Levi-Civita and Kronecker delta identity, proof with determinants

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Homework Help Overview

The discussion revolves around the Levi-Civita and Kronecker delta identity, specifically focusing on a proof involving determinants as presented in a textbook on tensor calculus. The original poster expresses confusion regarding the author's explanation in the proof, particularly about summing deltas and the significance of the number three in the context of determinants.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to clarify the author's statements regarding the summation of deltas and the treatment of determinants in the proof. They question whether the deltas should be summed individually or as part of the entire determinant expression. Additionally, they seek to understand the origin of the number three in the context of the proof.

Discussion Status

Some participants have engaged with the original poster's questions, offering insights into the manipulation of the Kronecker delta in relation to the determinants. There appears to be a productive exchange of ideas, although no consensus has been reached regarding the interpretation of the proof.

Contextual Notes

The discussion references an attachment containing equations and details pertinent to the problem, which may influence the understanding of the proof. The original poster's approach to writing out the determinants is noted as a potential barrier to clarity in the proof's progression.

Pifagor
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Homework Statement



I'm trying to understand a proof of the LC-KD identity involving determinants (see attachment), from the book Introduction to Tensor Calculus and Continuum Mechanics by Herinbockel.
What is the author saying in the last line of text? How can we sum the deltas in the upper right corner, shouldn't we sum the three determinants as a whole, since we're doing that on the other side of the equation? And how does the sum 3 come into the picture anyway?

Homework Equations



See attachment.

The Attempt at a Solution



It seems I should write out the whole determinant for all i,j,k,r,s,t, but that would not make the proof any easier than doing it by "brute force" in the first place!

Thanks for any help.
 

Attachments

Last edited:
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Welcome to PF!

Hi Pifagor! Welcome to PF! :smile:
Pifagor said:
See attachment.

erm :redface: …

no attachment! :biggrin:​
 
.../four letter word /... now it should be OK
 
that's better! :biggrin:

if you use the δit on the top right to replace all the is by ts in its cofactor, you get minus the cofactor for δii

the same for δis, so you have (3 - 1 - 1) times that cofactor :wink:
 

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