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or Minkowski Area. The values in the Levi-Civita symbol are unbiased of any metric tensor and coordinate system. Also, the precise expression "symbol" emphasizes that it's not a tensor on account of how it transforms concerning coordinate devices; even so it can be interpreted for a tensor density.
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This tensor shouldn't be baffled Using the tensor density field stated higher than. The presentation On this portion carefully follows Carroll 2004.
The Levi-Civita symbol lets the determinant of the square matrix, along with the cross product of two vectors in 3-dimensional Euclidean House, for being expressed in Einstein index notation.
Using the capital pi notation Π for normal multiplication of figures, an explicit expression to the symbol is:[citation necessary]
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On a pseudo-Riemannian manifold, just one might outline a coordinate-invariant covariant tensor industry whose coordinate representation agrees Along with the Levi-Civita image anywhere the coordinate system is these kinds of that The idea with the tangent House is orthonormal with regard towards the metric and matches a selected orientation.
where gab could be the illustration from the metric in that coordinate program. We are able to in the same way think about a contravariant Levi-Civita tensor by elevating the indices Together with the metric as standard,
Up to three sets of CTs per phase might be paralleled for every meter with no want for a meter multiplier