# Levi-Civita symbol

The **Levi-Civita symbol**, also called the permutation symbol or the antisymmetric symbol, is a collection of numbers \( \varepsilon_{i_1 i_2 \ldots i_n} \) defined from the sign of a permutation of the natural numbers 1, 2, ..., n: it equals +1 when its indices are an even permutation of the ordered sequence, −1 for an odd permutation, and 0 when any two indices are equal.<sup>[1](https://www.astro.yale.edu/vdbosch/astro320_notes.pdf)</sup> It is named after the Italian mathematician [Tullio Levi-Civita](https://www.edgepedia.org/tullio-levi-civita) (1873–1941), one of the founders of tensor calculus.<sup>[2](https://www.youtube.com/watch?v=ChlRE4JRnzs)</sup> The symbol appears throughout linear algebra, vector and tensor calculus, and differential geometry, most familiarly as the component form of the cross product and the determinant.<sup>[1](https://www.astro.yale.edu/vdbosch/astro320_notes.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \varepsilon_{i_1 \ldots i_n} = +1 \) for an even permutation of \( 12 \ldots n \), −1 for an odd permutation, 0 if not a permutation of distinct indices<sup>[3](https://pages.physics.ua.edu/staff/fabi/ph354/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf)</sup> |
| Count of values | Of the \( n^n \) index choices, \( n!/2 \) give +1, \( n!/2 \) give −1, and the rest give 0<sup>[3](https://pages.physics.ua.edu/staff/fabi/ph354/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf)</sup> |
| Determinant | \( \det A = \varepsilon_{i_1 \ldots i_n} A^1_{i_1} A^2_{i_2} \cdots A^n_{i_n} \) for an \( n \times n \) matrix \( A \)<sup>[3](https://pages.physics.ua.edu/staff/fabi/ph354/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf)</sup> |
| Epsilon-delta identity | \( \varepsilon_{ijk}\varepsilon_{lmk} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl} \)<sup>[4](https://sosovski.github.io/Constitutive_modeling/lectures/A03-axial-permutation-appendix.html)</sup> |
| Cross product | \( (\mathbf{A} \times \mathbf{B})^i = \varepsilon_{ijk} A^j B^k \) in three dimensions<sup>[5](https://www.sas.rochester.edu/pas/undergraduate/reu/files/mathprimer-grad07.pdf)</sup> |
| Tensor status | Not a tensor; it transforms with an extra factor of \( \det S \), making it a tensor density<sup>[3](https://pages.physics.ua.edu/staff/fabi/ph354/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf)</sup> |

## Definition

The n-dimensional Levi-Civita symbol \( \varepsilon_{i_1 i_2 \ldots i_n} \) takes n indices, each running from 1 to n, so it is fully specified by \( n^n \) values.<sup>[3](https://pages.physics.ua.edu/staff/fabi/ph354/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf)</sup> Its defining property is total antisymmetry: interchanging any two indices negates the symbol,<sup>[1](https://www.astro.yale.edu/vdbosch/astro320_notes.pdf)</sup>

\[ \varepsilon_{\ldots i_p \ldots i_q \ldots} = -\varepsilon_{\ldots i_q \ldots i_p \ldots}. \]

When all indices are distinct, \( \varepsilon_{i_1 i_2 \ldots i_n} = (-1)^p \, \varepsilon_{1,2,\ldots,n} \), where p is the parity of the permutation, the number of pairwise interchanges needed to unscramble the indices back into ascending order.<sup>[1](https://www.astro.yale.edu/vdbosch/astro320_notes.pdf)</sup> For example, the list (3, 4, 5, 2, 1) has parity p = 3, so \( \varepsilon_{34521} = -1 \).<sup>[1](https://www.astro.yale.edu/vdbosch/astro320_notes.pdf)</sup>

In three dimensions the definition reads explicitly:<sup>[1](https://www.astro.yale.edu/vdbosch/astro320_notes.pdf)</sup>

\[ \varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is } (1,2,3), (2,3,1), \text{, or } (3,1,2), \\ -1 & \text{if } (i,j,k) \text{ is } (3,2,1), (1,3,2), \text{, or } (2,1,3), \\ 0 & \text{if any two indices are equal.} \end{cases} \]

The symbol is nonzero in only \( n! \) of its \( n^n \) cases: +1 in \( n!/2 \) of them and −1 in \( n!/2 \).<sup>[3](https://pages.physics.ua.edu/staff/fabi/ph354/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf)</sup> A practical way to evaluate any component is to write the indices in ascending order, set that value to +1, and count adjacent transpositions.<sup>[5](https://www.sas.rochester.edu/pas/undergraduate/reu/files/mathprimer-grad07.pdf)</sup>

## Determinants and the epsilon-delta identity

The symbol gives a compact formula for determinants. For an \( n \times n \) matrix A,<sup>[3](https://pages.physics.ua.edu/staff/fabi/ph354/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf)</sup>

\[ \det A = \varepsilon_{i_1 i_2 \ldots i_n} A^1_{i_1} A^2_{i_2} \cdots A^n_{i_n}, \]

with implied summation over all indices. Although this sum contains \( n^n \) terms rather than the familiar \( n! \), only \( n! \) are nonzero, and each surviving term is a product of n matrix elements taken from all rows and all columns, signed by the parity of the row permutation.<sup>[3](https://pages.physics.ua.edu/staff/fabi/ph354/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf)</sup> For a 3 × 3 matrix this is \( \det A = \varepsilon_{ijk} A_{i1} A_{j2} A_{k3} \).<sup>[6](https://www.physics.muni.cz/~petrk/skripta_eng.pdf)</sup>

Products of two symbols reduce to determinants of Kronecker deltas:<sup>[6](https://www.physics.muni.cz/~petrk/skripta_eng.pdf)</sup>

\[ \varepsilon_{ijk}\varepsilon_{lmn} = \det \begin{pmatrix} \delta_{il} & \delta_{im} & \delta_{in} \\ \delta_{jl} & \delta_{jm} & \delta_{jn} \\ \delta_{kl} & \delta_{km} & \delta_{kn} \end{pmatrix}. \]

Contracting one index gives the most used identity in index manipulations,<sup>[4](https://sosovski.github.io/Constitutive_modeling/lectures/A03-axial-permutation-appendix.html)</sup>

\[ \varepsilon_{ijk}\varepsilon_{lmk} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}, \]

and further contractions give \( \varepsilon_{ijk}\varepsilon_{jkl} = 2\delta_{il} \) and \( \varepsilon_{ijk}\varepsilon_{ijk} = 6 \) in three dimensions.<sup>[6](https://www.physics.muni.cz/~petrk/skripta_eng.pdf)</sup> These identities are the working tools for proving vector identities without expanding components.<sup>[5](https://www.sas.rochester.edu/pas/undergraduate/reu/files/mathprimer-grad07.pdf)</sup>

## Vector calculus

The cross product is the symbol's best-known application. In components,<sup>[5](https://www.sas.rochester.edu/pas/undergraduate/reu/files/mathprimer-grad07.pdf)</sup>

\[ (\mathbf{A} \times \mathbf{B})^i = \varepsilon_{ijk} A^j B^k. \]

Antisymmetry of \( \varepsilon_{ijk} \) immediately yields \( \mathbf{A} \times \mathbf{B} = -\mathbf{B} \times \mathbf{A} \), and the epsilon-delta identity yields the double cross product rule \( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (\mathbf{A} \cdot \mathbf{C})\mathbf{B} - (\mathbf{A} \cdot \mathbf{B})\mathbf{C} \) and the [scalar triple product](https://www.edgepedia.org/scalar-triple-product) identity \( \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \mathbf{B} \cdot (\mathbf{C} \times \mathbf{A}) = \mathbf{C} \cdot (\mathbf{A} \times \mathbf{B}) \).<sup>[6](https://www.physics.muni.cz/~petrk/skripta_eng.pdf)</sup> The same machinery handles differential operators; for example, \( \nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} \) follows from a two-line index calculation.<sup>[5](https://www.sas.rochester.edu/pas/undergraduate/reu/files/mathprimer-grad07.pdf)</sup> In continuum mechanics, the axial vector \( \mathbf{w} \) of a skew-symmetric tensor W is extracted as \( w_i = -\tfrac{1}{2}\varepsilon_{ijk} W_{jk} \).<sup>[4](https://sosovski.github.io/Constitutive_modeling/lectures/A03-axial-permutation-appendix.html)</sup>

## Symbol versus tensor

Although the Levi-Civita symbol is written in tensor notation, it does not transform as a tensor. If \( \varepsilon_{i_1 \ldots i_n} \) were an \( (0,n) \)-tensor, a change of basis by a matrix S would have to give the usual transformation law; instead, the transformed components pick up an extra factor,<sup>[3](https://pages.physics.ua.edu/staff/fabi/ph354/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf)</sup>

\[ \tilde{\varepsilon}_{j_1 \ldots j_n} = (\det S)\, \varepsilon_{j_1 \ldots j_n}. \]

This is why the word symbol is used: the values \( \{1, -1, 0\} \) hold in every basis, which no genuine tensor with these components could manage.<sup>[3](https://pages.physics.ua.edu/staff/fabi/ph354/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf)</sup> The object is instead a tensor density: the lower-index version has weight +1 and the upper-index version weight −1.<sup>[7](https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_%28Crowell%29/04%3A_Tensors/4.07%3A_Things_that_Arent_Quite_Tensors)</sup> In tensor-calculus terminology, the permutation systems \( e_{ijk} \) and \( e^{ijk} \) are relative tensors whose transformation rules differ from those of true tensors only by the factor \( \det J \).<sup>[8](https://grinfeld.org/books/An-Introduction-To-Tensor-Calculus/Chapter17.html)</sup>

In general relativity this volume form supplies the invariant volume element used in integration, and the Hodge star dual of a differential form is built from the Levi-Civita duality operation combined with the metric to raise indices.<sup>[9](https://people.physics.illinois.edu/stone/bookmaster.pdf)</sup>

## Historical context

The symbol carries the name of Tullio Levi-Civita, who studied under [Gregorio Ricci-Curbastro](https://www.edgepedia.org/gregorio-ricci-curbastro) at Padua and co-authored with him the 1901 memoir "Méthodes de calcul différentiel absolu et leurs applications" in the *Mathematische Annalen*, the systematic exposition of what is now called tensor calculus.<sup>[10](https://archimede.dimai.unifi.it/archimede/matematicaitaliana/schede_opere/67leviricci01.html)</sup> That calculus, popularized by Einstein's general theory of relativity, is the setting in which the distinction between the numerical symbol and the metric-dependent volume tensor becomes necessary.<sup>[11](https://link.springer.com/article/10.1007/s00283-026-10560-z)</sup>

## References

1. [The Levi-Civita Symbol, Appendix E, ASTRO 320 notes, Yale University](https://www.astro.yale.edu/vdbosch/astro320_notes.pdf)
2. [The Levi-Civita Symbol, Tensor Calculus #14 (video description), Angewandte Mathematik für Ingenieure](https://www.youtube.com/watch?v=ChlRE4JRnzs)
3. [A Gentle Introduction to Tensors (B. Porat, 2014), Section 1.13](https://pages.physics.ua.edu/staff/fabi/ph354/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf)
4. [Appendix A: Skew Tensors, Permutation, and Tensor Algebra, Constitutive Models in Solid Mechanics](https://sosovski.github.io/Constitutive_modeling/lectures/A03-axial-permutation-appendix.html)
5. [Math Primer: Using the Levi-Civita Symbol, University of Rochester](https://www.sas.rochester.edu/pas/undergraduate/reu/files/mathprimer-grad07.pdf)
6. [Basics of Vector and Tensor Algebra, Masaryk University lecture notes](https://www.physics.muni.cz/~petrk/skripta_eng.pdf)
7. [Things that Aren't Quite Tensors, Physics LibreTexts (General Relativity, Crowell)](https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_%28Crowell%29/04%3A_Tensors/4.07%3A_Things_that_Arent_Quite_Tensors)
8. [The Levi-Civita Symbol and the Cross Product, An Introduction to Tensor Calculus, Chapter 17](https://grinfeld.org/books/An-Introduction-To-Tensor-Calculus/Chapter17.html)
9. [Differential Calculus on Manifolds (M. Stone), Chapter 11, University of Illinois](https://people.physics.illinois.edu/stone/bookmaster.pdf)
10. [La matematica italiana 1800–1950: Tullio Levi Civita e Gregorio Ricci Curbastro](https://archimede.dimai.unifi.it/archimede/matematicaitaliana/schede_opere/67leviricci01.html)
11. [From Differential Geometry to Relativity: Levi-Civita's Lectures on the Absolute Differential Calculus, The Mathematical Intelligencer](https://link.springer.com/article/10.1007/s00283-026-10560-z)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Multilinear and tensor algebra › Tensor manipulation and transformations*

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