### Levi-Civita symbol

Intro to the Levi-Civita symbol and an example with a cross product.

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In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the **Levi-Civita symbol** represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, …, *n*, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the **permutation symbol**, **antisymmetric symbol**, or **alternating symbol**, which refer to its antisymmetric property and definition in terms of permutations.

In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the **Levi-Civita symbol** represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, …, *n*, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the **permutation symbol**, **antisymmetric symbol**, or **alternating symbol**, which refer to its antisymmetric property and definition in terms of permutations.

The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis:

where *each* index *i*_{1}, *i*_{2}, …, *i*_{n} takes values 1, 2, …, *n*. There are n indexed values of *ε*_{i1i2…in}, which can be arranged into an n-dimensional array. The key definitive property of the symbol is *total antisymmetry* in all the indices. When any two indices are interchanged, equal or not, the symbol is negated:

If any two indices are equal, the symbol is zero. When all indices are unequal, we have:

where p (called the parity of the permutation) is the number of interchanges of indices necessary to unscramble *i*_{1}, *i*_{2}, …, *i*_{n} into the order 1, 2, …, *n*, and the factor (−1) is called the sign or signature of the permutation. The value *ε*_{12…n} must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose *ε*_{12…n} = +1, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article.

The term "n-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol n matches the dimensionality of the relevant vector space in question, which may be Euclidean or non-Euclidean, pure space or spacetime. The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate systems, however it can be interpreted as a tensor density.

The Levi-Civita symbol allows the determinant of a square matrix, and the cross product of two vectors in three-dimensional Euclidean space, to be expressed in index notation.

Intro to the Levi-Civita symbol and an example with a cross product.

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Everyone has their favorite method of calculating cross products. Today I go over the way I was taught, and then a more formal way of doing cross products by ...

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