line element

Stand at a point and take an infinitesimally small step described by coordinate increments dx^i. How long, in actual physical distance, is that step? On a flat plane with Cartesian coordinates the answer is the Pythagorean ds = sqrt(dx^2 + dy^2). The line element is the general answer to that question on any space: a formula for the squared length of a tiny displacement, written ds^2, and it is the most compact way to package the geometry of a space.

The line element is the metric tensor wearing its working clothes: ds^2 = g_{ij} dx^i dx^j, with the Einstein summation over i and j. Each term tells you how much a step in one coordinate, or a combined step in two coordinates, contributes to real length. Once you have ds^2 you recover everything metric: the length of a finite curve is the integral of ds along it (integrate sqrt(g_{ij} (dx^i/dt)(dx^j/dt)) dt over the parameter t), areas and volumes come from products of the appropriate ds pieces, and angles come from the cross terms. In effect, writing down the line element is the standard way physicists specify a geometry — you give ds^2 and the whole metric, and all of differential geometry follows.

Line elements are everywhere in applied calculus and physics. The arc-length integral you met in first-year calculus, ds = sqrt(1 + (dy/dx)^2) dx, is a one-dimensional line element. In curvilinear coordinates the scale factors that appear in gradient, divergence, and curl formulas are exactly the square roots of the diagonal metric entries read off the line element. In special relativity the spacetime line element ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 (Minkowski) is the invariant interval all observers agree on, and replacing it with a position-dependent g_{ij} is how general relativity describes gravity. The honest caveat: in relativity ds^2 can be negative or zero, so 'ds' is not always a real length — its sign classifies the separation as time-like, space-like, or null.