Canonical Measure and Volume Form
In analogy with the case of surfaces, a metric tensor on an n-dimensional paracompact manifold M gives rise to a natural way to measure the 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 Λ on the space C0(M) of compactly supported continuous functions on M. More precisely, if M is a manifold with a (pseudo-)Riemannian metric tensor g, then there is a unique positive Borel measure μg such that for any coordinate chart (U,φ),
for all ƒ supported in U. Here det g is the determinant of the matrix formed by the components of the metric tensor in the coordinate chart. That Λ is well-defined on functions supported in coordinate neighborhoods is justified by Jacobian change of variables. It extends to a unique positive linear functional on C0(M) by means of a partition of unity.
If M is in addition oriented, then it is possible to define a natural volume form from the metric tensor. In a positively oriented coordinate system (x1,...,xn) the volume form is represented as
where the dxi are the coordinate differentials and the wedge ∧ denotes the exterior product in the algebra of differential forms. 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.
Read more about this topic: Metric Tensor
Famous quotes containing the words canonical, measure, volume and/or form:
“If God bestowed immortality on every man then when he made him, and he made many to whom he never purposed to give his saving grace, what did his Lordship think that God gave any man immortality with purpose only to make him capable of immortal torments? It is a hard saying, and I think cannot piously be believed. I am sure it can never be proved by the canonical Scripture.”
—Thomas Hobbes (1579–1688)
“He gave man speech, and speech created thought,
Which is the measure of the universe;”
—Percy Bysshe Shelley (1792–1822)
“I dare say I am compelled, unconsciously compelled, now to write volume after volume, as in past years I was compelled to go to sea, voyage after voyage. Leaves must follow upon each other as leagues used to follow in the days gone by, on and on to the appointed end, which, being Truth itself, is One—one for all men and for all occupations.”
—Joseph Conrad (1857–1924)
“Women over fifty already form one of the largest groups in the population structure of the western world. As long as they like themselves, they will not be an oppressed minority. In order to like themselves they must reject trivialization by others of who and what they are. A grown woman should not have to masquerade as a girl in order to remain in the land of the living.”
—Germaine Greer (b. 1939)