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Metric tensor

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 $x i$ is chosen, the metric tensor appears as a matrix, conventionally denoted G. The notation $g i j$ 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:

$L = \int_a^b \sqrt{ g_{ij}{dx^i\over dt}{dx^j\over dt}}dt$

The angle $θ$ between two tangent vectors, $U=u^i{\partial\over \partial x_i}$ and $V=v^i{\partial\over \partial x_i}$ , is defined as:

$\cos \theta = \frac{g_{ij}u^iv^j} {\sqrt{ \left| g_{ij}u^iu^j \right| \left| g_{ij}v^iv^j \right|}}$

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

G = J T J

where $J$ denotes the Jacobian of the embedding and $J T$ its transpose.

Examples

The Euclidean metric

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:

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

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

Polar coordinates: $(x 1, x 2) = (r,θ)$

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

Cylindrical coordinates: $(x 1, x 2, x 3) = (r,θ, z)$

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

Spherical coordinates: $(x 1, x 2, x 3) = (r,φ,θ)$

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

Flat Minkowski space: $(x 1, x 2, x 3, x 4) = (t, x, y, z)$

$g = \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$

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