= Solution
The <Jacobian determinant> is the oriented <scalar triple product> of the three <coordinate basis> vectors:
$$
J=\det\frac{\partial(x,y,z)}{\partial(u,v,\phi)}=(\mathbf h_u\times\mathbf h_v)\cdot\mathbf h_\phi.
$$
Here $\mathbf h_u\times\mathbf h_v=(u^2+v^2)(-\sin\phi,\cos\phi,0)$, so
$$
\boxed{J=uv(u^2+v^2),\qquad dV=uv(u^2+v^2)\,du\,dv\,d\phi.}
$$
The sign is nonnegative on the stated range, and the absolute <Jacobian determinant> also equals $h_uh_vh_\phi$ as expected for <orthogonal coordinates>.
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