This is a useful mathematical tool
to make dealing with complex vector expression
s a lot easier. Suppose we have two vectors a
) and b
). If we wished to evaluate their scalar product a
we would write
a.b = a1b1 + a2b2 + a3b3.
We could write this more efficiently using a summation
a.b = Σ aibi.
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.
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: ai
is exactly the same as aq
. Sometimes it is necessary to use this fact: suppose we have
ai = bicjcj
and we wish to evaluate the expression
If we did a straight substitution
we would get
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
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