If $T_{i j}$ is a second rank tensor such that $b_{i} T_{i j} c_{j}=0$ for every vector $\mathbf{b}$ and every vector c, show that $T_{i j}=0$.

Let $S$ be a closed surface with outward normal $\mathbf{n}$ that encloses a three-dimensional region having volume $V$. The position vector is $\mathbf{x}$. Use the divergence theorem to find

$$\int_{S}(\mathbf{b} \cdot \mathbf{x})(\mathbf{c} \cdot \mathbf{n}) d S$$

for constant vectors $\mathbf{b}$ and $\mathbf{c}$. Hence find

$$\int_{S} x_{i} n_{j} d S$$

and deduce the values of

$$\int_{S} \mathbf{x} \cdot \mathbf{n} d S \text { and } \int_{S} \mathbf{x} \times \mathbf{n} d S$$