Einstein rule

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A convention for writing in a condensed form (without the summation symbol $\sum$) a finite sum in which every term contains the summation index twice: once as an upper, and once as a lower index. Thus, the sums $\sum_{i=1}^nx^ie_i$ and $\sum_{i,j=1}^nx^iy^ja_{ij}$ are written in the form $x^ie_i$ and $x^iy^ia_{ij}$, respectively; here $1\leq i,j\leq n$. The requirement that the indices should be written on different levels is sometimes dropped.

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


Also called the Einstein (summation) convention or simply the summation convention. It is mainly used in physics and differential geometry.

How to Cite This Entry:
Einstein rule. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article