Jacobi determinant derivative formula ($d\log \det A=\mathrm{tr}(A^{-1}dA)$)

For a differentiable invertible matrix $A(t)$, $\frac{d}{dt}\log \det A=\mathrm{tr}(A^{-1}\dot{A})$ wherever a consistent logarithm is defined. In particular a positive spatial metric tensor satisfies $h^{ij}{\dot{h}}_{ij}=\dot{h}/h$. This separates its local volume change from shape change.