What's the transformation law for the permutation

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The transformation law for permutation symbols indicates that the sign of a permutation changes when the order of elements is reversed, resulting in a negative sign for the reversed permutation. In contrast, the transformation law for Levi-Civita symbols states that exchanging any two indices results in a change of sign. For instance, swapping indices in the Levi-Civita symbol εijk yields -εjik. This transformation is crucial in tensor calculus, particularly for simplifying calculations related to determinants and cross products. Understanding these laws is essential for advanced mathematical applications.
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What's the transformation law for the permutation (or Levi-Civita) symbols?
 
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Tranformation between what and what?
 


The transformation law for permutations states that the sign of a permutation changes when the order of the elements is reversed. In other words, if we have a permutation σ = (1 2 3), then reversing the order would result in σ' = (3 2 1) and the sign of σ' would be -1, whereas the sign of σ is +1.

The transformation law for the permutation or Levi-Civita symbols is slightly different. It states that the symbol changes sign when any two indices are exchanged. For example, if we have the Levi-Civita symbol εijk and we exchange the indices i and j, the resulting symbol would be -εjik. This transformation law is important in tensor calculus and is used to simplify calculations involving determinants and cross products.
 

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