Question

Write the tensor transformation equations for ${\mathit{\epsilon }}_{\mathbf{i}\mathbf{j}\mathbf{k}}{\mathit{\epsilon }}_{\mathbf{m}\mathbf{n}\mathbf{p}}$to show that this is a (rank 6) tensor (nota pseudo tensor). Hint:Write (6.1) for each$\mathit{\epsilon }$and multiply them, being careful not to re-use a pair of summation indices.

Short answer

The tensor transformation equation for

$$\begin{gathered} {{\epsilon }^{\text{'}}}_{\alpha \beta \gamma }{{\epsilon }^{\text{'}}}_{\zeta n\theta }={(detA)}^2{a}_{\alpha i}{\alpha }_{\beta j}{a}_{\gamma k}{a}_{\zeta m}{a}_{nm}{a}_{\theta p}{\epsilon }_{ijk}{\epsilon }_{mnp} \\ =a_{\alpha i}{\alpha }_{\beta j}{a}_{\gamma k}{a}_{\zeta m}{a}_{nm}{a}_{\theta p}{\epsilon }_{ijk}{\epsilon }_{mnp} \end{gathered}$$.

Step-by-step solution

Given information.

Matrix definitions are given.

Definition of a rotation matrix.

The rotation matrix is defined in this way.

$\left[\begin{array}{cc}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{array}\right]$

Define an orthogonal transformation.

It A is an orthogonal matrix, that defines an orthogonal transformation.

${{\epsilon }^{\text{'}}}_{\alpha \beta \gamma }=(detA)a_{\alpha i}{\alpha }_{\beta j}{a}_{\gamma k}{\epsilon }_{ijk}$

Using the information$A=\pm 1$ make further simplifications.

$$\begin{gathered} {{\epsilon }^{\text{'}}}_{\alpha \beta \gamma }{{\epsilon }^{\text{'}}}_{\zeta n\theta }={(detA)}^2{a}_{\alpha i}{\alpha }_{\beta j}{a}_{\gamma k}{a}_{\zeta m}{a}_{nm}{a}_{\theta p}{\epsilon }_{ijk}{\epsilon }_{mnp} \\ =a_{\alpha i}{\alpha }_{\beta j}{a}_{\gamma k}{a}_{\zeta m}{a}_{nm}{a}_{\theta p}{\epsilon }_{ijk}{\epsilon }_{mnp} \end{gathered}$$

Hence, the tensor transformation equation is shown below:

$$\begin{gathered} {{\epsilon }^{\text{'}}}_{\alpha \beta \gamma }{{\epsilon }^{\text{'}}}_{\zeta n\theta }={(detA)}^2{a}_{\alpha i}{\alpha }_{\beta j}{a}_{\gamma k}{a}_{\zeta m}{a}_{nm}{a}_{\theta p}{\epsilon }_{ijk}{\epsilon }_{mnp} \\ =a_{\alpha i}{\alpha }_{\beta j}{a}_{\gamma k}{a}_{\zeta m}{a}_{nm}{a}_{\theta p}{\epsilon }_{ijk}{\epsilon }_{mnp} \end{gathered}$$