Definition:Einstein Summation Convention/Historical Note

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Historical Note on Einstein Summation Convention

When Albert Einstein was working with summations in the context of vectors and tensors, he noticed that the subscript or superscript to denote the index appeared twice.

So he had the idea of omitting the summation signs and instead using, for example, $x_i y_i$ to mean summation over the repeated subscripts over the entire applicable range.

Einstein himself humorously referred to this as:

... a great discovery in mathematics ...

A different (possibly apocryphal) story relates that this convention came about through a printer setting the type noticed the redundancy and suggested its omission to Einstein.