Understand Vector Product with Levi Civita Symbols

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SUMMARY

The discussion centers on the vector product using Levi Civita symbols and their relationship with Kronecker deltas. The user seeks clarification on how the product of two Levi Civita symbols can be expressed as Kronecker deltas, as well as the proof provided in the referenced URL. Additionally, the discussion involves the dot product of two vector products, specifically (a × b) · (c × d), which simplifies to c · ((d · b)a - (d · a)b) when using Levi Civita symbols. This mathematical framework is essential for understanding advanced vector calculus in physics.

PREREQUISITES
  • Understanding of vector calculus
  • Familiarity with Levi Civita symbols
  • Knowledge of Kronecker delta notation
  • Basic principles of tensor algebra
NEXT STEPS
  • Study the properties of Levi Civita symbols in detail
  • Learn about Kronecker delta and its applications in tensor calculus
  • Explore vector identities involving cross products and dot products
  • Review proofs related to vector products and their geometric interpretations
USEFUL FOR

This discussion is beneficial for graduate physics students, educators in vector calculus, and anyone interested in advanced mathematical concepts related to physics and engineering.

Septim
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Hello I am a graduate student and it is my first year in physics. Our instructor showed vector product with Levi Civita symbol and I did not understand the part where the product of two Levi Civita symbols is expressed as Kronecker Deltas. Additionally I did not get the proof on the URL http://folk.uio.no/patricg/teaching/a112/levi-civita/ can you explain these. Thanks for your contribution.
 
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img8.png


now take dot product of two vector products
(aXb).(cXd), which can be expressed further as
(aXb).(cXd)=c.((d.b)a - (d.a)b)

plug in the values of vector product using above equation(which includes levi civita symbols)
and then finally this would lead to the expression
img19.png
 

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