= Radial deformation of an elastic solid
{title2=$Dy=(r/R)I+(r'-r/R)\boldsymbol\theta\otimes\boldsymbol\theta$}
A spherical radial deformation is $y(x)=r(R)x/R$, with $R=|x|$ and $\boldsymbol\theta=x/R$. Differentiation gives the displayed symmetric gradient. Its radial eigenvalue is $r'$ and its two tangential eigenvalues are $r/R$; when these are positive, they are the <principal stretches>. The determinant is $r'(r/R)^2$. Positivity distinguishes an orientation-preserving radial elastic deformation from a map that collapses an interior sphere.
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