Use Stokes' Theorem to evaluate ${}_s$ curl $F*$ ds$.$

$F(x,y,z)=3ycos(z)1e^{x}sin(z)jx{e}^{y}k$

$Sis$ the hemisphere $x^2{y}^2{z}^2=16,z0,$ oriented upward. Stetes' Theorem ters us that if C is the boundary curve of a perface S$.,$ then

${}_s$curfF$*ds={\displaystyle \underset{c}{\overset{\phantom{}}{}}}F*dr$

Since $S$ is the hemisphere $x^2{y}^2{z}^2=16,z0$ oriented upward, then the boundary curve $C$ is the circle in the $x{y}^{\text{'}}-$plane, $x^2{y}^2=,z=0,$ oriented in the counterclockwise direction when viewed from above. A vector cequation of C ts

$r(t)=cos(t)i,sin(t)j0k$

with $0t2.$