Answer
$0$
Work Step by Step
The tangential form for Green Theorem is given as:
Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy$
So, $\dfrac{\partial M}{\partial y}=\dfrac{\partial^2 f}{\partial y^2}$ and $\dfrac{\partial N}{\partial x}=- \dfrac{\partial^2 f}{\partial x^2}$
Thus, $\oint_C \dfrac{\partial f}{\partial y} \ dx - \dfrac{\partial f}{\partial x} \ dy=\iint_{R} (- \dfrac{\partial^2 f}{\partial x^2} - \dfrac{\partial^2 f}{\partial y^2}) \ dx \ dy =0$
