Answer
The line integral $\int_Cf_y dx-f_x dy$ is independent of the path in any simple region $D$
Work Step by Step
We need to show that $\int_Cf_y dx-f_x dy=0$, which is independent of the path.
Here, $f$ is a harmonic function.
Green's Theorem states that $\int_C Adx+B dy=\iint_D (\dfrac{\partial B}{\partial x}-\dfrac{\partial A}{\partial y})dA$
Here, $D$ is the region enclosed inside the counter-clockwise oriented loop $C$.
Here, we have $\nabla^2 f=0 \implies \nabla(\nabla f)=0$
This gives: $\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}=0$ ....(1)
Now, we have $\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 $
From the equation (1), we have
$\int_Cf_y dx-f_x dy=\iint_D (0)dA=0$
We can conclude that the line integral $\int_Cf_y dx-f_x dy$ is independent of the path in any simple region $D$
Hence, the result has been proved.
