Properties
A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called a permutation tensor. It is actually a pseudotensor because under an orthogonal transformation of jacobian determinant −1 (i.e., a rotation composed with a reflection), it acquires a minus sign. As the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector.
Under a general coordinate change, the components of the permutation tensor are multiplied by the jacobian of the transformation matrix. This implies that in coordinate frames different from the one in which the tensor was defined, its components can differ from those of the Levi-Civita symbol by an overall factor. If the frame is orthonormal, the factor will be ±1 depending on whether the orientation of the frame is the same or not.
In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual.
In these examples, superscripts should be considered equivalent with subscripts.
Read more about this topic: Levi-Civita Symbol
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