Einstein Summation Convention

This is an introduction to the Einstein summation convention used widely in the theoretical contexts in Physics.

The definition of the convention is simple: repeating upper and lower indices implies summation, unless stated otherwise, i.e.

\[x^iy_i = \sum_{i\in I}x^iy_i\]

where set \(I\) is the set of possible values of the indices.

Please notice that there are other versions of Einstein summation convention. One version may consider all the dummy indices as summation. However, in this context, only the repeated upper and lower indices should be interpreted as summation, expression as \(x_iy_i\) does not apply to the summation convention.