Chapter 14.8, Problem 37E

Singular radial field Consider the radial field

$F=\frac{r}{\left|r\right|}=\frac{\langle x,y,z\rangle }{{\left(x^2+y^2+z^2\right)}^{1/2}}.$

a.    Evaluate a surface integral to show that ${\displaystyle {\iint }_{S}F\cdot ndS=4\pi a^2}$, where S is the surface of a sphere of radius a centered at the origin.

b.    Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a) is finite. Evaluate the triple integral of the Divergence Theorem as an improper integral as follows. Integrate div F over the region between two spheres of radius a and 0 < g < a. Then let g →0⁺ to obtain the flux computed in part (a).