Line element

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A line element in mathematics can most generally be thought of as the square of the change in a position vector in an affine space equated to the square of the change of the arc length. An easy way of visualizing this relationship is by parametrizing the given curve by Frenet's formulas. As such, a line element is then naturally a function of the metric, and can be related to the curvature tensor.

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

planar: $d s 2 = d x 2 + d y 2$

spatial: $d s 2 = d x 2 + d y 2 + d z 2$

Other line elements are given by:

flat polar: $ds^2= dr^2 +r^2 d \theta\ ^2$

spherical polar: $ds^2=dr^2+r^2 d \theta\ ^2+ r^2 \sin^2 \theta\ d \phi\ ^2$

cylindrical polar: $ds^2=dr^2+ r^2 d \theta\ ^2 +dz^2$

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

$ds^2= f ( \chi\ , \psi\ )d \chi\ ^2 + g ( \chi\ , \psi\ )d \chi\ d \psi\ + h ( \chi\ , \psi\ ) d \psi\ ^2$

[edit] Line elements in physics

Line elements are used in physics, especially in theories of gravitation such as general relativity, where spacetime is modelled as a curved manifold with a metric.