Levi-Civita Tensor: Index Interchange Identity

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SUMMARY

The Levi-Civita tensor identity, specifically the equation \(\epsilon_{ijk} a_j b_k = -\epsilon_{ijk} a_k b_j\), holds true. This conclusion is derived from the property of the Levi-Civita symbol that allows for the interchange of indices, demonstrated by the transformation \(\epsilon_{ijk} a_j b_k = -\epsilon_{ikj} a_j b_k\). By relabeling the dummy indices \(j\) and \(k\), the identity is confirmed as valid.

PREREQUISITES
  • Understanding of tensor notation and properties
  • Familiarity with the Levi-Civita symbol
  • Basic knowledge of index manipulation in mathematical expressions
  • Concept of dummy indices in tensor calculus
NEXT STEPS
  • Study the properties of the Levi-Civita tensor in depth
  • Explore applications of the Levi-Civita symbol in physics, particularly in electromagnetism
  • Learn about tensor calculus and its relevance in general relativity
  • Investigate the role of antisymmetric tensors in various mathematical frameworks
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Mathematicians, physicists, and students studying advanced calculus or theoretical physics, particularly those focusing on tensor analysis and its applications in various fields.

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Does the following identity hold?:

[tex] \epsilon_{ijk} a_j b_k = -\epsilon_{ijk} a_k b_j[/tex]
 
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Yes.
 


The answer is yes; clearly we have:
[tex] \epsilon_{ijk} a_j b_k = -\epsilon_{ikj} a_j b_k[/tex]

from which we can obtain your identity by relabeling the dummy indices j and k.
 

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