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:
“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)
“To be a good enough parent one must be able to feel secure in one’s parenthood, and one’s relation to one’s child...The security of the parent about being a parent will eventually become the source of the child’s feeling secure about himself.”
—Bruno Bettelheim (20th century)
“Those who, while they disapprove of the character and measures of a government, yield to it their allegiance and support are undoubtedly its most conscientious supporters, and so frequently the most serious obstacles to reform.”
—Henry David Thoreau (1817–1862)