Metric tensor

In mathematics, inRiemannian geometrymetric tensor istensorrank 2 thatusedmeasure distanceangle. Oncelocal basischosen,therefore appears asmatrix ),conventionally notated as (see also metric). The notation conventionally used forcomponents ofmetric tensor. Infollowing, we useEinstein summation convention.

The length ofsegment ofcurve parameterized by t, fromto b,defined as:

The angle between two tangent vectorss, ,defined as:

To computemetric tensor fromsetequations relatingspace cartesian space (g_ij = δ_ij: see Kronecker deltamore details), computejacobian ofsetequations,multiply (outer product)transposethat jacobian byjacobian.

Example

Giventwo-dimensional Euclidean metric tensor:

The length ofcurve reduces tofamiliar Calculus formula:

Some basic Euclidean metrics

Polar coordinates:

Cylindrical coordinates:

Spherical coordinates: