Metric tensor Guide, Meaning , Facts, Information and Description

In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space.

Once a local coordinate system is chosen, the metric tensor appears as a matrix, conventionally notated as G. The notation is conventionally used for the components of the metric tensor (i.e. the elements of the matrix). In the following, we use the Einstein notation for implicit sums.

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:

The induced metric tensor for a smooth embedding of a manifold into Euclidean space can be computed by the formula

where denotes the Jacobian of the the embedding and its transpose.

Table of contents

1 Examples 1.1 The Euclidean metric 2 See also

Examples

The Euclidean metric

Given a two-dimensional Euclidean metric tensor:

The length of a curve reduces to the familiar calculus formula:

The Euclidean metric in some other common coordinate systems can be written as follows.

Polar coordinates:

Cylindrical coordinates:

Spherical coordinates:

See also

• pseudo-Riemannian metric

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