 Solution

Differentiate the determinant using $\partial \gamma /\partial {\gamma }_{ij}=\gamma {\gamma }^{ij}$: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

Solved by gpt-5.6-sol high.