How to Expand and Simplify the Expression of Kronecker Delta?

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Discussion Overview

The discussion revolves around the expansion and simplification of the expression involving the Kronecker delta, specifically the expression $$\delta_{ij}\delta_{ij}$$. Participants explore the implications of this expression in the context of linear algebra and continuum mechanics.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant, Jason, presents their solution to the problem, expressing uncertainty about whether they have satisfied the requirement to "expand" the expression.
  • Another participant confirms that Jason's working is correct under the assumption of 3 dimensions and the use of the Einstein summation convention.
  • Jason seeks clarification on what is meant by "Expand" in the context of the problem.
  • A later reply suggests that expansion involves writing out every term in the sum, which then simplifies to a compact number as shown in Jason's solution.

Areas of Agreement / Disagreement

Participants generally agree on the correctness of the mathematical manipulation presented by Jason, but there is uncertainty regarding the interpretation of the term "Expand" and what is specifically required by the problem.

Contextual Notes

The discussion does not resolve the ambiguity surrounding the term "Expand," and there may be assumptions about dimensionality and notation that are not explicitly stated.

jasonmcc
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Hi, I'm working on a problem stated as:
Expand the following expression and simplify where possible
$$
\delta_{ij}\delta_{ij}
$$

I'm pretty sure this is correct, but not sure that I am satisfying the expand question. I'm not up to speed in linear algebra (taking a continuum mechanics course) - the question could be asking for $\hat{e}$ or matrix type expansion...

my solution:
\begin{alignat}{3}
\delta_{ij}\delta_{ij} & = & \delta_{ij}\delta_{ji}\\
& = & \delta_{ii}\\
& = & 3
\end{alignat}

Any suggestions for how to expand - or does this answer the question?
Thanks, Jason
 
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Your working is correct if you're in 3 dimensions and you're using the Einstein summation convention.
 
Thanks. What do you think they mean by "Expand", then?
 
When you expand a sum like that, you're writing out every term, essentially. And then because they combine as you've shown, it collapses down to a nice compact number.
 

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