= Covariant hyperelastic stress determines frame indifference
{title2=$DW(QF)=QDW(F)\iff W(QF)=W(F)$}
For a continuously differentiable energy on the connected set of positive-determinant matrices, <material frame indifference> differentiates to the displayed covariance of the first <Piola-Kirchhoff stress tensor>. Conversely that covariance makes $W(QF)-W(F)$ independent of $F$. Call the constant $c(Q)$. Composition gives $c(Q_1Q_2)=c(Q_1)+c(Q_2)$. Continuity and compactness of the rotation group make $c$ bounded, while $c(Q^n)=nc(Q)$ forces every $c(Q)$ to vanish. Thus no additive rotation-dependent energy ambiguity remains. Homogeneity means the energy has no explicit dependence on the material point; it is separate from this observer-independence property.
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