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 a_{1}, a_{2}, ..., 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_{ij}k_{j} = k_{i}

a·b = a_{i}b_{i}

(a×b)_{i} = ε_{ijk}a_{j}b_{k}

ε_{ijk}ε_{ilm} = δ_{jm}δ_{kl} - δ_{jl}δ_{km}

I_{ij} = δ_{ij}

(AB)_{ij} = A_{ik}B_{kj}

det(A) = ε_{ijk}A_{i1}A_{j2}A_{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).