2.1 Summation Convention, Dummy Indices

Consider the sum

We can write it in a compact form as

It is obvious that the index i, j, or m in Eq. (2.2) is dummy in the sense that the sum is independent of the letter used. This is analogous to the dummy variable in an integral of a function over a finite interval:

The three dots in the term on the right-hand side of Eq. (2.1) stand for the ( n ? 3) missing terms. The common convention is to denote such missing terms by three, and not any other number of, dots. The first digit in an equation number stands for the chapter and the second for the equation number in the chapter.

We can simplify the writing of Eq. (2.2) by adopting the following convention, sometimes called Einstein s summation convention. Whenever an index is repeated once in the same term, it implies summation over the specified range of the index. Using the summation convention, Eq. (2.2) shortens to

where indices i, j, and m take values 1 through n. Note that expressions such as a _i b _i x _i are not defined according to this convention. That is, an index should never be repeated more than once in the same term for the summation convention to be implied. Therefore, an expression of the form must retain the summation sign.

In the following, unless otherwise specified, we shall...