In Purcell's, Electricity and Magnetism, 61, is stated that the following limit : $$\lim_{V_{i} \to 0}\frac{\int_{S_{i}} \overrightarrow{F}.\overrightarrow{ds}}{V_{i}}$$
[ Where $\overrightarrow{F}$ is a vector field, $S_{i}$ a small closed surface containing the point $i$ and $V_{i}$ the volume inside $S_{i}\,$] , which is the definition of $\,\mathrm{div}F\,$, is independent of the shape of $S_{i}$. Why is it this true? Why do these two shapes:
have the same limit defined above?
P.S. the second shape is essentially the first one completed
The numerator in the limit may be rewritten as $\iint_{\partial S_i}\mathbf F\cdot\mathbf n\,dS_i$, which by the divergence theorem is equivalent to $\iiint_{V_i}(\nabla\cdot\mathbf F)\,dV_i$. Thus the limit gives the mean value of $\nabla\cdot\mathbf F$ in $V_i$, and the limit as $V_i\to0$ is just $(\nabla\cdot\mathbf F)(i)$, which is independent of $S_i$.