Metric tensorThe metric tensor (see also metric), conventionally notated as , is a tensor of rank 2 (making it a matrix once a basis is chosen), that is used to measure distance and angle in a Riemannian geometry. The notation is conventionally used for the components of the metric tensor (that is, 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:
ExampleGiven a two-dimensional Euclidean metric tensor:
Some basic Euclidean metricsPolar coordinates: