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 (math), conventionally denoted ''G''. The notation

g_{ij}

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

\theta

between two tangent vector (spatial)s,

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, \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}

Flat Minkowski space:

(x^0, x^1, x^2, x^3)=(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}

==See also== *pseudo-Riemannian metric Riemannian geometry

Metric tensor

please do not delete this page. although there is an alternative approach to differential geometry, the component-based approach is fundamental to understanding the 'modern' approach, and the metric tensor is the fundamental definition of Riemannian geometry. What are the goals of this encyclopedia? what should they be? to be esoteric and create what some very few people might find to be 'elegant' and 'precise', or to make information accessible? I believe that it is the latter. Furthermore, I do not believe that the two goals are mutually exclusive. I believe, rather, that writting in a clear language that can readily be understood is a form of eloquence and 'perfection', and should be a priority. I am reminded of early medicine, when the professors turned the science of medicine into an esoteric and pedantic rite in pursuit of the luster of exclusive power. I would hate to see mathematics go the same way. :I like your attitude. I don't suppose you know what a tensor product is? See my comment on Talk:Tensor product. By the way, User:Kevin baas, you should sign your entries on talk with ~~~~, which is automatically replaced with a signature like the following. -- User:Tim Starling 04:23 Mar 14, 2003 (UTC) perhaps we should explain the implicit summation and products of differentials more? - User:Gauge 05:41, 31 Jul 2004 (UTC) == Support for keeping this page == This page is simple, clear, and essentially self-contained. Browsing from the General Relativity entry, I was much happier with this page than with most other tensor-related explanations, which were so extravagantly reference-dependent as to be useless. I have a pretty solid general math and physics background, and doubt that a much more demanding presentation would serve a significant number of readers. User:PeterPearson 19:50, 4 Feb 2005 (UTC) I wonder if anybody has thought about the divide and conquer approach for writing mathematical objects like equations. It requires only the existence of parallel computers normally used in business (say stores). Benjamin Cuong P. Nghiem bcnghiem@hotmail.com

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