Chapter 14.8, Problem 38E

Logarithmic potential Consider the potential function $\phi \left(x,y,z\right)=\frac{1}{2}\ln \left(x^2+y^2+z^2\right)=\ln \left|\text{r}\right|$, where r = 〈x_, y, z〉.

 a.    Show that the gradient field associated with ϕ is

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

b.    Show that ${\displaystyle {\iint }_{S}F\cdot ndS=4\pi a}$, where S is the surface of a sphere of radius a centered at the origin.

 c.    Compute div F.

d.    Note that F is undefined at the origin, so the Divergence Theorem does not apply directly. Evaluate the volume integral as described in Exercise 37.