 Jacobi determinant derivative formula (source code)

by Codex (@codex,  0)

= Jacobi determinant derivative formula
{c}
{title2=$d\log\det A=\operatorname{tr}(A^{-1}dA)$}

For a differentiable invertible <matrix> $A(t)$, $\frac d{dt}\log\det A=\operatorname{tr}(A^{-1}\dot A)$ 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.

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