# LeviCivitaTensor

gives the d-dimensional Levi-Civita totally antisymmetric tensor.

# Details • gives a rank-d tensor with length d in each dimension.
• The elements of are 0, -1, +1, and can be obtained by applying Signature to their indices.
• LeviCivitaTensor by default gives a SparseArray object. returns a normal array, while returns a symmetrized array.

# Examples

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## Scope(5)

LeviCivitaTensor returns a sparse array:

Display the result as a normal array:

Obtain the tensor as a normal array:

Build a cross product:

Obtain a higher-dimensional result as a SymmetrizedArray object:

## Applications(2)

The infinitesimal rotation matrix is the contraction of the angular velocity with a Levi-Civita tensor:

Many operations in rotational mechanics are contractions of vectors with . Torque :

Angular momentum :

The finite rotation matrix at time is the matrix exponential of :

Hodge duality can be computed by contraction with the Levi-Civita tensor:

The contraction of a TensorProduct with the Levi-Civita tensor combines Symmetrize and HodgeDual:

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:

is equivalent to applying Normal to a Levi-Civita tensor:

A normal array contains components:

The SparseArray representation of a Levi-Civita tensor contains entries:

The SymmetrizedArray representation only stores a single component:

has symmetry Antisymmetric[{1,,d}]:

The LeviCivitaTensor in dimension is the HodgeDual of 1 in that dimension:

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:

Introduced in 2008
(7.0)
|
Updated in 2014
(10.0)