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Old May 7th 2019, 05:44 AM   #1
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Index notation

Does anyone know how to demonstrate this identity?

Let $\displaystyle D=det(a_{ij})$

$\displaystyle \epsilon_{ijk}\epsilon_{pqr}D=

\begin{vmatrix}
a_{ip} & a_{iq} & a_{ir} \\
a_{jp} & a_{jq} & a_{jr} \\
a_{kp} & a_{kq} & a_{kr}
\end{vmatrix}
$

The book demonstrates this way:

"If (at least) two of i, j, k or two of p, q, r are equal, then both sides of (1.7.21) are 0. If i, j, k and p, q, r are both cyclic or acyclic, then each side of (1.7.21) is equal to D. If i, j, k are cyclic but p, q, r are acyclic or vice versa, then each side of (1.7.21) is equal to -D. Thus, in all possible cases, result (1.7.21) is verified."

But for me it doesn't make sense, this sum would give 0.
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Old May 8th 2019, 03:08 AM   #2
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I've never seen that before. However, calculating the determinant of the RHS could yield a result which can be compared more easily with the LHS. Perhaps that helps constrain the $\displaystyle \epsilon$ values?

Another option is just to examine each case by substituting indices into the formula. For example, for the first case, try setting j = i and q = p and see how the results change?
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Last edited by benit13; May 8th 2019 at 03:10 AM.
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Old May 8th 2019, 09:12 AM   #3
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I'd like to see if there is another possibility before expanding the terms into sums.

I'm trying to use this identity:

$\displaystyle \epsilon_{ijk}D=\epsilon_{pqr}a_{ip}a_{jq}a_{kr}$

But without success.

These results are in the book "Continuum Mechanics" by Chandrasekharaiah
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Old May 8th 2019, 10:27 AM   #4
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Originally Posted by mscfd View Post
I'd like to see if there is another possibility before expanding the terms into sums.

I'm trying to use this identity:

$\displaystyle \epsilon_{ijk}D=\epsilon_{pqr}a_{ip}a_{jq}a_{kr}$

But without success.

These results are in the book "Continuum Mechanics" by Chandrasekharaiah
This is the right way to proceed, can you show some working as to why you can't develop your identity?
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Old May 8th 2019, 11:30 AM   #5
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Studiot,


I'm afraid you didn't understand my intention. I didn't create this topic to ask for opinions on the ways I'm going to try to solve this problem. I created this topic to talk about possible ways to solve this problem. This is not my "homework"...
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Old May 8th 2019, 11:46 AM   #6
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Epsilon i,j,k is a tensor. and the indices are dummies.

So you can peform a tensor contraction on it.

So what is its square?

Look here

https://math.stackexchange.com/quest...is-determinant

or here

https://www.physicsforums.com/thread...entity.834083/
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Old May 8th 2019, 12:35 PM   #7
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I think i, j, k and p, q, r are free suffixes, right?

$\displaystyle \epsilon_{ijk}=\hat{e}_i . (\hat{e}_j \times \hat{e}_k) \in\{-1,0,1\}$

I think the best way is substituting indices into the formula.

I'll do it and post it later.
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