Paper 3, Section II, C

(a) Define a rank two tensor and show that if two rank two tensors $A_{i j}$ and $B_{i j}$ are the same in one Cartesian coordinate system, then they are the same in all Cartesian coordinate systems.

The quantity $C_{i j}$ has the property that, for every rank two tensor $A_{i j}$, the quantity $C_{i j} A_{i j}$ is a scalar. Is $C_{i j}$ necessarily a rank two tensor? Justify your answer with a proof from first principles, or give a counterexample.

(b) Show that, if a tensor $T_{i j}$ is invariant under rotations about the $x_{3}$-axis, then it has the form

$\left(\begin{array}{ccc} \alpha & \omega & 0 \\ -\omega & \alpha & 0 \\ 0 & 0 & \beta \end{array}\right)$

(c) The inertia tensor about the origin of a rigid body occupying volume $V$ and with variable mass density $\rho(\mathbf{x})$ is defined to be

$I_{i j}=\int_{V} \rho(\mathbf{x})\left(x_{k} x_{k} \delta_{i j}-x_{i} x_{j}\right) \mathrm{d} V$

The rigid body $B$ has uniform density $\rho$ and occupies the cylinder

$\left\{\left(x_{1}, x_{2}, x_{3}\right):-2 \leqslant x_{3} \leqslant 2, x_{1}^{2}+x_{2}^{2} \leqslant 1\right\}$

Show that the inertia tensor of $B$ about the origin is diagonal in the $\left(x_{1}, x_{2}, x_{3}\right)$ coordinate system, and calculate its diagonal elements.