course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Methods |
IB |
43 |
|
4.II.16H |
2005 |
Define an isotropic tensor and show that
For
$$\int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}{i{1}} \ldots \hat{x}{i{n}}$$
is an isotropic tensor for any
$$\begin{aligned} &\int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}{i} \hat{x}{j}=a \delta_{i j}, \quad \int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}{i} \hat{x}{j} \hat{x}{k}=0 \ &\int \mathrm{d} S(\hat{\mathbf{x}}) \hat{x}{i} \hat{x}{j} \hat{x}{k} \hat{x}{l}=b\left(\delta{i j} \delta_{k l}+\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}\right) \end{aligned}$$
for some
Explain why
where
[The general isotropic tensor of rank 4 has the form