= Nanson's formula
{c}
{title2=$dS=JF^{-T}dS_0$}
For an orientation-preserving <deformation gradient> $F$, oriented area transforms by its cofactor matrix: $(Fa)\times(Fb)=(\det F)F^{-T}(a\times b)$. Taking the <dot product> with $Fc$ proves this from the determinant identity for the scalar triple product. Applying it to tangent vectors of a surface proves <Nanson's formula>. Equating current and reference <tractions> then gives the material-first <nominal stress tensor> $P=JF^{-1}\sigma$ for symmetric <Cauchy stress tensor> $\sigma$.
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