Let $~F = \langle x,y,z \rangle~$ be a vector field on $ R^3$ . Let $S$ be the graph of $ g(x,y)=1- y^2 + x^2 $ over the unit disc $D$ in $ R^2$ and let $S$ be equipped with the upward orientation. Calculate the flux of $F$ across $S$.
From my understanding of the question, I used divergence/gauss theorem to get divergence as $3$ & integrate that over the area of the disc $D$, which amounts to $3 \pi $ but I'm not sure if I've understood the part about the graph over the unit disc correctly. Should I be integrating it over the disc only or the surface defin]ed with $g(x,y)$?
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