Once we write the Levi-Civita image as being a determinant like we did above, the fully antisymmetric house that it have will become apparent: swapping any two indices corresponds to interchanging their corresponding rows in the matrix as a result of which the determinant, which can be the Levi-Civita alone, alterations sign.

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= δ i l ( δ j m δ k n − δ j n δ k m ) − δ i m ( δ j l δ k n − δ j n δ k l ) + δ i n ( δ j l δ k m − δ j m δ k l ) .

or Minkowski Area. The values from the Levi-Civita image are impartial of any metric tensor and coordinate program. Also, the precise expression "symbol" emphasizes that it's not a tensor as a consequence of how it transforms in between coordinate systems; having said that it could be interpreted for a tensor density.

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The subsequent are examples of the final identification over specialised to Minkowski Room (With all the unfavorable indication arising through the odd number of negatives during the signature with the metric more info tensor in either indicator convention):

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The covariant Levi-Civita tensor (often known as the Riemannian quantity variety) in any coordinate technique that matches the chosen orientation is

exactly where gab would be the representation with the metric in that coordinate procedure. We could in the same way consider a contravariant Levi-Civita tensor by raising the indices With all the metric as common,

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