= Solution
The coordinate <tangent vectors> are the partial <derivatives> of the position <vector>:
$$
\boxed{\begin{aligned}
\mathbf h_r&=(\sin\alpha\cos\beta,\sin\alpha\sin\beta,\cos\alpha),\\
\mathbf h_\alpha&=(r\cos\alpha\cos\beta,r\cos\alpha\sin\beta,-r\sin\alpha),\\
\mathbf h_\beta&=(-(1+r\sin\alpha)\sin\beta,(1+r\sin\alpha)\cos\beta,0).
\end{aligned}}
$$
Their pairwise <dot products> vanish. Their lengths, the <scale factors of orthogonal coordinates>, are $1,r,1+r\sin\alpha$. Since $R<1$, the last is positive throughout the body. Thus these <circular torus coordinates> are \b[orthogonal wherever $r>0$]; at $r=0$ the cross-section angle degenerates, as in ordinary <polar coordinates>.
The ordered <Jacobian determinant> is
$$
\boxed{J=\det\frac{\partial(x,y,z)}{\partial(r,\alpha,\beta)}=\mathbf h_r\cdot(\mathbf h_\alpha\times\mathbf h_\beta)=r(1+r\sin\alpha).}
$$
Its sign is positive for $r>0$, so the <volume> element is $dV=r(1+r\sin\alpha)\,dr\,d\alpha\,d\beta$. The coordinate degeneracy on the central <circle> has zero <volume> and does not obstruct <volume> <integration>.
\Image[/past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1-q1-torus-coordinates.png]
{title=Circular torus coordinates and their orthogonal tangent directions}
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