- #1

TheFerruccio

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**Description of Problem**

Verify that [itex]\epsilon_{ijk}\epsilon_{iqr}=\delta_{jq}\delta_{kr}-\delta_{kq}\delta_{jr}[/itex]

**Attempt at Solution**

I know that the term is only positive if there is an even permutation in both terms, or an odd permutation in both terms. This means that the second indices equal and the third indices equal, creating the positive set of deltas. The term is negative if 2nd = 3rd and 3rd = 2nd in each one, respectively, creating the negative set of deltas.

This is right. However, I do not know how to write that out mathematically. How do I expand out the Levi-Civita symbol to show that this is the case? Do I expand it out as a sum over i and write all the different possible cases? Is there a more elegant way of doing this?