Examples
On non-orientable manifolds, one cannot define a volume form due to the non-orientability, but one can define a volume element, which is formally a density, and may also be called a pseudo-volume form, due to the additional sign twist (tensoring with the sign bundle).
A change of variables in multi-dimensional integration is achieved by incorporation of a factor of the absolute value of the determinant of the Jacobian matrix. The use of the absolute value introduces a sign-flip for improper coordinate transformations; as such, an integrand is an example of a pseudotensor density according to the first definition.
Read more about this topic: Pseudotensor
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