Volume Form - Relation To Measures

Relation To Measures

See also: Density on a manifold

Given a volume form ω on an oriented manifold, the density |ω| is a volume pseudo-form on the nonoriented manifold obtained by forgetting the orientation. Densities may also be defined more generally on non-orientable manifolds.

Any volume pseudo-form ω (and therefore also any volume form) defines a measure on the Borel sets by

The difference is that while a measure can be integrated over a (Borel) subset, a volume form can only be integrated over an oriented cell. In single variable calculus, writing considers as a volume form, not simply a measure, and indicates "integrate over the cell with the opposite orientation, sometimes denoted ".

Further, general measures need not be continuous or smooth: they need not be defined by a volume form, or more formally, their Radon–Nikodym derivative with respect to a given volume form needn't be absolutely continuous.

Read more about this topic:  Volume Form

Famous quotes containing the words relation to, relation and/or measures:

    “A theory of the middle class: that it is not to be determined by its financial situation but rather by its relation to government. That is, one could shade down from an actual ruling or governing class to a class hopelessly out of relation to government, thinking of gov’t as beyond its control, of itself as wholly controlled by gov’t. Somewhere in between and in gradations is the group that has the sense that gov’t exists for it, and shapes its consciousness accordingly.”
    —Lionel Trilling (1905–1975)

    “Concord is just as idiotic as ever in relation to the spirits and their knockings. Most people here believe in a spiritual world ... in spirits which the very bullfrogs in our meadows would blackball. Their evil genius is seeing how low it can degrade them. The hooting of owls, the croaking of frogs, is celestial wisdom in comparison.”
    —Henry David Thoreau (1817–1862)

    “I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
    —Henry David Thoreau (1817–1862)