Metric tensor

The metric tensor (see also metric), conventionally notated as , is a tensor of rank 2 (making it a matrix once a basis is chosen), that is used to measure distance and angle in a Riemannian geometry. The notation is conventionally used for the components of the metric tensor (that is, the elements of the matrix). (In the following, we use the Einstein summation convention).

The length of a segment of a curve parameterized by t, from a to b, is defined as:

The angle between two tangent vectorss, and , is defined as:

To compute the metric tensor from a set of equations relating the space to cartesian space (g_ij = δ_ij: see Kronecker delta for more details), compute the jacobian of the set of equations, and multiply (outer product) the transpose of that jacobian by the jacobian.

Example

Given a two-dimensional Euclidean metric tensor:

The length of a curve reduces to the familiar Calculus formula:

Some basic Euclidean metrics

Polar coordinates:

Cylindrical coordinates:

Spherical coordinates: