summation convention (idea)

This is a useful mathematical tool to make dealing with complex vector expressions a lot easier. Suppose we have two vectors a = (a₁,a₂,a₃) and b = (b₁,b₂,b₃). If we wished to evaluate their scalar product a.b we would write

a.b = a1b1 + a2b2 + a3b3.

We could write this more efficiently using a summation

       3
a.b =  Σ aibi.
      i=1

Even for a relatively simple expression such as this, the sigma notation is rather cumbersome. Hence at higher levels of mathematics, the sigma is dropped, and the summation is assumed from the context. We would therefore write

a.b = aibi.

Although this is a lot better, we have to be careful by what we mean. For a general vector expression,

• if a suffix appears once, no summation is implied,
• if a suffix appears twice, a summation is implied,
• if a suffix appears three or more times, there's something wrong.

For example,

aibjcjdkdk = ai(b1c1 + b2c2 + b3c3)(d1d1 + d2d2 + d3d3).

A suffix that appears twice is known as a dummy suffix, as it can be replaced by anything: a_ib_i is exactly the same as a_qb_q. Sometimes it is necessary to use this fact: suppose we have

ai = bicjcj

and we wish to evaluate the expression

aidiejfj.

If we did a straight substitution for a_i we would get

bicjcjdiejfj

but this is meaningless: it is not clear what we are meant to sum over. The correct thing to do is to relabel the first pair of j's beforehand. Then we get

bickckdiejfj

which makes sense.

In pure mathematics, often we are interested in an n-dimensional space, so the sum changes from 1,2,3 to 1,2,...,n. A generalization of summation convention is used in tensor algebra.