- #1
- 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?