How to decide whether an equation in index notation is valid.

Question

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?

Answer 1

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.

Answer 2

$$ k_{ijkl} = \underbrace{a_{i}b_{kl}c_{njm}d_{mn}}_{x_{ijkl}} + \underbrace{e_{ik}e_{jn}f_{n}}_{y_{ijk}} $$ 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 $i$.