= Symmetric Biot stress
{title2=$T^{(1)}=\tfrac12(PR+R^TP^T)$}
Use the material-first <nominal stress> $P_{Ii}$ and $F=RU$. The <symmetric Biot stress> conjugate to the <Biot strain tensor> is $T^{(1)}=\operatorname{sym}(PR)$. To prove it, set $M=P^T$ and differentiate $F=RU$. Conservation of <angular momentum> makes $MF^T$ symmetric, so the contraction of $R^TM$ with $R^T\dot R\,U$ is zero: it is a contraction of a symmetric <tensor> with a skew <tensor> after cyclically moving $U$. Thus $M:\dot F=\operatorname{sym}(R^TM):\dot U=T^{(1)}:\dot E^{(1)}$. No assumption that $PR$ itself is symmetric is needed.
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