## Encyclopedia > Metric tensor

Article Content

# Metric tensor

The metric tensor (see also metric), conventionally notated as $G$, 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 $g_{ij}$ 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:

$L = \int_a^b \sqrt{ g_{ij}dx^idx^j}$

The angle between two tangent vectors, $U$ and $V$, is defined as:

$\cos \theta = \frac{g_{ij}U^iV^j} {\sqrt{ \left| g_{ij}U^iU^j \right| \left| g_{ij}V^iV^j \right|}}$

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.

$G = J^T J$

Example Given a two-dimensional Euclidean metric tensor:

$G = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$

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

$L = \int_a^b \sqrt{ (dx^1)^2 + (dx^2)^2}$

Some basic Euclidean metrics Polar coordinates: $(x^1, x^2)=(r, \theta)$

$G = \begin{bmatrix} 1 & 0 \\ 0 & (x^1)^2\end{bmatrix}$

Cylindrical coordinates: $(x^1, x^2, x^3)=(r, \theta, z)$

$G = \begin{bmatrix} 1 & 0 & 0\\ 0 & (x^1)^2 & 0 \\ 0 & 0 & 1\end{bmatrix}$

Spherical coordinates: $(x^1, x^2, x^3)=(r, \phi, \theta)$

$G = \begin{bmatrix} 1 & 0 & 0\\ 0 & (x^1)^2 & 0 \\ 0 & 0 & (x^1\sin x^2)^2\end{bmatrix}$

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Reformed churches ... in Scotland include the Church of Scotland, the established church in Scotland and smaller denominations such as the Free Church of Scotland[?] and the F ...