Metric tensor

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In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space.

Once a local basis is chosen, the metric tensor appears as a matrix, conventionally notated as G (see also metric). The notation g_ij is conventionally used for the components of the metric tensor. (i.e. 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: