Paper 3, Section II, C

Write down the most general isotropic tensors of rank 2 and 3. Use the tensor transformation law to show that they are, indeed, isotropic.

Let $V$ be the sphere $0 \leqslant r \leqslant a$. Explain briefly why

$T_{i_{1} \ldots i_{n}}=\int_{V} x_{i_{1}} \ldots x_{i_{n}} \mathrm{~d} V$

is an isotropic tensor for any $n$. Hence show that

$\int_{V} x_{i} x_{j} \mathrm{~d} V=\alpha \delta_{i j}, \quad \int_{V} x_{i} x_{j} x_{k} \mathrm{~d} V=0 \quad \text { and } \int_{V} x_{i} x_{j} x_{k} x_{l} \mathrm{~d} V=\beta\left(\delta_{i j} \delta_{k l}+\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}\right)$

for some scalars $\alpha$ and $\beta$, which should be determined using suitable contractions of the indices or otherwise. Deduce the value of

$\int_{V} \mathbf{x} \times(\boldsymbol{\Omega} \times \mathbf{x}) \mathrm{d} V$

where $\boldsymbol{\Omega}$ is a constant vector.

[You may assume that the most general isotropic tensor of rank 4 is

$\lambda \delta_{i j} \delta_{k l}+\mu \delta_{i k} \delta_{j l}+\nu \delta_{i l} \delta_{j k}$

where $\lambda, \mu$ and $\nu$ are scalars.]