Answer
$\int_Cf_y dx-f_x dy=0$ and independent of the path.
Work Step by Step
Our aim to show that $\int_Cf_y dx-f_x dy=0$, which is independent of the path.
Here, we have $f$, a harmonic function.
Apply Green's Theorem as: $\int_C adx+b dy=\iint_D (\dfrac{\partial b}{\partial x}-\dfrac{\partial a}{\partial y})dA$
where, $D$ shows the region enclosed inside the counter-clockwise oriented loop $C$.
$\int_Cf_y dx+(-f_x) dy=\iint_D (\dfrac{\partial f_x}{\partial x}-\dfrac{\partial (-f_y)}{\partial y})dA=\iint_D(\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2})dA $
Therefore,
$\int_Cf_y dx-f_x dy=\iint_D (0)dA=0$
It can be found that $\int_Cf_y dx-f_x dy$ is independent of the path in any simple region $D$.
