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Let alpha=xdy-ydx and let M be a compact domain in the plane R^2. Show that the integral along the boundary of M of alpha is twice the surface area of M.

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Use Green's theorem:

$$\int_{\partial M} xdy - ydx = \int_M d( xdy - ydx) = \int_M dx \wedge dy - dy \wedge dx = 2 \int_M dx \wedge dy $$The last integral is the surface area of $M$.

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