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:
“A German immersed in any civilization different from his own loses a weight equivalent in volume to the amount of intelligence he displaces.”
—José Bergamín (1895–1983)
“As for the terms good and bad, they indicate no positive quality in things regarded in themselves, but are merely modes of thinking, or notions which we form from the comparison of things with one another. Thus one and the same thing can be at the same time good, bad, and indifferent. For instance music is good for him that is melancholy, bad for him who mourns; for him who is deaf, it is neither good nor bad.”
—Baruch (Benedict)