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The Levi-Civita image is most frequently Utilized in 3 and four Proportions, and also to some extent in two dimensions, so they're offered listed here in advance of defining the overall circumstance. Two dimensions[edit]

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or Minkowski space. The values with the Levi-Civita image are independent of any metric tensor and coordinate system. Also, the precise expression "image" emphasizes that it is not a tensor as a result of how it transforms concerning coordinate devices; even so it might be interpreted being a tensor density.

The expression "n-dimensional Levi-Civita image" refers to The point that the amount of indices over the image n matches the dimensionality on the vector Room in problem, which can be Euclidean or non-Euclidean, by way of example, R 3 displaystyle mathbb R ^ three

From this behavior, we can certainly deduce what $epsilon^ ij4k $ might be with no dealing with the computations Yet again: it is simply the results of exchanging the last two indices of $epsilon^ ijk4 $,

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This tensor really should not be perplexed While using the tensor density field pointed out previously mentioned. The presentation With this portion intently follows Carroll 2004.

For each sequence of nonempty open sets There's a disjoint sequence of nonempty open sets "under" it

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The covariant visit Levi-Civita tensor (also known as the Riemannian volume type) in almost any coordinate procedure that matches the selected orientation is

The important thing defining property with the image is complete antisymmetry from the indices. When any two indices are interchanged, equivalent or not, the image is negated:

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