$(1$ point$)$ Use Stokes' Theorem to evaluate ${}_S$curlF$*dS$

where $F(x,y,z)=($:$2ycos(z),-e^{x}sin(z),4x{e}^y$:$)$ and $S$ is the hemisphere $x^2+y^2+z^2=4,z0,$ oriented upwards.

Since the hemisphere is oriented upwards the boundary curve must be transversed counter$-$clockwise $$ when viewed from above.

A parametrization for the boundary curve $C$ can be given by:

$r(t)=2cos(t)i+$ j$+k,0t2$pi

$($use the most natural parametrization$)$

${}_S$curlF$*dS={\displaystyle \underset{0}{\overset{b}{}}}$

${}_S$curlF$*dS=$