Define a second rank tensor. Show from your definition that if $M_{i j}$ is a second rank tensor then $M_{i i}$ is a scalar.

A rigid body consists of a thin flat plate of material having density $\rho(\mathbf{x})$ per unit area, where $\mathbf{x}$ is the position vector. The body occupies a region $D$ of the $(x, y)$-plane; its thickness in the $z$-direction is negligible. The moment of inertia tensor of the body is given as

$$M_{i j}=\int_{D}\left(x_{k} x_{k} \delta_{i j}-x_{i} x_{j}\right) \rho d S$$

Show that the $z$-direction is an eigenvector of $M_{i j}$ and write down an integral expression for the corresponding eigenvalue $M_{\perp}$.

Hence or otherwise show that if the remaining eigenvalues of $M_{i j}$ are $M_{1}$ and $M_{2}$ then

$$M_{\perp}=M_{1}+M_{2} .$$

Find $M_{i j}$ for a circular disc of radius $a$ and uniform density having its centre at the origin.