The volume of a closed ball recovered from the outward flux of the position field

Given: A radius $R>0$, the ball $B_R$, the spherical octant presentation of The closed ball is an elementary solid region, presented by the eight spherical octants, one of its octant parametrizations $\phi (\varphi ,\theta )=R(\sin \varphi \cos \theta ,\sin \varphi \sin \theta ,\cos \varphi )$, and the position field $X(x,y,z)=(x,y,z)$.

For the spherical parametrization $\phi (\varphi ,\theta )=R(\sin \varphi \cos \theta ,\sin \varphi \sin \theta ,\cos \varphi )$, one has ${\phi }_{\varphi }\times {\phi }_{\theta }=R\sin \varphi \phi (\varphi ,\theta )$ (The cross product in ${\mathbb{R}}^3$, The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine).

The content of a glued elementary solid is one third of the outward flux of the position field through its boundary (The volume of a glued elementary solid is a third of the outward flux of the position field).

For an elementary solid region $E$ and a $C^1$ field $F$, ${\iiint }_E\mathrm{div}F={\iint }_{\mathrm{\partial }E}⟨F,n⟩$ (The divergence theorem on an elementary solid region).

The divergence of a field is the sum of its coordinate partial derivatives (Divergence and curl of a $C^1$ vector field).

For a bounded Jordan set and an integrable function whose sections are integrable outside a content-zero exceptional set, Jordan Fubini computes the multiple integral by the corresponding iterated section integrals (Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable).

If $a<b$, $G$ is differentiable on $[a,b]$, and $G^{\mathrm{\prime }}=f$ is integrable there, then ${\int }_a^{b}f=G(b)-G(a)$ (The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$).

$(\sin t{)}^{\mathrm{\prime }}=\cos t$ and $(\cos t{)}^{\mathrm{\prime }}=-\sin t$ (The derivatives of sine and cosine are cosine and minus sine).

${\sin }^{2}t+{\cos }^{2}t=1$ (Parity and the Pythagorean identity for sine and cosine).

The flux in the orientation induced by a patch $\phi$ is ${\int }_D(F\circ \phi )\cdot ({\phi }_u\times {\phi }_v)$ (Unit normal fields, orientations, and flux through a regular surface patch).

For a finite patch presentation, the total flux is the sum of the patch fluxes (Finitely patched regular surfaces, their area, scalar integrals, and flux).

The closed three-dimensional ball of radius $r$ has volume $4\pi r^3\mathrm{/}3$ (A closed three-dimensional ball of radius $r\ge 0$ has volume $4\pi r^3\mathrm{/}3$).

$⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).