Oh sorry, right, but my original question is why is it written in a textbook that every cyclic permutation is even while the cyclic permutations ε132=ε213=ε321 are odd?micromass said:Why would ##\varepsilon_{321}## be even?
A cyclic permutation is a rearrangement of the elements in a set or sequence in which the first element becomes the last, the second becomes the first, and so on. This means that the elements are shifted in a circular pattern.
The notation "εijk" represents the Levi-Civita symbol, which is used to denote the parity (even or odd) of a permutation. It is defined as 1 if the indices ijk are an even permutation of 123, -1 if they are an odd permutation, and 0 if any two indices are the same.
The parity of a cyclic permutation can be determined by counting the number of times the elements are shifted in a circular pattern. If the number of shifts is even, the permutation is even and the Levi-Civita symbol will be 1. If the number of shifts is odd, the permutation is odd and the symbol will be -1.
No, the Levi-Civita symbol "εijk" can only be used for a 3-element cyclic permutation. For larger permutations, different notations and methods must be used to determine the parity.
Cyclic permutations are used in many areas of science, including mathematics, physics, and chemistry. They are often used to describe the rotation of objects or particles, as well as in mathematical proofs and calculations involving permutations and combinations.