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:
“F.R. Leavis’s “eat up your broccoli” approach to fiction emphasises this junkfood/wholefood dichotomy. If reading a novel—for the eighteenth century reader, the most frivolous of diversions—did not, by the middle of the twentieth century, make you a better person in some way, then you might as well flush the offending volume down the toilet, which was by far the best place for the undigested excreta of dubious nourishment.”
—Angela Carter (1940–1992)
“For Mercy has a human heart,
Pity, a human face;
And Love, the human form divine,
And Peace, the human dress.”
—William Blake (1757–1827)