2.I.2C

Explain briefly why the second-rank tensor

$\int_{S} x_{i} x_{j} d S(\mathbf{x})$

is isotropic, where $S$ is the surface of the unit sphere centred on the origin.

A second-rank tensor is defined by

$T_{i j}(\mathbf{y})=\int_{S}\left(y_{i}-x_{i}\right)\left(y_{j}-x_{j}\right) d S(\mathbf{x})$

where $S$ is the surface of the unit sphere centred on the origin. Calculate $T(\mathbf{y})$ in the form

$T_{i j}=\lambda \delta_{i j}+\mu y_{i} y_{j}$

where $\lambda$ and $\mu$ are to be determined.

By considering the action of $T$ on $\mathbf{y}$ and on vectors perpendicular to $\mathbf{y}$, determine the eigenvalues and associated eigenvectors of $T$.

*Typos? Please submit corrections to this page on GitHub.*