After we generate the Levi-Civita image being a determinant like we did over, the fully antisymmetric assets that it possess gets to be apparent: swapping any two indices corresponds to interchanging their corresponding rows in the matrix resulting from which the determinant, that's the Levi-Civita itself, adjustments indicator.
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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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These properties are important for being familiar with the habits on the tensor in several coordinate devices.
From this conduct, we can easily deduce what $epsilon^ ij4k $ would be without under-going the computations Yet again: it is just the results of exchanging the last two indices of $epsilon^ ijk4 $,
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This tensor shouldn't be puzzled While using the tensor density area stated previously mentioned. The presentation Within this part closely follows Carroll 2004.
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The next are samples of the final identification earlier mentioned specialised to Minkowski Area (Using the adverse sign arising within the odd amount of negatives inside the signature of the metric tensor in both sign convention):
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On the pseudo-Riemannian manifold, a person could determine a coordinate-invariant covariant tensor area whose coordinate illustration agrees Using the Levi-Civita image where ever the coordinate program is such that The idea of your tangent Area is orthonormal with regard into the metric and matches a selected orientation.
where by gab will be the representation on the metric in that coordinate procedure. We can easily similarly contemplate a contravariant Levi-Civita tensor by increasing the indices While using the metric as usual,
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