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:

    “The adolescent does not develop her identity and individuality by moving outside her family. She is not triggered by some magic unconscious dynamic whereby she rejects her family in favour of her peers or of a larger society.... She continues to develop in relation to her parents. Her mother continues to have more influence over her than either her father or her friends.”
    —Terri Apter (20th century)

    “Skepticism is unbelief in cause and effect. A man does not see, that, as he eats, so he thinks: as he deals, so he is, and so he appears; he does not see that his son is the son of his thoughts and of his actions; that fortunes are not exceptions but fruits; that relation and connection are not somewhere and sometimes, but everywhere and always; no miscellany, no exemption, no anomaly,—but method, and an even web; and what comes out, that was put in.”
    —Ralph Waldo Emerson (1803–1882)

    “One encounters very capable fathers abashed by their piano-playing daughters. Three measures of Schumann make them red with embarrassment.”
    —Alfred Döblin (1878–1957)