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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 Room. The values of your Levi-Civita symbol are impartial of any metric tensor and coordinate process. Also, the specific term "symbol" emphasizes that it is not a tensor as a consequence of the way it transforms among coordinate programs; on the other hand it can be interpreted as a tensor density.

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This tensor should not be perplexed Together with the tensor density area talked about over. The presentation Within this segment carefully follows Carroll 2004.

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but recognize that When the metric signature incorporates an odd amount of unfavorable eigenvalues q, then the signal of the components of the tensor differ in the common Levi-Civita symbol:

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Over a pseudo-Riemannian manifold, one particular could define a coordinate-invariant covariant tensor discipline whose coordinate illustration agrees Using the Levi-Civita image anywhere the coordinate system is such that The premise of your tangent Area is orthonormal with respect on the metric and matches a particular orientation.

wherever gab may be the representation of the metric in that coordinate procedure. We could in the same way consider a contravariant Levi-Civita tensor by elevating the indices Using the metric as standard,

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