Question

Show that ${\mathit{\delta }}_{\mathbf{i}\mathbf{j}}{\mathit{ϵ}}_{\mathbf{k}\mathbf{l}\mathbf{m}}$is an isotropic tensor of rank 5. Hint: Combine equations (5.4) and (5.7).

Short answer

The expression${\delta }_{ij}{\in }_{klm}$ is an isotropic tensor.

Step-by-step solution

Given Information

An isotropic tensor of rank 5is .${\delta }_{ij}{\in }_{klm}$

Definition of Isomorphic tensor

Isotropic tensor is defined as a tensor with components that remain unaltered when the coordinate system is rotated arbitrarily.

Simplify the isotropic tensor

Use the result given in previous question.

$\begin{array}{c}{ϵ}_{\rho \lambda \sigma }^{\text{'}}=a_{\rho i}{a}_{\lambda j}{a}_{\sigma k}{ϵ}_{ijk}\\ ={ϵ}_{\rho \lambda \sigma }\det A\\ ={ϵ}_{\rho \lambda \sigma }\end{array}$

It is known that${\delta }_{\alpha \beta }^{\text{'}}={\delta }_{\alpha \beta }$

Use the above result to get${\delta }_{\alpha \beta }^{\text{'}}{\in }_{\rho \lambda \sigma }^{\text{'}}={\delta }_{\alpha \beta }{\in }_{\rho \lambda \sigma }$

Hence proved that${\delta }_{ij}{\in }_{klm}$ is an isotropic tensor.