Flux of a vectioral field across a sphere for $r \to 0^+$

Question

It's given the vectorial field $\vec{F}(x,y,z)$ of class $C^1$ in $\mathbb{R}^3$. Let be $\phi (r)$ its outgoing flux across the sphere centered in $(0,0,0)$ and with radius $r>0$. Then for $r \to 0^+$, to what tends $\phi(r)$?

Answer 1

There are an $R>0$ and an $M>0$ such that $$|{\bf F}({\bf x})|\leq M\qquad \bigl(|{\bf x}|\leq R\bigr)\ .$$ For $0<r<R$ then one has $$\phi(r)=\int_{\partial B_r}{\bf F}({\bf x})\cdot{\bf n}({\bf x})\>{\rm d}\omega$$ and therefore $$|\phi(r)|\leq M \cdot4\pi r^2\qquad(r<R)\ .$$ It follows that $\lim_{r\to0+}\phi(r)=0$.