Invariants of A Volume Form
Volume forms are not unique; they form a torsor over non-vanishing functions on the manifold, as follows. Given a non-vanishing function f on M, and a volume form, is a volume form on M. Conversely, given two volume forms, their ratio is a non-vanishing function (positive if they define the same orientation, negative if they define opposite orientations).
In coordinates, they are both simply a non-zero function times Lebesgue measure, and their ratio is the ratio of the functions, which is independent of choice of coordinates. Intrinsically, it is the Radon–Nikodym derivative of with respect to . On an oriented manifold, the proportionality of any two volume forms can be thought of as a geometric form of the Radon–Nikodym theorem.
Read more about this topic: Volume Form
Famous quotes containing the words volume and/or form:
“There is a note in the front of the volume saying that no public reading ... may be given without first getting the author’s permission. It ought to be made much more difficult to do than that.”
—Robert Benchley (1889–1945)
“But as to women, who can penetrate
The real sufferings of their she condition?
Man’s very sympathy with their estate
Has much of selfishness and more suspicion.
Their love, their virtue, beauty, education,
But form good housekeepers, to breed a nation.”
—George Gordon Noel Byron (1788–1824)