Use the Fundamental Theorem to deduce the formula for the area of an ellipse.

Question

Use the Fundamental Theorem (Green's Theorem) to deduce the formula for the area of an ellipse. Hint: find a 1-form whose exterior derivative is $ dxdy $.

Answer 1

Green's Theorem states that, for a simply connected region $D$:

$$\oint_{\partial D} (P dx + Q dy) = \iint_D \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right )$$

Choose $P=-y$ and $Q=x$ to get for the area $A(D)$

$$A(D) = \frac{1}{2} \oint_{\partial D} (-y \,dx + x \,dy)$$

For an ellipse, $x=a \cos{t}$, $y=b \sin{t}$, and we get

$$A(D) = \frac{1}{2} a b \int_0^{2 \pi} dt (\cos^2{t} + \sin^2{t}) = \pi \,a\,b$$