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I am given the following equation in index notation: $k_{ijkl} = a_{i}b_{kl}c_{njm}d_{mn} + e_{ik}e_{jn}f_{n}$. I am told that this is a valid equation, but can anyone explain why? It doesn't violate the summation convention, and there's no obvious illegal characters in there, but how does one decide whether an equation as complicated as this is valid?

wrb98
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    What does "valid" mean here? – hmakholm left over Monica Feb 03 '17 at 10:08
  • Could you add some context? – Janik Feb 03 '17 at 10:08
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    Basically, it's part of an introductory worksheet to index notation. Valid would be something like the dot product in index notation, but invalid would be something like $a_{i}b_{i}c_{i}$ because the same index appears more than twice in a single term (i.e. it is not allowed by the rules of index notation) – wrb98 Feb 03 '17 at 10:10

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$$ k_{ijkl} = \underbrace{a_{i}b_{kl}c_{njm}d_{mn}}_{x_{ijkl}} + \underbrace{e_{ik}e_{jn}f_{n}}_{y_{ijk}} $$ While the second term is missing the index $l$, this is not bad, the right hand side is defined for any $(i,j,k,l)$.

Compare it to a vector $x = (x_i)$ with $x_i = 2$. It is just constant regarding $i$.

mvw
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Hint: Check if all non-free indices are on both sides. Left side we have $i,j,k,l$. On the right-hand side, we have for the first term $i,k,l,j$ ($m$ and $l$ are repeated indices, which by itself is not valid as far as I remember) and for the second term we have $i,k,j$ ($n$ is a repeated index). Hence, this expression is not valid.

MrYouMath
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  • I thought the non-repeated indices had to appear on both sides. $i,j,k,l$ appear on the LHS and the first term of the RHS. Do they have to be in every term of the RHS? – wrb98 Feb 03 '17 at 10:15
  • As far as I remember they have to be in each additive term. It would not make sense if you had different indices in each term. – MrYouMath Feb 03 '17 at 10:16