= 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. Thus
$$
\boxed{\oint_C M\,dx+N\,dy=0\text{ for every closed }C\quad\Longleftrightarrow\quad N_x=M_y.}
$$
Simple 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.
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