Metric tensor

The metric tensor(see also metric), conventionally notated as $G$, is a 2-dimensional tensor (making it a matrix), that is used to measure distance and angle in a Riemannian geometry. $g_{ij}$ is conventionally used to notate the components of the metric tensor. (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:

$L=\int _{a}^{b}{\sqrt {g_{ij}dx^{i}dx^{j}}}$

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

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

To compute the metric tensor from a set of equations relating the space to cartesian space(g_ij = δ_ij: 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: $(x^{1},x^{2},x^{3})=(r,\phi ,\theta )$

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