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:
Example
Given a two-dimensional Euclidean metric tensor:
Some basic Euclidean metrics
Polar coordinates: