It makes sense, based on a visual image (for example in Complex Analysis: Selected Topics by Mario Gonzalez page 116) , for a harmonic function $\Omega:\mathbb{R}^2 \rightarrow \mathbb{R}$ that for simple, closed and smooth curves C
$\int_{C}^{} (\frac{\partial \Omega}{\partial x}dy - \frac{\partial \Omega}{\partial y}dx) = 0$
but is that always the case?
As @r9m pointed out in the comments section the assumption in the question follows from Green's theorem.