Homework $6-$ Optional: Problem $42$

$(1$ point$)$

Let $M$ be the capped cylindrical surface which is the union of two surfaces, a cylinder given by $x^2+y^2=16,0z1,$ and a hemispherical cap defined by $x^2+y^2+(z-1{)}^2=16,z1.$ For the vector field $F=(zx+z^{2}y+3y,z^{3}yx+2x,z^4{x}^2),$ compute ${}_M(gradF)*dS$ in any way you like.

${}_M(gradF)*dS=$