# 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: