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i understand how to change 2 tensors into Kronecker delta symbols but unsure how they managed to transform back to just one. If someone could add all the steps to get to the answer that would be amazing :)

  • $\begingroup$ Do you know about (anti)symmetric matrices? $\endgroup$ – hkBst Nov 27 '18 at 17:30
  • $\begingroup$ no but I do know about matrices in general $\endgroup$ – complexityyy Nov 27 '18 at 17:53

By the replacement property: \begin{align*} (\delta_{ad}\delta_{be}-\delta_{ae}\delta_{bd})\varepsilon_{efa} &=\delta_{ad}\delta_{be}\varepsilon_{efa}-\delta_{ae}\delta_{bd}\varepsilon_{efa} \\ &=\delta_{be}\varepsilon_{efd}-\delta_{bd}\varepsilon_{afa} \\ &=\varepsilon_{bfd}-0, \end{align*} since, if an index is repeated, the Levi-Civita symbol is zero.

  • $\begingroup$ thank you so much! $\endgroup$ – complexityyy Nov 27 '18 at 17:52
  • $\begingroup$ You're welcome! $\endgroup$ – Adrian Keister Nov 27 '18 at 17:54

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