Evaluate $\iint_s\text{curl}\textbf F\cdot \textbf {n}dS$

Question

Let $\textbf F=<xy,yz,zx>$ and $S$ be the upper half of the ellipsoid $\displaystyle \frac {x^2}{4}+\frac {y^2}{9}+z^2=1$. Evaluate $\iint_s\text {curl}\textbf F\cdot \textbf {n}dS$

I know the procedure to solve this question, but I can't think of a parametrization of $C$, the boundary curve.

Hints are appreciated, thanks!

Answer 1

Even easier, what is the flux of $\text{curl}\,\mathbf F$ across the filled-in ellipse? But, to answer your question, an ellipse is a stretched circle, so try $x=a\cos t$, $y=b\sin t$ for appropriate $a$ and $b$.