$(16$ points$)$ Let F$($x$,$y$,$z$)=$x$^(2)+$y$^(2)-$z$^(2).$ Let S be the surface given as the solution

set of the equation F$($x$,$y$,$z$)=8$S$,$T$\_($p$)$S at the point

p$=(2,2,0).$

$($b$)(4$ points$)$ Let c$($t$)=(3$cost$,3$sinttpi $.$ Show that the curve c lies

in the surface S$.($c$)(6$ points$)$ Show that the velocity vector c$^(\text{'})($t$)$ of the curve from part $($b$)$ is per$-$

pendicular to the gradient vector gradF$($c$($t$)).$