at a point P...

Suppose you are looking at a field [m[x,y],n[x,y]] which has one singualrity at a point P and no other singularities. While studying the singulairty, you center a circle of radius r on the point P and no other singularites. While studying the singularity, you center a circle of radius r on the point P and parameterize this circle in the counter-clockwise direction as Cr;{x[t],y[t]}. You calculate (integralCr) -n[x[t],y[t]]x'[t] + m[x[t], y[t]]y'[t]dt and find that it is equal to -1 + e^(r^2)

What is this integral telling you about the vector field at the point P

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