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[QUE/ME-02011]

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Using definition of Levi-Civita symbol, prove the following identities. \begin{eqnarray} \epsilon_{i\,j\,k}\,\epsilon_{i\,j\,k} \ &=&\ 6 \\ \epsilon_{i\,j\,k}\,\epsilon_{l\,j\,k} \ &=&\ 2\,\delta_{il} \\ \epsilon_{i\,j\,k}\,\epsilon_{l\,m\,k}\ &=&\ (\,\delta_{il}\,\delta_{jm}\,-\,\delta_{im}\,\delta_{jl}) \\ \epsilon_{i\,j\,k}\,\epsilon_{l\,m\,n}\ &=&\ \left(\begin{array}{ccc} \delta_{il}&\delta_{im}&\delta_{in}\\ \delta_{jl}&\delta_{jm}&\delta_{jn}\\ \delta_{kl}&\delta_{km}&\delta_{kn} \end{array} \right) \end{eqnarray}

Use result on \(\vec{A}\times(\vec{B}\times{\vec{C})}\) to give an alternate proof of identity (3) .

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