Exercise 12

Sometimes, the value in Green's Theorem is in converting a double integral into a line integral. Recall that the area of a region $R$ in the plane can be found by calculating $\iint_R 1\, dA$.

1. Find a smooth vector field $\vF$ such that the circulation density of $\vF$ is $1$ everywhere.

2. Write out both integrals in Green's Theorem for the vector field you selected in part.

3. For positive real numbers $a,b$, the ellipse $x^2/a^2 + y^2/b^2 = 1$ can be parametrized as $\vr(t) = \langle a\cos(t),b\sin(t)\rangle$ with $0\leq t\leq 2\pi$. Find the area of this ellipse by calculating a line integral.

4. Find another vector field $\vG$ (different from the one you found in part) that has circulation density $1$ everywhere, and calculate the area of the ellipse in part by calculating the circulation of $\vG$ along the ellipse.

5. How many possible vector fields are there with circulation density $1$ everywhere? Why?