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: