I am trying to calculate the flux through the unit sphere centered at the origin given a vector field $F:\mathbb{R}^3 \setminus \{(0,0,0)\} \rightarrow \mathbb{R}^3$ with $\operatorname{div} F=1/(x^2+y^2+z^2)$.
I can't apply the divergence theorem directly because of the discontinuity so what I have done instead is to consider an inner sphere of infinitesimal radius $\epsilon>0$ and write
$\iiint_V \operatorname{div} F dV$= (flux through unit sphere)-(flux through inner sphere).
However, I really don't know how to go on from here.
Can I prove that the flux through the sphere of radius $\epsilon$ is $0$ or is this not even true?
I would appreciate some help.
Best Answer
For points ${\bf x}=(x_1,x_2,x_3)\in{\mathbb R}^3$ write $|{\bf x}|=:r$, and consider the vector field $${\bf v}({\bf x}):=\left({c\over r^2}+{1\over r}\right)\>{{\bf x}\over r}\qquad({\bf x}\ne{\bf 0})\ ,$$ where $c$ is an arbitrary constant. One computes$${\partial v_1\over\partial x_1}=-\left({3c\over r^4}+{2\over r^3}\right){x_1\over r}x_1+\left({c\over r^3}+{1\over r^2}\right)\ ,$$ which then leads to $${\rm div}\,{\bf v}({\bf x})={1\over r^2}\qquad({\bf x}\ne{\bf 0})\>,$$ as required. Note that the part ${\displaystyle{c\over r^2}{{\bf x}\over r}}$ of ${\bf v}$, the gravitational field of a mass point at ${\bf 0}$, has zero divergence. The field ${\bf v}$ is radial and with constant radial component on spheres $S_R$ centered at ${\bf 0}$. The flux $\Phi$ of ${\bf v}$ through such a sphere therefore is given by $$\Phi(S_R)=\left({c\over R^2}+{1\over R}\right)\>{\rm area}(S_R)=4\pi(c+R)\ .$$It follows that this flux is not determined by the data given in the problem.