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Why is Einstein summation of tensors (summation of repeated indices) named after Einstein?

"Einstein rule" in the Encyclopedia of Mathematics only says:

This rule was proposed by A. Einstein (1916).

What's the full citation?

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    $\begingroup$ @Conifold Was he the first? Also, please turn your comment into an answer. $\endgroup$
    – Geremia
    Commented May 22 at 23:42

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Einstein proposed the convention in Die Grundlage der allgemeinen Relativitätstheorie (Annalen der Physik, 354 (1916) no.7, 769-822). It is introduced on p.158 of its English translation. Einstein's source, which introduced what is now called tensor calculus, Méthodes de calcul différential absolu et leurs applications (1900) by Ricci-Curbastro and Levi-Civita, did not use the convention, and neither did Einstein's earlier joint paper with Grossman (who introduced him to the tensor calculus), Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation (1913). Einstein explained it thus:

"A glance at the equations of this paragraph shows that there is always a summation with respect to the indices which occur twice under a sign of summation ( e.g. the index $\nu$ in (5)), and only with respect to indices which occur twice. It is therefore possible, without loss of clearness, to omit the sign of summation. In its place we introduce the convention: If an index occurs twice in one term of an expression, it is always to be summed unless the contrary is expressly stated."

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  • $\begingroup$ Thanks. I was wondering if perhaps Levi-Civita or another pure mathematician used the conversion first, but it seems Einstein holds priority. $\endgroup$
    – Geremia
    Commented May 23 at 0:27

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