$(1$ point$)$ Verify that Stokes' Theorem is true for the vector field $F=4$yzi$-yj+3xk$ and the surface $S$ the part of the paraboloid $z=6-x^2-y^2$ that lies above the plane $z=2,$ oriented upwards. To verify Stokes' Theorem we will compute the expression on each side. First compute ${}_S$curlF$*dS$

$$

where

$x_1=y_1=$

$x_2=$

$y_2=$

${}_S$curlF$*dS=$

$$

Now compute ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr$

The boundary curve $C$ of the surface $S$ can be parametrized by: $,$

$$ :

$($Use the most natural parametrization$)$

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr={\displaystyle \underset{0}{\overset{2}{}}}$

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr=$

$dt$

$$