Answer
$$0$$
Work Step by Step
The tangential form for Green's Theorem can be written as:
Counterclockwise Circulation:
$\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy$
Now $\dfrac{\partial^2 f}{\partial y^2} =\dfrac{\partial M}{\partial y}$ and $- \dfrac{\partial^2 f}{\partial x^2} =\dfrac{\partial N}{\partial x}$
Now $\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 $
or, $\oint_C \dfrac{\partial f}{\partial y} \ dx - \dfrac{\partial f}{\partial x} \ dy=0$
