- #1

hotvette

Homework Helper

- 996

- 5

## Homework Statement

[tex] \text{Show that } \epsilon_{ijk} \epsilon_{mjk} = 2\delta_{im} [/tex]

## Homework Equations

[tex]

\begin{equation*}

\epsilon_{ijk} \epsilon_{mnp} =

\left| \! \begin{array}{ccc}

\delta_{im} & \delta_{in} & \delta_{ip}

\\ \delta_{jm} & \delta_{jn} & \delta_{jp}

\\ \delta_{km} & \delta_{kn} & \delta_{kp} \end{array} \! \right|

\end{equation*}[/tex]

## The Attempt at a Solution

[tex]

\begin{align*}

&\text{Evaluating the first term of the determinant, I get:}

\\

&\delta_{im}

\left| \! \begin{array}{cc}

\delta_{jj} & \delta_{jk}

\\ \delta_{kj} & \delta_{kk} \end{array} \! \right|

= \delta_{im}(\delta_{jj}\delta_{kk}-\delta_{jk}\delta_{kj}) = \delta_{im}(9-3) = 6\delta_{im}

\\

& \text{But I'm not sure that's correct. If I just look at } \delta_{ii} \text{ , I get} \\

& \delta_{ii} = \delta_{11} + \delta_{22} + \delta_{33} = 3

\text{ which means }

\delta_{jj}\delta_{kk} = 3(3) = 9

\end{align*} [/tex]

Am I doing something wrong?