Integral over ellipsoid

Question

Let $\Gamma$ be the ellipsoid $\{(x,y,z) \in \Bbb{R}^3 : x^2+\frac{y^2}{4}+\frac{z^2}{9}=1\}$. Let $\omega=z dx \wedge dy - y dz \wedge dx$ and compute $\int_\Gamma \omega$. So I thought about parameterizing the ellispoid and doing spherical coordinates but they have different radii so I was wondering how Id go about this directly? Would my $x$ bounds be $0$ to $1$, my $y$ bounds be $0$ to $2$ and same for $z$? And for $z$ in the integral so I wolve for $z$ in the ellipsoids equation and same for $x$ and $y$?

Answer 1

Too long for a comment. Since there are still questions unanswered in the comment section:

The two-form you are asked to inegrate is $$ \omega=z\,dx \wedge dy - y\,dz \wedge dx\,. $$ You should be able to see that your $d\omega$ is zero. If not ask a specific question about this calculation. Since this $\omega$ is defined in all of $\mathbb R^3$ the Poincare lemma is more than applicable and tell us that $\omega$ is not only closed but also exact. The one-form $\eta$ such that $d\eta=\omega$ is obviously

$$\eta=-zy\,dx$$

We do not even have to know it explicitly. Since $\Gamma$ is a closed surface its manifold boundary is the empty set and Stokes $$ \int_\Gamma\omega=\int_\Gamma\,d\eta=\int_{\partial \Gamma}\eta =\int_{\emptyset}\eta $$ tells us what?