Symbol of Levi-Civita of order NR

Definition

The symbol of Levi-Civita of order NR , $\backslash \; epsilon\_\; \{i\_1\; i\_2\; \backslash \; ldots\; i\_N\}$, also called completely antisymmetric pseudo-tensor of order NR , is a generalization of the Symbole of Levi-Civita of order 3.

Each index can take an unspecified value among NR . This symbol mainly is worth 0, except if the list of the indices is made of NR distinct values. In this case, the symbol is worth 1 or -1, the change of sign corresponding to a odd Permutation of the list of the indices.

Let us suppose for example that the list of the indices is T, X, there, Z to define a symbol of order 4. There is a priori $4^4\; =\; 256$ possible values of the symbol. The symbol $\backslash \; epsilon\_\; \{ttxz\}$ is worth 0 because the index T figure twice. So arbitrarily one chooses the sign + for $\backslash \; epsilon\_\; \{txyz\}$, then one will have $\backslash \; epsilon\_\; \{txzy\}\; =\; -1$, $\backslash \; epsilon\_\; \{tyxz\}\; =\; +1$, $\backslash \; epsilon\_\; \{tyzx\}\; =\; -1$, etc 4! /2 = 12 values are worth +1, 12 values are worth -1.

Tensor dualisor

The symbol of Levi-Civita of order NR is not a tensor. Its components do not depend on the frame of reference chosen, and by convention $\backslash \; epsilon^\; \{i\_1\; i\_2\; \backslash \; ldots\; i\_N\}\; =\; \backslash \; epsilon\_\; \{i\_1\; i\_2\; \backslash \; ldots\; i\_N\}$. On the other hand, a simple factor of standardization based on the determinant of the metric Tenseur makes it possible to define the Tensor dualisor, or tensor of Levi-Civita.

Formulas

to write

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