= Solution
On $r=0$, the screen metric is $h_{ij}$ and $U=\partial_\lambda$. Therefore the <null expansion> is
$$
\theta=\frac12h^{ij}\frac{\partial h_{ij}}{\partial\lambda}.
$$
The derivative formula for the <determinant> gives
$$
\frac{\partial h}{\partial\lambda}
=h\,h^{ij}\frac{\partial h_{ij}}{\partial\lambda},
$$
and hence
$$
\boxed{\frac{\partial\sqrt h}{\partial\lambda}=\theta\sqrt h}.
$$
Thus $\theta$ is the fractional rate of change of an infinitesimal transverse area carried along the generators: positive expansion enlarges the beam and negative expansion focuses it.
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