= Jacobian determinant as a divergence
{c}
{title2=$J[u]=\nabla\cdot V$}
For a smooth map $u:\mathbb R^3\to\mathbb R^3$, set $V_i=\epsilon_{ijk}\epsilon_{abc}(\partial_j u_a)(\partial_k u_b)u_c/6$. In <Einstein summation convention>, applying the <product rule> leaves only the term differentiating $u_c$: the terms with second derivatives cancel against antisymmetry. The remaining <Levi-Civita symbol> contraction is $\det Du$, so $\nabla\cdot V=J[u]$. Equivalently this vector field is one third of the cofactor matrix transposed, applied to $u$.
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