Define an isotropic tensor and show that $\delta_{i j}, \epsilon_{i j k}$ are isotropic tensors.

For $\hat{\mathbf{x}}$ a unit vector and $\mathrm{d} S(\hat{\mathbf{x}})$ the area element on the unit sphere show that

$$\int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}_{i_{1}} \ldots \hat{x}_{i_{n}}$$

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

$$\begin{aligned} &\int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}_{i} \hat{x}_{j}=a \delta_{i j}, \quad \int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}_{i} \hat{x}_{j} \hat{x}_{k}=0 \\ &\int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}_{i} \hat{x}_{j} \hat{x}_{k} \hat{x}_{l}=b\left(\delta_{i j} \delta_{k l}+\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}\right) \end{aligned}$$

for some $a, b$ which should be determined.

Explain why

$$\int_{V} \mathrm{~d}^{3} x\left(x_{1}+\sqrt{-1} x_{2}\right)^{n} f(|\mathbf{x}|)=0, \quad n=2,3,4$$

where $V$ is the region inside the unit sphere.

[The general isotropic tensor of rank 4 has the form $a \delta_{i j} \delta_{k l}+b \delta_{i k} \delta_{j l}+c \delta_{i l} \delta_{j k} .$ ]