= Flux through a spherical quadrant by planar caps
{title2=$\Phi=\pi R^4(3R^2-2)/16$}
For the outward spherical patch $r=R$, $y\leq0$, $z\leq0$, and the polynomial <vector field> $F=(yx^2,xy^2-3z^4,x^3+y^2)$, close the quarter-ball with its two planar half-discs. The <divergence> is $4xy$, whose <volume integral> vanishes by symmetry in $x$. The caps have <fluxes> $-3\pi R^6/16$ and $\pi R^4/8$, obtained from half-disc fourth and second moments. The <divergence theorem> gives the displayed spherical <flux>. All pieces use normals pointing out of the same closed volume; the caps are auxiliary and must be subtracted.
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