- #1

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- Homework Statement:
- Solve for ##\epsilon_{ij \ell} \, \epsilon_{km \ell} \, \epsilon_{ijm} \, a_k##

- Relevant Equations:
- Refer Below ##\longrightarrow##

Below is my attempted solution:

$$\epsilon_{ij \ell} \, \epsilon_{km \ell} \, \epsilon_{ijm} \, a_k$$

$$\Rightarrow (\delta_{ik} \, \delta_{jm} - \delta_{im} \, \delta_{jk}) \epsilon_{ijm} \, a_k$$

$$\Rightarrow \delta_{ik} \, \delta_{jm} \, \epsilon_{ijm} \, a_k - \delta_{im} \, \delta_{jk} \epsilon_{ijm} \, a_k$$

$$\Rightarrow a_i \, \delta_{jm} \, \epsilon_{ijm} - a_j \delta_{im} \, \epsilon_{ijm}$$

From here on out, I am not sure what I am doing. Therefore, any guidance or assistance would be greatly appreciated.

$$\Rightarrow a_m \epsilon_{ijm} - a_m \epsilon_{ijm} = 0$$

Perhaps, my biggest question is would eliminating the Kronecker delta and getting ##a_m## be mathematically legal?

Also, I have heard from someone else that the above equation results to zero because of anti-symmetry. How is one able to determine that?

Thank you for the help!

$$\epsilon_{ij \ell} \, \epsilon_{km \ell} \, \epsilon_{ijm} \, a_k$$

$$\Rightarrow (\delta_{ik} \, \delta_{jm} - \delta_{im} \, \delta_{jk}) \epsilon_{ijm} \, a_k$$

$$\Rightarrow \delta_{ik} \, \delta_{jm} \, \epsilon_{ijm} \, a_k - \delta_{im} \, \delta_{jk} \epsilon_{ijm} \, a_k$$

$$\Rightarrow a_i \, \delta_{jm} \, \epsilon_{ijm} - a_j \delta_{im} \, \epsilon_{ijm}$$

From here on out, I am not sure what I am doing. Therefore, any guidance or assistance would be greatly appreciated.

$$\Rightarrow a_m \epsilon_{ijm} - a_m \epsilon_{ijm} = 0$$

Perhaps, my biggest question is would eliminating the Kronecker delta and getting ##a_m## be mathematically legal?

Also, I have heard from someone else that the above equation results to zero because of anti-symmetry. How is one able to determine that?

Thank you for the help!