How do I solve A×((B×C)×D) in index notation? I ended up with three Levi-Civita symbols and have no idea how to contract them. Thanks for the help!
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We will use the contraction $\varepsilon_{ijk}\varepsilon_{ilm}=\delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl}$ and the fact that the Levi-Civita is cyclic. Let $\bf V=A\times((B\times C)\times D)$. We then have $$\begin{eqnarray} V_i=&&\varepsilon_{ijk}A_j[\mathbf{((B\times C)\times D)}]_k =\\ &&\varepsilon_{ijk}A_j\varepsilon_{klm}(\mathbf{B\times C})_l D_m =\\ &&\varepsilon_{ijk}A_j\varepsilon_{klm}\varepsilon_{lno}B_n C_o D_m=\\ &&\varepsilon_{kij}\varepsilon_{klm}\varepsilon_{lno}A_jB_n C_o D_m =\\ &&(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})\varepsilon_{lno}A_jB_n C_o D_m=\\ &&\varepsilon_{ino}A_jB_n C_o D_j-\varepsilon_{jno}A_jB_n C_o D_i=\\ &&(\mathbf{A\cdot D})(\mathbf{B\times C})_i-\mathbf{A\cdot(B\times C)}D_i \end{eqnarray} $$ So, $\bf V=(A\cdot D)(B\times C)-A\cdot(B\times C)D$.
Mateus Sampaio
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That really helps! I didn't realize they can be further contracted. – Anne Oct 30 '13 at 03:18