What's the transformation law for the permutation

Click For Summary
SUMMARY

The transformation law for permutations states that the sign of a permutation changes when the order of its elements is reversed. For example, a permutation σ = (1 2 3) has a sign of +1, while its reverse σ' = (3 2 1) has a sign of -1. The transformation law for Levi-Civita symbols indicates that the symbol changes sign when any two indices are exchanged; for instance, exchanging indices in εijk results in -εjik. This law is crucial in tensor calculus for simplifying calculations involving determinants and cross products.

PREREQUISITES
  • Understanding of permutation groups
  • Familiarity with Levi-Civita symbols
  • Basic knowledge of tensor calculus
  • Concept of determinants in linear algebra
NEXT STEPS
  • Study the properties of permutation groups in abstract algebra
  • Learn about the applications of Levi-Civita symbols in physics
  • Explore tensor calculus and its relevance in differential geometry
  • Investigate the role of determinants in linear transformations
USEFUL FOR

Mathematicians, physicists, and students studying linear algebra or tensor calculus who seek to understand the implications of permutation and Levi-Civita symbols in their work.

elessar_telkontar
Messages
16
Reaction score
0
What's the transformation law for the permutation (or Levi-Civita) symbols?
 
Physics news on Phys.org
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.
 

Similar threads

Network Analysis, Kirchhoff's Laws, changing current source to voltage
  • · Replies 4 ·
Replies
4
Views
2K
Applying Ampere's Law to a Transformer
  • · Replies 5 ·
Replies
5
Views
4K
Galilean transform and Lorentz transform questions
  • · Replies 2 ·
Replies
2
Views
2K
Time dependent canonical transformation
  • · Replies 3 ·
Replies
3
Views
2K
Understanding Velocity Transformation
  • · Replies 8 ·
Replies
8
Views
2K
Importance/Use of Gauss' Law and Ampere's Circuital Law
  • · Replies 30 ·
2
Replies
30
Views
2K
Verifying Kirchoff's law in an AC RLC circuit
  • · Replies 7 ·
Replies
7
Views
2K
Apply the Delta Wye Transform to a Circuit
  • · Replies 13 ·
Replies
13
Views
4K
A problem about the laws of conservation: Bullet colliding with a wood block
  • · Replies 28 ·
Replies
28
Views
2K
Ohm's Law confusion -- How to handle an incandescent bulb?
  • · Replies 7 ·
Replies
7
Views
2K