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Lets consider a n-manifold. A differential n-form describing a signed volume element will then transform as:

[tex]f(x^i) dx^1 \wedge dx^2 \wedge \cdots \wedge dx^n = f(y^i) \;\text{det}\left( \frac{\partial x^i}{\partial y^j}\right) dy^1 \wedge dy^2 \wedge \cdots \wedge dy^n,[/tex]

which we can compare to the unsigned volume element in Euclidean space that transforms as:

[tex]f(x^i) dx^1 dx^2 \cdots dx^n = f(y^i) \left|\text{det}\left( \frac{\partial x^i}{\partial y^j}\right)\right|dy^1 dy^2 \cdots dy^n.[/tex]

Clearly we can get a sign wrong when integrating and conisdering coordinate changes from systems of different orientation. What I dont see is the need for the factor of

[tex]\sqrt{|g|}[/tex]

on Riemannian manifolds. Will this factor only fix the sign error or have I missunderstood something basic?