= Solution
A <homogeneous hyperelastic material> has a stored energy per reference <volume> $W(F)$ depending on the local <deformation gradient> but not explicitly on the material point. Its first Piola-Kirchhoff response is
$$
\boxed{\widehat T(F)=DW(F),\qquad T_{i\alpha}=\partial W/\partial F_{i\alpha}.}
$$
A superposed rigid rotation changes the observer's description without changing the energy of the <deformation>. <Material frame indifference> is therefore
$$
\boxed{W(QF)=W(F)\quad(F\in\mathbb M_+^{3\times3},\ Q\in\mathrm{SO}(3)).}
$$
This left-rotation invariance is distinct from a <material isotropy> assumption on right rotations. Differentiate it in an arbitrary matrix direction $H$:
$$
DW(QF):(QH)=DW(F):H.
$$
The <Frobenius inner product> then gives $Q^T\widehat T(QF)=\widehat T(F)$, proving $\widehat T(QF)=Q\widehat T(F)$.
For the converse, that stress covariance makes the <derivative> in $F$ of $W(QF)-W(F)$ zero. The positive-<determinant> matrices form a <connected set>, as also follows from <polar decomposition>, so this difference is a constant $c(Q)$ independent of $F$. Composition gives $c(Q_1Q_2)=c(Q_1)+c(Q_2)$. For fixed $F$, <continuity> of $W$ and <compactness> of $\mathrm{SO}(3)$ make $c$ bounded. But $c(Q^n)=nc(Q)$ for every positive integer, forcing $c(Q)=0$. Thus \b[stress covariance and frame indifference of the energy are equivalent]. <Compactness> removes the possible additive energy constant in the converse.
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