Chapter 14.8, Problem 47E

A beautiful flux integral Consider the potential function ϕ(x, y, z) = G(p), where G is any twice differentiable function and $\rho =\sqrt{x^2+y^2+z^2}$; therefore, G depends only on the distance from the origin.

a.    Show that the gradient vector field associated with ϕ  is $F=\nabla \phi =G^{\prime }\left(\rho \right)\frac{r}{\rho }$, where r = 〈x, y, z〉 and ρ = |r|.

b.    Let S be the sphere of radius a centered at the origin and let D be the region enclosed by S. Show that the flux of F across S is ${\displaystyle {\iint }_{s}F\cdot ndS=4\pi a^2{G}^{\prime }\left(a\right)}$.

  c.    Show that $\nabla \cdot F=\nabla \cdot \nabla \phi =\frac{2G^{\prime }\left(\rho \right)}{\rho }+G^{″}\left(\rho \right)$.

d.    Use part (c) to show that the flux across S (as given in part (b)) is also obtained by the volume integral ${\displaystyle {\iiint }_D\nabla \cdot FdV}$. (Hint: use spherical coordinates and integrate by parts.)