Relation To Measures
See also: Density on a manifoldGiven 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:
“You see, I am alive, I am alive
I stand in good relation to the earth
I stand in good relation to the gods
I stand in good relation to all that is beautiful
I stand in good relation to the daughter of Tsen-tainte
You see, I am alive, I am alive”
—N. Scott Momaday (b. 1934)
“The instincts of the ant are very unimportant, considered as the ant’s; but the moment a ray of relation is seen to extend from it to man, and the little drudge is seen to be a monitor, a little body with a mighty heart, then all its habits, even that said to be recently observed, that it never sleeps, become sublime.”
—Ralph Waldo Emerson (1803–1882)
“I hear America singing, the varied carols I hear,
Those of mechanics, each one singing his as it should be blithe and
strong,
The carpenter singing his as he measures his plank or beam,
The mason singing his as he makes ready for work, or leaves off
work,”
—Walt Whitman (1819–1892)