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Metric tensor

The metric tensor (see also metric), conventionally notated as <math>G</math>, is a 2-dimensional tensor (making it a matrix once a basis is chosen), that is used to measure distance and angle in a Riemannian geometry. The notation <math>g_{ij}</math> 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:

<math>L = \int_a^b \sqrt{ g_{ij}dx^idx^j}</math>

The angle between two tangent vectors, <math>U</math> and <math>V</math>, is defined as:

<math>
\cos \theta = \frac{g_{ij}U^iV^j} {\sqrt{ \left| g_{ij}U^iU^j \right| \left| g_{ij}V^iV^j \right|}} </math>

To compute the metric tensor from a set of equations relating the space to cartesian space (gij = δij: 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.

<math>G = J^T J</math>

     Example  

Given a two-dimensional Euclidean metric tensor:

<math>G = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}</math>

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

<math>L = \int_a^b \sqrt{ (dx^1)^2 + (dx^2)^2} </math>

     Some basic Euclidean metrics  

Polar coordinates: <math>(x^1, x^2)=(r, \theta)</math>

<math>G = \begin{bmatrix} 1 & 0 \\ 0 & (x^1)^2\end{bmatrix}</math>

Cylindrical coordinates: <math>(x^1, x^2, x^3)=(r, \theta, z)</math>

<math>G = \begin{bmatrix} 1 & 0 & 0\\ 0 & (x^1)^2 & 0 \\ 0 & 0 & 1\end{bmatrix}</math>

Spherical coordinates: <math>(x^1, x^2, x^3)=(r, \phi, \theta)</math>

<math>G = \begin{bmatrix} 1 & 0 & 0\\ 0 & (x^1)^2 & 0 \\ 0 & 0 & (x^1\sin x^2)^2\end{bmatrix}</math>