$(10$ points$)$ Let vec$(C)be$ the curve formed $by$ the intersection $of$ the plane

$y-z=2$

and the ellipsoid

$2x^2+y^2+z^2+4y=6$

oriented $in$ the standard $(counterclockwise)$ direction when viewed from above. Let

vec$(F)(x,y,z)=($:$yz,-zx,xy$:$)$

$be$ a vector field $in{R}^3.$ Use the Stokes' theorem $to$ evaluate the line integral