Let $\vec F(x,y,z)=(x^2+y^2+\sin(xy)$, $e^x$+$2xy, -yz$). Calculate the flow of $\nabla\times\vec F$ through the following surface: $$ S=\{(x,y,z) x^2 + 2y^2 + 3z^2=10, y\ge 0\} $$
I figured that if you added $S_{aux} =\{(x,y,z)\in \Bbb R^3 : x^2 + 3z^2 =10\}$ to S you had a closed surface and therefore could use the divergence theorem. This would mean that:
$$
\iint_{S+S_{aux}}\nabla\times\vec F \mathrm{d}s = \iiint_{V} \nabla\circ(\nabla\times\vec F) \mathrm{d}v.
$$
Since $\nabla\circ(\nabla\times\vec F)=0$, then:
$$
\iiint_{V} \nabla\circ(\nabla\times\vec F)\mathrm{d}v = 0.
$$
Therefore
$$\iint_{S+S_{aux}}\nabla\times\vec F \mathrm{d}s = 0,
$$
and since
$$\iint_{S+S_{aux}}\nabla\times\vec F \mathrm{d}s = \iint_S\nabla\times\vec F \mathrm{d}s + \iint_{S_{aux}}\nabla\times\vec F \mathrm{d}s = 0$$ we have $$\iint_S\nabla\times\vec F \mathrm{d}s = -\iint_{S_{aux}}\nabla\times\vec F \mathrm{d}s.
$$
Could you please tell me if my thinking is correct? Thank you very much!
I'm certain that the point of the problem is to use Stokes' theorem. $\partial S$ is the intersection of the plane $y=0$ with $S$ so it's the ellipse $x^2+3z^2=10, y=0.$ On this curve, $F$ simplifies to $(x^2, e^x, 0)$ When you compute the circulation of $F$ over $\partial S$ there will only be a contibution from the first coordinate.