= Solution
A <homogeneous hyperelastic material> has a <strain energy density> per unit reference <volume> $W(F)$ depending on the <deformation gradient>, with no explicit dependence on the <reference configuration>. For orientation-preserving gradients $F\in GL^+(3)$, its first <Piola-Kirchhoff stress tensor> is the <derivative>:
$$
\boxed{\widehat T(F)=D_FW(F),\qquad T_{iA}=\frac{\partial W}{\partial F_{iA}}(F).}
$$
Equivalently the infinitesimal stored-energy change is $DW(F)[H]=T:H$. Reference and spatial indices are distinguished here by upper- and lowercase index letters.
The <material symmetry group> may be written $G_W=\{H\in GL^+(3):W(FH)=W(F)\text{ for every }F\in GL^+(3)\}$. A volume-preserving relabelling convention restricts the candidate $H$ to $SL(3)$; the <material isotropy> criterion is the same. The material has \b[<material isotropy>] when $SO(3)\subseteq G_W$, meaning $W(FQ)=W(F)$ for every proper <rotation> $Q$. If only rotational <symmetries> are included in the definition of the group, <material isotropy> means that group equals $SO(3)$. This right-rotation invariance is distinct from <material frame indifference>, the left-rotation invariance $W(QF)=W(F)$.
The <polar decomposition in continuum mechanics> gives the unique factorizations
$$
\boxed{F=RU=VR,\qquad R\in SO(3),\quad U=(F^TF)^{1/2},\quad V=(FF^T)^{1/2},}
$$
where $U,V$ are <positive-definite symmetric matrices>. Their common positive <eigenvalues> $\lambda_1,\lambda_2,\lambda_3$ are the \b[<principal stretches>], equivalently the <singular values> of $F$. They describe the local stretch magnitudes after separating out a rigid <rotation>.
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