Page 19 - Introduction to Continuum Mechanics

4 Indicial Notation

The index i in Eq. (2A1.2), or; in Eq. (2A1.3), or m in Eq. (2A1.4) is a dummy index in the
sense that the sum is independent of the letter used.
We can further simplify the writing of Eq.(2Al.l) if we adopt the following convention:
Whenever an index is repeated once, it is a dummy index indicating a summation with the
index running through the integers 1,2,..., n.
This convention is known as Einstein's summation convention. Using the convention,
Eq. (2A1.1) shortens to

We also note that

It is emphasized that expressions such as a ib ix i are not defined within this convention. That
is, an index should never be repeated more than once when the summation convention is used.
Therefore, an expression of the form

must retain its summation sign.
In the following we shall always take n to be 3 so that, for example,

a ix i = a mx m = a 1x 1 + a 2x 2 + a 3x 3
a ii = a mm = a 11 + a 22 + a 33
a ie i = a 1 e i1 + a 2 e 2 + a 3 e 3
The summation convention obviously can be used to express a double sum, a triple sum,
etc. For example, we can write

simply as

Expanding in full, the expression (2A1.8) gives a sum of nine terms, i.e.,

For beginners, it is probably better to perform the above expansion in two steps, first, sum
over i and then sum over j (or vice versa), i.e.,

a ijx ix j = a 1jx 1x j + a 2jx 2x j + a 3jx 3x j