# All Science Fair Projects

## Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

# Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

# Metric tensor

(Redirected from Riemannian metric)

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 xi is chosen, the metric tensor appears as a matrix, conventionally denoted G. The notation gij 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 = JTJ

where J denotes the Jacobian of the embedding and JT 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: (x1,x2) = (r,θ)

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

Cylindrical coordinates: (x1,x2,x3) = (r,θ,z)

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

Spherical coordinates: (x1,x2,x3) = (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: (x1,x2,x3,x4) = (t,x,y,z)

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