Question: Let \(\mathrm{S}\) be the surface formed by the part of the paraboloid \(z=1-x^{2}-y^{2}\) lying above the xy-plane and let \(\mathrm{F}=x \mathrm{i}+y \mathbf{j}+2(1-z) \mathrm{k}\). Calculate the flux of \(\mathrm{F}\) across \(S\), taking the upward direction as the one for which the flux is positive. Do this in two ways:

1. (5 points) by direct calculation of \(\iint_{S} \mathbf{F} \cdot \mathrm{n} d S\);
2. (5 points) by computing the flux of \(\mathrm{F}\) across a simpler surface and using the divergence theorem.

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