I know the proof of the Green Riemann identity $$\int P dx+ Q dy=\int\int\big[\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\big] dx\ dy$$ on a rectangle. Could someone give me a detailed proof of the general case (a $C^1$ connected domain in $\mathbb R^2$, or $\mathbb C$)? I thought of an approximation technique, but I would like to know a canonical proof in order to see which ideas are developed.
My idea was to approximate the boundary with rectangles, and doing this by using the implicit function theorem (one should always be able to do this with a change of coordinates): is this always possible (and successful)?
Thank you in advance.