= Solution
{c}
In standard <spherical coordinates>, parametrize the <two-sphere> by
$$
\mathbf x(\theta,\phi)=R(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta).
$$
The <coordinate basis> transforms from the fixed Cartesian basis according to $\mathbf e_a=(\partial x^i/\partial q^a)\mathbf e_i$. Therefore
$$
\begin{aligned}
\mathbf e_\theta&=\partial_\theta\mathbf x
=R(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta),\\
\mathbf e_\phi&=\partial_\phi\mathbf x
=R(-\sin\theta\sin\phi,\sin\theta\cos\phi,0).
\end{aligned}
$$
Their <inner products> are $\mathbf e_\theta^2=R^2$, $\mathbf e_\theta\cdot\mathbf e_\phi=0$ and $\mathbf e_\phi^2=R^2\sin^2\theta$. Thus the <induced metric> is
$$
\boxed{(g_{ij})=R^2\begin{pmatrix}1&0\\0&\sin^2\theta\end{pmatrix},\qquad
ds^2=R^2(d\theta^2+\sin^2\theta\,d\phi^2).}
$$
The source's intermediate display places an extra $R^2$ outside entries already labelled $g_{\theta\theta},\ldots$. Interpreting those entries as the actual <metric tensor> components would double-count the factor: for example, $R=2$ gives $g_{\theta\theta}=4$ directly from $\mathbf e_\theta^2$, not $16$. Its final matrix is the correct <induced metric>; the entries of its intermediate matrix must instead denote dimensionless coefficients.
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