ABSTRACT
Because the various gj(x) and Jacobian are (strictly) positive by assumption, then the n-form o is nonzero at every point x EM, that is a volume.
Conversely, if there is a volume on M, let us prove there is an atlas {(Up 9'I)}IEl such that every coordinate change shows a (strictly) positive Jacobian. Let (Up 9'1), (Uj' tpj) be charts ofthe atlas,
o be a volume element on M, hi: 9'1(UI)~R:(xl,...,Xn)Hhl(xl,...,xn).
