brotherbobby
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- TL;DR Summary
- The Levi-Civita symbol in three dimensions ##\epsilon_{ijk}## is sometimes defined as a determinant of various kronecker symbols as under :
$$\epsilon_{ijk}\epsilon_{pqr}=\begin{vmatrix}\delta_{ip} & \delta_{iq} & \delta_{ir}\\ \delta_{jp} & \delta_{jq} & \delta_{jr}\\ \delta_{kp} & \delta_{kq} & \delta_{kr}\\ \end{vmatrix}.
$$ Can this be shown to reduce to its more usual definition where its value is ##+1,\,-1,\,0## according to whether the indices are cyclic or repeat? [equation (1) below]
The two definitions :
The Levi Civita Alternating Symbol is defined as below, taken from here. I put the relevant image above to the right to save you the trouble of having to look through the wikipage.
\begin{equation}
\epsilon_{ijk} = \begin{cases}+1, & \text{if } (i,j,k) \text{ is an even permutation of } (1,2,3) \\-1, & \text{if } (i,j,k) \text{ is an odd permutation of } (1,2,3) \\ 0, & \text{otherwise}\end{cases}
\label{usual}
\end{equation}
But here's my problem.
\begin{equation}
\epsilon_{ijk}\;\epsilon_{lmn}=\begin{vmatrix}\delta_{il} & \delta_{im} & \delta_{in}\\ \delta_{jl} & \delta_{jm} & \delta_{jn}\\ \delta_{kl} & \delta_{km} & \delta_{kn}\\ \end{vmatrix}
\label{kronecker}
\end{equation}
The issue :
I have been through the referred text on the wikipage, marked in the image as ##{\color{blue}{^{[4]}}}##.
Nowhere does it say how does ##\ref{kronecker}## follow from ##\ref{usual}##.
I am at a loss to prove myself, where I have tried taken values like ##i=1, j=2, k=1\quad l=1, m=2, n=3##. The answers came to 1 on both sides alright, but that is not a proof. I admit I could go on and take all values from ##1,2,3## and indeed show that the second reduces to the first definition in each case. Still, won't make it a proof.
Does a proof exist whereby ##\ref{kronecker}## can be shown to reduce to ##\ref{usual}##?
Request : A hint or clue would be of immense help.
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