 Solution

Assume the stated continuous derivatives on an open simply connected region. If $N_x-M_y=0$, Green's theorem gives zero circulation around every simple closed curve and its enclosed region; a piecewise smooth closed curve can be decomposed into such loops, giving the same conclusion.

Conversely, suppose every closed-curve integral vanishes. Apply Green's theorem to an arbitrarily small disk about any interior point $P$. Its integral of $N_x-M_y$ is zero. Divide by the disk area and shrink its radius to zero; continuity makes the average approach $(N_x-M_y)(P)$, so that value must vanish. ThusSimple connectivity is needed for the global sufficiency statement, ensuring that a loop does not enclose a missing portion of the domain. Local necessity already follows from small disks.