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Metric tensor

In the [mathematical] field of [differential geometry] , a metric tensor is a type of function defined on a [manifold] (such as a [surface] in space) which takes as input a pair of [tangent vector] s v and w and produces a [real number] ( [scalar] ) g ( v , w ) in a way that generalizes many of the familiar properties of the [dot product] of [vectors] in [Euclidean space] . In the same way as a dot product, metric tensors are used to define the length of, and angle between, tangent vectors.

A metric tensor is defined to be a [nondegenerate] [symmetric bilinear form] on each tangent space that varies [smoothly] from point to point. It is an example of a [tensor field] . Relative to a [local coordinate system] , a metric tensor takes on the form of a [symmetric matrix] whose entries transform [covariantly] under changes to the coordinate system, which is to say that the metric tensor is a covariant [symmetric tensor] .

Introduction
[Carl Friedrich Gauss] in his 1827 [''Disquisitiones generales circa superficies curvas''] ( General investigations of curved surfaces ) considered a surface [parametrically] , with the [Cartesian coordinates] x , y , and z of points on the surface depending on two auxiliary variables u and v . Thus a parametric surface is (in contemporary terms) a [vector valued function]

:{r}(u,v) = \langle x(u,v), y(u,v), z(u,v)\rangle

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