or Minkowski Place. The values in the Levi-Civita symbol are impartial of any metric tensor and coordinate program. Also, the specific expression "symbol" emphasizes that it's not a tensor as a consequence of how it transforms between coordinate devices; even so it can be interpreted to be a tensor density.
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Examples of Levi-Civita properties in four Proportions incorporate the symmetric and antisymmetric nature with the tensor, its relation to your metric tensor, and its transformation Houses beneath coordinate transformations.
The Levi-Civita symbol allows the determinant of the square matrix, as well as the cross item of two vectors in a few-dimensional Euclidean Place, to get expressed in Einstein index notation.
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ε i j k ε l m n =
The typical letters to denote the Levi-Civita symbol would be the Greek lower circumstance epsilon ε or ϵ, or considerably less typically the Latin lower circumstance e.
Over a pseudo-Riemannian manifold, 1 may perhaps outline a coordinate-invariant covariant tensor industry whose coordinate representation agrees with the Levi-Civita image wherever the coordinate technique is this kind of that The premise in the tangent House is orthonormal with respect to your metric and matches a selected orientation.