Past exam of the solid mechanics course of the University of Oxford / 2010 / 2 / b / Solution

For $R>0$, differentiation gives ${\partial }_{j}R={\theta }_j$ and ${\partial }_j{\theta }_i=({\delta }_{ij}-{\theta }_i{\theta }_j)/R$. HenceThe 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 $r^{\prime }$ and each tangent direction by $r/R$. With positive $r,r^{\prime }$, it is itself symmetric positive definite, so its eigenvalues are its principal stretches: