Explain step by step and show all work. Question $2.$ Let $S$ be a simple closed surface in $R^3$ that encloses a solid region with volume equal to $22.$

Find the flux of the vector field

$F(x,y,z)=(x+sin(yz),f(x,y,z)+y,cos(xy)-4z-f(x,y,z))$

across the surface $S$ in the inward direction, where $f(x,y,z)$ is an arbitrary smooth function such that gradf is perpendicular to the vector $(0,1,-1).$