= Solution
For $R>0$, differentiation gives $\partial_jR=\theta_j$ and $\partial_j\theta_i=(\delta_{ij}-\theta_i\theta_j)/R$. Hence
$$
\boxed{Dy=\frac rR(I-\theta\otimes\theta)+r'\theta\otimes\theta.}
$$
The two matrices are the <orthogonal projections> onto the tangent plane of the reference sphere and its radial line. The <gradient> multiplies the radial direction by $r'$ and each tangent direction by $r/R$. With positive $r,r'$, it is itself symmetric positive definite, so its <eigenvalues> are its principal stretches:
$$
\boxed{\lambda_1=r',\qquad\lambda_2=\lambda_3=r/R.}
$$
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