Metric tensor

In mathematics, in a Riemannian geometry the metric tensor is a tensor of rank 2 that is used to measure distance and angle. Once a local basis is chosen, it therefore appears as a matrix ),conventionally notated as (see also metric). The notation is conventionally used for the components of the metric tensor. 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: