[NOTES/ME-02007]-Einstein Summation Convention

We describe the Einstein summation convention and give some examples.

1. Summation convention If \(\vec{x}=(x_1, x_2,x_3)\) is vector, square of its length is given by \[ |\vec{x}|^2 = \sum_{i=1}^3 x_i^2.\] We can rewrite it as \[ |\vec{x}|^2 = \sum_{i=1}^3 x_i x_i.\] In this form the index \(i\) is repeated and is summed over all values. The {\it Einstein summation convention } says all repeated indices are automatically summed over all possible values. With this convention we write \[ |\vec{x}|^2 = x_i x_i.\]
2. Dummy indexThe index which is summed over all values is called a {\it dummy index}. A dummy index can be replaced with any other index taking the same set of values. Thus we can write \(|\vec{x}|^2 \) as \(x_i x_i\), or as \(x_j x_j\). Obviously the two expressions are equal.
3. Free index must balance An index which appears only once in an expression is not summed, is called a {\it free index.} Every term of an equation (or an expression) the free indices must balance.
4. A relation having having a free index If an index appears as a free index in an equation, it is understood, by convention, that the hold for all values of the free index. As an example, matrix multiplication of a column vector \(u\) by a matrix, \(v=A u\), is normally written as \begin{equation} v_i =\sum_{j=1}^N A_{ij} u_j, i=1,\ldots, N, \end{equation} With the above convention we will write it as \begin{equation} v_i = A_{ij} u_j \end{equation} In the above equation \(i\)is a free index. It is understood that the above equation holds for all values of the free index \(i\).