Explain how to relate the integral curvilinear of field vector $\displaystyle F(x,y) = (x-y)i + (x+y)j$ with the area the region limited by a regular curve, simple and closed ? Apply the result and calculate the area of the ellipse:

$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$