This is a useful
mathematical tool to make dealing with complex
vector expressions a lot easier. Suppose we have two vectors
a = (a
1,a
2,a
3) and
b = (b
1,b
2,b
3). 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.