= Solution
Differentiate the position vector with respect to the three coordinates:
$$
\begin{aligned}
\mathbf r_\rho&=(\sinh\rho\cos u\cos v,\sinh\rho\cos u\sin v,\cosh\rho\sin u),\\
\mathbf r_u&=(-\cosh\rho\sin u\cos v,-\cosh\rho\sin u\sin v,\sinh\rho\cos u),\\
\mathbf r_v&=(-\cosh\rho\cos u\sin v,\cosh\rho\cos u\cos v,0).
\end{aligned}
$$
Their pairwise <dot products> vanish. For example, $\mathbf r_\rho\cdot\mathbf r_u=-\sinh\rho\cosh\rho\sin u\cos u+\sinh\rho\cosh\rho\sin u\cos u=0$. Thus these are <orthogonal coordinates> wherever the coordinate tangents are nonzero. Their <scale factors of orthogonal coordinates> are
$$
h_\rho^2=h_u^2=\sinh^2\rho\cos^2u+\cosh^2\rho\sin^2u=\sinh^2\rho+\sin^2u,\qquad h_v=\cosh\rho\cos u.
$$
The absolute <Jacobian determinant> is their product, giving
$$
\boxed{dV=\cosh\rho\cos u(\sinh^2\rho+\sin^2u)\,d\rho\,du\,dv.}
$$
The ordered coordinate system is left handed: $\mathbf r_\rho\cdot(\mathbf r_u\times\mathbf r_v)=-h_\rho h_uh_v$. The volume measure uses the absolute value. The axis and the focal circle contain coordinate degeneracies, as expected for <oblate spheroidal coordinates>. This is the <hyperbolic parametrization of oblate spheroidal coordinates>. The local TeX has an OCR substitution $\sinh^2u$ in the volume formula; the original PDF correctly prints $\sin^2u$.
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