Jacobian determinant ($\det J_f$)

The Jacobian determinant is the determinant of the square Jacobian matrix. Its absolute value is the local volume-scaling factor in the change of variables formula.

The Jacobian matrix and determinant describe respectively the linear derivative and its signed volume factor.

For a smooth map $u:{\mathbb{R}}^3\to {\mathbb{R}}^3$, set $V_i={ϵ}_{ijk}{ϵ}_{abc}({\partial }_j{u}_a)({\partial }_k{u}_b)u_c/6$. 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 $u$.