TITLE: Divergence of a vector field on a sequence of spheres
QUESTION [11 upvotes]: I'm studying for my exams and I found this problem in the book "Advanced Calculus", written by Friedman:
"Consider a sequence of spheres $S_n$ in $\mathbb{R}^3$ with center $P_n$ and radius $r_n$, such that $P_n \to P, r_n \to 0$ as $ n \to \infty.$ Let $\textbf{F}$ be a continuously differentiable vector field in a neighborhood of $P$. Prove that
$$(\nabla \cdot \textbf{F})(P)=\lim_{n \to \infty}\frac{1}{V_n}\iint\limits_{S_n}\textbf{F}\cdot \textbf{n}\ dS,$$
where $V_n$ is the volume of the ball with boundary $S_n$."
This is what I did:
Observe that, by the Divergence Theorem,
$$\frac{1}{V_n}\iint\limits_{S_n}\textbf{F}\cdot \textbf{n}\ dS = \frac{1}{V_n}\iiint\limits_{D_n}\nabla\cdot\textbf{F} \ dV,$$
where $D_n$ is the region bounded by $S_n$.
Now, notice that $\dfrac{1}{V_n}\displaystyle \iiint\limits_{D_n}\nabla\cdot\textbf{F} \ dV$ is the mean value of $\nabla \cdot \textbf{F}$ on $D_n$. Since F is continuously differentiable, we have that $\nabla \cdot \textbf{F}$ is continuous, so, for each $n$, there is a point $P_n$ such that $$\frac{1}{V_n} \iiint\limits_{D_n}\nabla\cdot\textbf{F} \ dV = (\nabla \cdot \textbf{F})(P_n)$$
So, 
$$\lim_{n \to \infty}\frac{1}{V_n}\iint\limits_{S_n}\textbf{F}\cdot \textbf{n} \ dS = \lim_{n \to \infty}\frac{1}{V_n}\iiint\limits_{D_n}\nabla\cdot\textbf{F} \ dV \\ =\lim_{n \to \infty}(\nabla \cdot \textbf{F})(P_n) = (\nabla \cdot \textbf{F})(P),$$
as we wanted.
I have two problems here:
1) I don't know how to prove that because $\nabla \cdot \textbf{F}$ is continuous, we certainly have that there is a point $P_n$ such that $\frac{1}{V_n} \iiint\limits_{D_n}\nabla\cdot\textbf{F} \ dV = (\nabla \cdot \textbf{F})(P_n)$. I know it's true because I saw it in some calculus books, but I haven't found a proof.
2) After I finished, I realize that my points $P_n$ doesn't necessarily have to be the points $P_n$ center of $S_n$. So, because I know that the radius $r_n$ of $S_n$ goes to 0, can I assume that $P$ will be the limit of every sequence $(a_n)$ such that $a_k \in S_k$ $\forall k \in \mathbb{N}$?
I'd be really grateful if someone could help me with my questions above :) Thanks!

REPLY [2 votes]: Your proof is correct with the proviso that the points $P_n$ occurring in the proof are not the centers of the spheres $S_n$. 
The mean value theorem guarantees for any connected body $B$ and continuous function $f:\ B\to{\mathbb R}$ a point $\xi\in B$ with
$$\int\nolimits_Bf(x)\ {\rm d}x=f(\xi)\ {\rm vol}(B)\ .$$
You have applied this principle with $f(x):={\rm div}\,{\bf F}(x)\ .$