Line element
Encyclopedia
A line element ds in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

can most generally be thought of as the change in a position vector in an affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

expressing the change of the arc length
Arc length
Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...

. An easy way of visualizing this relationship is by parametrizing the given curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

by Frenet–Serret formulas. As such, a line element is then naturally a function of the metric, and can be related to the curvature tensor
Curvature tensor
In differential geometry, the term curvature tensor may refer to:* the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds;* the curvature of an affine connection or covariant derivative ;...

.

The most well known line elements are those of cartesian planar and spatial coordinates. They are given by

planar:

spatial:

Other line elements are given by:

flat polar:

spherical polar:

cylindrical polar:

The most general 2- dimensional (coordinates (χ,ψ)) metric is given by

## Line elements in physics

Line elements are used in physics, especially in theories of gravitation such as general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, where spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

is modelled as a curved manifold with a metric. For example, if a massive object causes some curvature in spacetime, the trajectory
Trajectory
A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...

of an object with negligible mass over that curvature would follow the line element.