Editing Metric tensor (section)

==Canonical measure and volume form==
In analogy with the case of surfaces, a metric tensor on an {{mvar|n}}-dimensional paracompact manifold {{mvar|M}} gives rise to a natural way to measure the {{mvar|n}}-dimensional [[volume]] of subsets of the manifold. The resulting natural positive [[Borel measure]] allows one to develop a theory of integrating functions on the manifold by means of the associated [[Lebesgue integral]].

A measure can be defined, by the [[Riesz representation theorem]], by giving a [[positive linear functional]] {{mvar|Λ}} on the space {{math|''C''<sub>0</sub>(''M'')}} of [[compact support|compactly supported]] [[continuous function]]s on {{mvar|M}}. More precisely, if {{mvar|M}} is a manifold with a (pseudo-)Riemannian metric tensor {{mvar|g}}, then there is a unique positive [[Borel measure]] {{math|''μ''<sub>''g''</sub>}} such that for any [[coordinate chart]] {{math|(''U'', ''φ'')}},
<math display="block">\Lambda f = \int_U f \, d\mu_g = \int_{\varphi(U)} f \circ \varphi^{-1}(x) \sqrt{\left|\det g\right|}\,dx</math>
for all {{mvar|f}} supported in {{mvar|U}}. Here {{math|det ''g''}} is the [[determinant]] of the matrix formed by the components of the metric tensor in the coordinate chart. That {{math|Λ}} is well-defined on functions supported in coordinate neighborhoods is justified by [[integration by substitution|Jacobian change of variables]]. It extends to a unique positive linear functional on {{math|''C''<sub>0</sub>(''M'')}} by means of a [[partition of unity]].

If {{mvar|M}} is also [[orientation (mathematics)|oriented]], then it is possible to define a natural [[volume form]] from the metric tensor. In a [[right-handed coordinate system|positively oriented coordinate system]] {{math|(''x''<sup>''1''</sup>, ..., ''x''<sup>''n''</sup>)}} the volume form is represented as
<math display="block">\omega = \sqrt{\left|\det g\right|} \, dx^1 \wedge \cdots \wedge dx^n</math>
where the {{math|''dx''<sup>''i''</sup>}} are the [[coordinate differential]]s and {{math|∧}} denotes the [[exterior product]] in the algebra of [[differential form]]s. The volume form also gives a way to integrate functions on the manifold, and this geometric integral agrees with the integral obtained by the canonical Borel measure.