The divergence at a point is the limit of outward flux per unit volume

Given: The open $O\subseteq {\mathbb{R}}^3$, the $C^1$ field $F$ on $O$, the point $p\in O$, and for each $m$ the finite gluing with union $E^{(m)}\subseteq O$ containing $p$, of positive content, with $\mathrm{diam}(E^{(m)})\to 0$.

The divergence of a $C^1$ field is $\mathrm{div}G={\sum }_{i<n}{\mathrm{\partial }}_i{G}_i$; a $C^1$ map has continuous first partial derivatives, so $\mathrm{div}G$ is continuous (Divergence and curl of a $C^1$ vector field, $C^k$ Euclidean maps and diffeomorphisms).

For $f=1$ the zero extension is $1_E$, so ${\int }_{E}1=\mathrm{cont}(E)$ (The Riemann integral of a bounded function over a bounded Jordan measurable set, A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content).

For a nonempty bounded $A$ in a metric space, $\mathrm{diam}(A)=\sup \{d(a,b):a,b\in A\}$ (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space), the metric on ${\mathbb{R}}^3$ being $d(a,b)=\parallel a-b{\parallel }_2$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

A map between metric spaces is continuous at a point when for every real $\epsilon >0$ there is a real $\delta >0$ such that points within $\delta$ of it have images within $\epsilon$ (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form).

A sequence of reals converges to $x$ when for every rational $\epsilon >0$ there is $K$ with $\mathrm{\mid }{x}_k-x\mathrm{\mid }<\epsilon$ for all $k\ge K$ (Limits and Cauchy sequences of reals).

For a finite gluing with union $E$ and outer presentation ${\mathrm{\Sigma }}^{\mathrm{o}\mathrm{u}\mathrm{t}}$ and a $C^1$ field $G$ on an open set containing $E$, ${\iiint }_E\mathrm{div}G={\iint }_{\mathrm{\partial }E}⟨G,n⟩$ (The divergence theorem for finite gluings of elementary solid regions).

For a finite gluing, $E$ is compact and Jordan measurable (Internal faces cancel and volume integrals add when elementary solid regions are glued, Finite gluings of elementary solid regions and their outward boundary presentation).

For integrable $f,g$ on a nondegenerate rectangle and scalars $\alpha ,\beta$: $\alpha f+\beta g$ is integrable with integral $\alpha \int f+\beta \int g$; if $f\le g$ then $\int f\le \int g$; and $\mid f\mid$ is integrable with $\mid \int f\mid \le \int \mid f\mid$ (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in ${\mathbb{R}}^m$).

Every continuous real function on a compact Jordan measurable set is Riemann integrable over it (A continuous real function on a compact Jordan measurable set is Riemann integrable over that set).