Integration
Densities play a significant role in the theory of integration on manifolds. Indeed, the definition of a density is motivated by how a measure dx changes under a change of coordinates (Folland 1999, Section 11.4, pp. 361-362).
Given a 1-density ƒ supported in a coordinate chart Uα, the integral is defined by
where the latter integral is with respect to the Lebesgue measure on Rn. The transformation law for 1-densities together with the Jacobian change of variables ensures compatibility on the overlaps of different coordinate charts, and so the integral of a general compactly supported 1-density can be defined by a partition of unity argument. Thus 1-densities are a generalization of the notion of a volume form that does not necessarily require the manifold to be oriented or even orientable. One can more generally develop a general theory of Radon measures as distributional sections of using the Riesz representation theorem.
The set of 1/p-densities such that is a normed linear space whose completion is called the intrinsic Lp space of M.
Read more about this topic: Density On A Manifold
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