Question: Verify Stoke’s Theorem by evaluating \[\iint\limits_{C}{\ F\ \cdot \ T\ d\,S}\] as a line integral:

\[\begin{aligned} & F\,\left( \,x,\ y,\ z\, \right)\quad =\quad \left( \,-\,y\ +\ z\, \right)\ i\quad +\quad \left( \,x\ -\ z\, \right)\ j\quad +\quad \left( \,x\ -\ y\, \right)\ k \\ & \\ & S:\quad z\ =\ 4\ -\ {{x}^{\,2}}\ -\ {{y}^{\,2}},\ 0\ \le \ z \\ \end{aligned}\]