Discussion
The scale factors corresponding to the covariant metric tensor \(g_{ij}\) in spherical polar coordinates are
Answer: D
The scale factors are \(1, r, r\sin\theta\).
The line element in spherical polar coordinates is
\[ \begin{aligned} ds^{2} &= dr^{2} + r^{2}d\theta^{2} \\ &\quad + r^{2}\sin^{2}\theta\,d\phi^{2} \end{aligned} \]
so the covariant metric is diagonal with \(g_{rr} = 1\), \(g_{\theta\theta} = r^{2}\), \(g_{\phi\phi} = r^{2}\sin^{2}\theta\). The scale (Lamé) factors are the square roots, \(h_{i} = \sqrt{g_{ii}}\): 1, \(r\) and \(r\sin\theta\).
\(1, r^{2}, r^{2}\sin^{2}\theta\) are the metric components themselves, not the scale factors. \(1, r^{2}, \sin^{2}\theta\) is a garbled version of them, and 1, 1, 1 belongs to Cartesian coordinates.
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