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If we have an expression like $$a^ib_i + c^{ij}d_{ij} + e^{ijk} f_{ijk}$$ should it be interpreted as $$\sum_{i,j,k} a^ib_i + c^{ij}d_{ij} + e^{ijk} f_{ijk} $$ or $$\sum_i a^ib_i + \sum_{i,j} c^{ij}d_{ij} + \sum_{i,j,k} e^{ijk} f_{ijk}~?$$ I'm thinking the second one is the answer: we append different summations to different summands.

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    $\begingroup$ Yes, second one. $\endgroup$
    – lalala
    Dec 15, 2019 at 20:29
  • $\begingroup$ I doubt it's possible to generate those sums using tensors since it implies adding tensors of different orders is allowed - which is not true. However, assuming it is possible, since addition is associative, I can relabel the sums - similar to the relabeling of the sum for the Christoffel symbol in the covariant derivative. I would argue $(2)$ doesn't make sense, and $(1)$ is equivalent to $(3)$. $\endgroup$ Dec 15, 2019 at 22:41
  • $\begingroup$ @CinaedSimson You're right; in that case, what I mean is something like $a^{ij} b_{ik} + c^{im} d_{im}$ $\endgroup$
    – finnlim
    Dec 15, 2019 at 22:47
  • $\begingroup$ @CinaedSimson There are many examples where this shows up. Think of the definition of the Ricci scalar. In the $\partial \Gamma$ terms three pairs of indices are contracted and in the $\Gamma\Gamma$ terms four pairs are contracted. $\endgroup$
    – MannyC
    Dec 16, 2019 at 0:31
  • $\begingroup$ @MannyC: the OP examples were all contractions - for $3$ tensors each with a different order and summed as if they generated by the same object. Regarding the Ricci tensor, the scalar curvature $S$ is the contraction between the metric tensor and Ricci tensor, namely, $S=\sum g^{ij}R_{ij}$. $\endgroup$ Dec 16, 2019 at 3:22

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It means the second thing. There are a couple of reasons why it needs to be this way.

In ordinary algebra, if $A=B$ and $C=D$, then $A+C=B+D$. Under your first interpretation, this would fail when you substituted sums for the symbols $A$...

The notation also involves a lot of implicit use of the metric to raise and lower indices. The metric is being multiplied by things, and this has higher priority than addition.

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