summation convention

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.

The summation convention was invented by Albert Einstein. In an interview he is supposed to have been asked what he thought was his greatest contribution to human knowledge. He did not mention his Nobel prize awarded discoveries or the theory of general relativity, but instead answered that it was the summation convention.

The summation convention is in fact very useful and makes calculations that would otherwise be unintelligible comprehensible.
Two symbols are commonly used to make the summation convention more useful: δ and ε.
δ_ij is defined to be 1 if i = j and 0 otherwise.
ε_{a(1)a(2)...a(n)} is defined to be 1 if a₁, a₂, ..., a_n is an even permutation of 1, 2, ..., n; -1 if it is an odd permutation and 0 otherwise.

We denote the ith component of the vector a by a_i and the element in the ith row and jth column of the matrix A by A_ij . The following identities hold and can be used in calculations (in some cases given only for 3 dimensions, but with the exception for identities involving vector product they generalise):

δ_ijk_j = k_i
a·b = a_ib_i
(a×b)_i = ε_ijka_jb_k
ε_ijkε_ilm = δ_jmδ_kl - δ_jlδ_km
I_ij = δ_ij
(AB)_ij = A_ikB_kj
det(A) = ε_ijkA_i1A_j2A_k3

By simply applying these identities and doing the calculations you can prove all sorts of weird and wonderful things (left as an exercise to the reader).