gives the d-dimensional Levi-Civita totally antisymmetric tensor.
- LeviCivitaTensor[d] gives a rank-d tensor with length d in each dimension.
- The elements of LeviCivitaTensor[d] are 0, -1, +1, and can be obtained by applying Signature to their indices.
- LeviCivitaTensor by default gives a SparseArray object. LeviCivitaTensor[d,List] returns a normal array, while LeviCivitaTensor[d,SymmetrizedArray] returns a symmetrized array.
Examplesopen allclose all
Basic Examples (1)
In dimension three, Hodge duality is often used to identify the cross product and TensorWedge of vectors:
Properties & Relations (7)
Components of the Levi-Civita tensor coincide with the value of Signature:
The SparseArray representation of a Levi-Civita tensor contains entries:
The SymmetrizedArray representation only stores a single component:
The determinant Det[m] is the contraction of m's rows or columns into the Levi-Civita tensor:
Cross in dimension is the contraction of vectors into the Levi-Civita tensor:
Wolfram Research (2008), LeviCivitaTensor, Wolfram Language function, https://reference.wolfram.com/language/ref/LeviCivitaTensor.html (updated 2014).
Wolfram Language. 2008. "LeviCivitaTensor." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/LeviCivitaTensor.html.
Wolfram Language. (2008). LeviCivitaTensor. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LeviCivitaTensor.html