After we generate the Levi-Civita image like a determinant like we did over, the absolutely antisymmetric property that it have turns into apparent: swapping any two indices corresponds to interchanging their corresponding rows inside the matrix as a consequence of which the determinant, which is the Levi-Civita alone, variations signal.
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δ i l δ i m δ i n δ j l δ j m δ j n δ k l δ k m δ k n
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The subsequent are examples of the final identification higher than specialized to Minkowski Area (with the negative indicator arising from your odd number of negatives during the signature on the metric tensor in either indicator convention):
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The covariant Levi-Civita tensor (generally known as the Riemannian volume kind) in almost any coordinate system that matches the selected orientation is
E α β γ δ = g α ζ g β η g γ θ g δ ι E ζ η θ ι . displaystyle E^ alpha beta gamma delta =g^ alpha zeta g^ beta eta g^ gamma theta g^ delta iota E_ zeta eta theta iota ,.
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