Page 20 - Introduction to Continuum Mechanics

Part A Free Indices 5

where

etc.
Similarly, the triple sum

will simply be written as

The expression (2A1.11) represents the sum of 27 terms.
We emphasize again that expressions such as a ii x i x j x j or a ijk x ix ix j x k are not defined in the
summation convention, they do not represent

2A2 Free Indices
Consider the following system of three equations

Using the summation convention, Eqs. (2A2.1) can be written as

which can be shortened into

An index which appears only once in each term of an equation such as the index i in
Eq. (2A2.3) is called a "free index." A free index takes on the integral number 1, 2, or 3 one
at a time. Thus Eq. (2A2.3) is short-hand for three equations each having a sum of three terms
on its right-hand side [i.e., Eqs. (2A2.1)].
A further example is given by