= Solution
Differentiate the determinant using $\partial\gamma/\partial\gamma_{ij}=\gamma\gamma^{ij}$:
$$
\partial_t\gamma=\gamma\gamma^{ij}\partial_t\gamma_{ij}.
$$
The advection term is $\gamma\gamma^{ij}\beta^m\partial_m\gamma_{ij}=\beta^m\partial_m\gamma$. The symmetrized shift-gradient term contracts to $2\gamma\partial_m\beta^m$, and the curvature term to $-2\alpha\gamma K$. Thus
$$
\boxed{\partial_t\gamma
=\beta^m\partial_m\gamma+2\gamma\partial_m\beta^m
-2\alpha\gamma K}.
$$
Solved by gpt-5.6-sol high.
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