
The animation illustrates the core of the proof of Greens theorem. The circulation around a small square is the differential quotient approximation of the curl of F. If you add up all the circulations, then only the boundary circulation survives due to cancellations in the interior. In the limit, when the squares become smaller and smaller, the sum of circulations becomes the double integral of the curl over the region. The total circulation is the line integral along the boundary. 
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Oliver Knill, Math21a, Multivariable Calculus, Fall 2005, Department of Mathematics, Faculty of Art and Sciences, Harvard University 