= Piola identity
{c}
{title2=$\operatorname{Div}\operatorname{cof}Dy=0$}
For a twice continuously differentiable <deformation map>, each row of its <cofactor matrix> has zero <divergence> in reference coordinates. In three dimensions write $(\operatorname{cof}Dy)_{iA}=\frac12\varepsilon_{ijk}\varepsilon_{ABC}y_{j,B}y_{k,C}$. Applying $\partial_A$ gives terms in which a symmetric pair of second <partial derivatives> contracts with an antisymmetric <Levi-Civita symbol>, so each term vanishes. This identity is the cancellation that makes the <Piola-Kirchhoff stress tensor> transform force balance correctly between configurations.
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