The outward flux of the inverse-square field through a sphere centred at the origin is $4\pi$

Given: A radius $R>0$, the spherical parametrization $\phi (\varphi ,\theta )=R(\sin \varphi \cos \theta ,\sin \varphi \sin \theta ,\cos \varphi )$ on $[0,\pi ]\times [0,2\pi ]$, and the inverse-square field $F$ on ${\mathbb{R}}^3\setminus \{0\}$.

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

A regular patch may degenerate on its parameter boundary, but on the parameter interior its cross product is nonzero and no interior parameter point shares its image with a distinct parameter point (Regular parametrized surface patches on compact Jordan parameter regions).

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 cross product is that of The cross product in ${\mathbb{R}}^3$.

$(\sin t{)}^{\mathrm{\prime }}=\cos t$ 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).

Sine is positive on $(0,\pi )$, cosine is strictly decreasing on $[0,\pi ]$, $\cos 0=1$, $\cos \pi =-1$, and $\theta \mapsto (\cos \theta ,\sin \theta )$ is injective on $[0,2\pi )$ (Signs, monotonicity intervals, and ranges of sine and cosine, Quarter-turn values and shifts by pi/2 and pi, $t\mapsto (\cos t,\sin t)$ is a bijection from $[0,2\pi )$ onto the real unit circle).

Jordan Fubini computes a multiple integral by 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 ${\int }_a^{b}f=G(b)-G(a)$).

$\mathrm{\parallel }x{\mathrm{\parallel }}_2=\sqrt{⟨x,x⟩}$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

The divergence theorem for an elementary solid region assumes a $C^1$ field on an open set containing the solid (The divergence theorem on an elementary solid region).