= Solution
{c}
The <curl> is $\nabla\times\mathbf F=(x,y,-2z)$. For the upward orientation of the paraboloid, the <vector surface area element> is $(2x,2y,1)\,dx\,dy$. Therefore <Stokes theorem> gives
$$
\begin{aligned}
\oint_C\mathbf F\cdot d\mathbf r
&=\iint_{x,y\geq0,\ x^2+y^2\leq4}\{2x^2+2y^2-2(4-x^2-y^2)\}\,dx\,dy\\
&=\int_0^{\pi/2}\int_0^2(4r^2-8)r\,dr\,d\theta
=\boxed{0}.
\end{aligned}
$$
The upward normal gives the prescribed anticlockwise boundary orientation viewed from above.
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