Use Stokes' Theorem $to$ evaluate:

$(a)4$ marks ${}_S$curlvec$(F)*dvec(S)$ where vec$(F)(x,y,z)=z{e}^y$vec$(i)+xcosyvec(j)+$xzsinyvec$(k),$

$Sis$ the hemisphere $x^2+y^2+z^2=16,y0,$ oriented $in$ the direction $of$ the positive

$y-$axis.

$(b)4$ marks $o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r),$ where $Cis$ oriented counterclockwise $as$ viewed from above.

vec$(F)(x,y,z)=(x+y^2)vec(i)+(y+z^2)vec(j)+(z+x^2)vec(k)$

$Cis$ the triangle with vertices $(1,0,0),(0,1,0),$ and $(0,0,1).$