Let $V$ be an arbitrary $3-$dim volume with smooth boundary $.$ Also let vec$(n)$ be the unit outward normal to $.$ The divergence theorem states that, if vector field in $3$ D is given by

vec$(F)=a(x,y,z)vec(i)+b(x,y,z)vec(j)+c(x,y,z)vec(k)$

we have

${}_V(grad*vec(F))dV={}_{}$vec$(F)*vec(n)dS={}_{}$vec$(F)*dvec(S)$

Now assume that $f$ is a scalar function with continuous partial derivatives. $1)$ Prove

${}_{}$fvec$(n)dS={}_V$gradfdV

A solid occupies a region $V$ with surface $$ and is immersed in a liquid with constant density $.$ We set up a coordinate system so that the $xy-$plane conincides with the surface of the liquid, and positive values of $z$ are measured downward into the liquid. Then the pressure at depth $z$ is $p=gz,$ where $g$ is the acceleration due to gravity. The total buoyant force on the solid due to the pressure distribution is given by the surface integral

vec$(F)=-{}_{}$pvec$(n)dS$

where vec$(n)$ is the unit outward normal. Use the result in part $1)$ to show that vec$(F)=-$Wvec$(k),$ where $W$ is the weight of the liquid displaced by the solid.