Volume Form - Invariants of A Volume Form

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:

    Although volume upon volume is written to prove slavery a very good thing, we never hear of the man who wishes to take the good of it, by being a slave himself.
    Abraham Lincoln (1809–1865)

    [With the Union saved] its form of government is saved to the world; its beloved history, and cherished memories, are vindicated; and its happy future fully assured, and rendered inconceivably grand.
    Abraham Lincoln (1809–1865)