I confirmed the contraction id by creating down Every case for all indices. But that's not a proof, merely a verification, and I believe verifying in the same style for 4th dimension will likely be cumbersome.

I'm using the Einstein notation, so multiple indices are summed) or they provide two 3-indices symbols with various indication

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In almost any arbitrary curvilinear coordinate process as well as from the absence of the metric to the manifold, the Levi-Civita image as defined higher than can be thought to be a tensor density discipline in two various ways.

or Minkowski space. The values with the Levi-Civita image are impartial of any metric tensor and coordinate system. Also, the precise phrase "symbol" emphasizes that it's not a tensor as a consequence of how it transforms amongst coordinate methods; having said that it could be interpreted as being a tensor density.

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but discover that In case the metric signature incorporates an odd quantity of adverse eigenvalues q, then the sign read more here in the parts of the tensor vary from your common Levi-Civita image:[nine]

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