Einstein summation convention
/ EYEN-styne /
Physics is full of sums that all look the same: a dot product u dot v is u1 v1 + u2 v2 + u3 v3, a matrix times a vector A x has entries that sum A(i,k) times x(k) over k, and once you go to four dimensions and curved space these sums sprout everywhere. Writing the summation sign over and over is tedious and clutters the page. Einstein's convention is a single, devastatingly effective bookkeeping rule that lets you drop almost every summation sign.
The rule is this: whenever an index appears exactly twice in a single term — once as an upper (superscript) index and once as a lower (subscript) index — you automatically sum over that index across its whole range, with no sigma written. So a^i b_i means the sum of a^1 b_1 + a^2 b_2 + ... + a^n b_n. The repeated index is called a dummy or summed index and its name is irrelevant: a^i b_i and a^k b_k are the same number. An index that appears once is a free index; it must match on both sides of an equation and labels which component you mean. The convention also enforces a discipline: a legal expression never has the same index appearing three times, and a summed pair should be one up and one down — a quiet reminder that you are pairing contravariant with covariant objects.
This is not just shorthand; it is the working language of tensor calculus, general relativity, continuum mechanics, and modern differential geometry. Because the rule pairs an upper index with a lower one, the sums it produces are exactly the coordinate-independent combinations (contractions) that have geometric meaning, which is why Einstein's notation makes correct equations easy to write and many wrong ones impossible to write. The one discipline to keep is honesty about up versus down: in flat Cartesian coordinates people get lazy and write everything downstairs, but in curved space or non-orthonormal coordinates the up/down distinction is real and must be respected.
The squared length of a vector in a space with metric g is written ds^2 = g_{ij} dx^i dx^j. Both i and j are repeated (each appears once up on the dx and once down on g), so this single short expression secretly means a double sum over i and j of g_{ij} dx^i dx^j across all coordinate pairs — in 3-D that is nine terms collapsed into four symbols.
Two repeated index pairs, two hidden sums — the convention turns nine terms into four symbols.
A repeated index that is NOT meant to be summed must be flagged explicitly (people write 'no sum on i'); and an index that legitimately appears three times in one term is a sign you made an algebra mistake, not a new kind of sum.