= Pauli-coordinate Haar measure on SU(2)
{c}
{title2=$d\rho=d^3u/|u_0|$}
The hemisphere graph $u_0=\pm\sqrt{1-|\mathbf u|^2}$ of <SU(2) as the three-sphere> has induced metric $g_{ij}=\delta_{ij}+u_iu_j/u_0^2$, whose <determinant> is $1/u_0^2$. Its volume density is therefore $d^3u/|u_0|$. Fixed left or right multiplication preserves the Euclidean norm on the four Pauli coefficients, so the round volume is <Haar measure>. Both hemispheres have volume $\pi^2$, giving total volume $2\pi^2$ and normalized density $d^3u/(2\pi^2|u_0|)$ on each hemisphere. The apparent equatorial divergence is an integrable coordinate singularity.
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