$2.$ The figure below shows the outwardly oriented surface $\backslash ($ S $\backslash ),$ which is a sphere of radius $5$ centered at the origin, with the top cut off, so the upper edge of the surface lies at $\backslash ($ z$=4\backslash ).$

$($a$)$ Explain which of the theorems from Problem $1$ can be applied to determine the curl or circulation of the vector field $\backslash (\backslash$mathbf$\{$F$\}\backslash )$ along the surface $\backslash ($ S $\backslash ).$

$($b$)$ Use your selected theorem from part $($a$)$ to evaluate $\backslash (\backslash$iint$\_\{$S$\}\backslash$operatorname$\{$curl$\}\backslash$mathbf$\{$F$\}\backslash$cdot d $\backslash$mathbf$\{$S$\}\backslash ),$ where $\backslash (\backslash$mathbf$\{$F$\}=\backslash$langle y$,-$x$,$ z$\backslash$rangle $\backslash ).$