- #1

- 555

- 19

I am confused about how I arrive at the contracted epsilon identity. [tex]\epsilon_{ijk} \epsilon_{imn} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km}[/tex]

1. Homework Statement

1. Homework Statement

Show that [tex]\epsilon_{ijk} \epsilon_{imn} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km}[/tex]

## Homework Equations

## The Attempt at a Solution

[/B]

From the relation between the Levi-civita symbol and the Kronecker delta, I compute [itex]\epsilon_{ijk} \epsilon_{imn}[/itex] by finding the determinant of the following matrix.

[itex]\epsilon_{ijk} \epsilon_{imn} = det \left[ \begin{array}{cccc} \delta_{ii} & \delta_{im} & \delta_{in} \\ \delta_{ji} & \delta_{jm} & \delta_{jn} \\ \delta_{ki} & \delta_{km} & \delta_{kn} \end{array} \right][/itex] which yields

[itex]\epsilon_{ijk} \epsilon_{imn} = \delta_{ii} (\delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km}) - \delta_{im} (\delta_{ji} \delta_{kn} - \delta_{jn} \delta_{ki}) + \delta_{in} (\delta_{ji} \delta_{km} - \delta_{jm} \delta_{ki})[/itex]

I am confused about how to progress.

Thanks for any help you can give.