$(1$ point$)$ Verify that Stokes' Theorem is true for the vector field $F=yi+zj+xk$ and the surface $S$ the hemisphere $x^2+y^2+z^2=4,y0$ oriented in the direction of the positive $y-$axis.

To verify Stokes' Theorem we will compute the expression on each side. First compute ${}_S$curlF$*dS$

curlF$=$

$$

The surface $S$ can be parametrized by

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

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

$-$ :

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

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