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 vectors,  and  , is defined as:
To compute the metric tensor from a set of equations relating the space to cartesian space (gij = δij: 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: 
Referenced By
Generalized orthogonal group | Generalized special orthogonal group | Glossary of tensor theory | List of differential geometry topics | List of physics topics M-Q | Metric | Minkowski space | Minkowski spacetime | Tensor notation
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