# 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:

Wolfram Research (2008), LeviCivitaTensor, Wolfram Language function, https://reference.wolfram.com/language/ref/LeviCivitaTensor.html (updated 2014).

#### Text

Wolfram Research (2008), LeviCivitaTensor, Wolfram Language function, https://reference.wolfram.com/language/ref/LeviCivitaTensor.html (updated 2014).

#### CMS

Wolfram Language. 2008. "LeviCivitaTensor." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/LeviCivitaTensor.html.

#### APA

Wolfram Language. (2008). LeviCivitaTensor. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LeviCivitaTensor.html

#### BibTeX

@misc{reference.wolfram_2022_levicivitatensor, author="Wolfram Research", title="{LeviCivitaTensor}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/LeviCivitaTensor.html}", note=[Accessed: 09-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_levicivitatensor, organization={Wolfram Research}, title={LeviCivitaTensor}, year={2014}, url={https://reference.wolfram.com/language/ref/LeviCivitaTensor.html}, note=[Accessed: 09-June-2023 ]}