Editing Pseudotensor (section)

==Examples==
On [[Orientable manifold|non-orientable manifolds]], one cannot define a [[volume form]] globally due to the non-orientability, but one can define a [[volume element]], which is formally a [[Density on a manifold|density]], and may also be called a ''pseudo-volume form'', due to the additional sign twist (tensoring with the sign bundle).  The volume element is a pseudotensor density according to the first definition.

A [[Integration by substitution|change of variable]]s in multi-dimensional integration may be achieved through the incorporation of a factor of the absolute value of the [[determinant]] of the [[Jacobian matrix and determinant|Jacobian matrix]].  The use of the absolute value introduces a sign change for improper coordinate transformations to compensate for the convention of keeping integration (volume) element positive; as such, an [[integrand]] is an example of a pseudotensor density according to the first definition.

The [[Christoffel symbols]] of an [[affine connection]] on a manifold can be thought of as the correction terms to the partial derivatives of a coordinate expression of a vector field with respect to the coordinates to render it the vector field's covariant derivative.  While the affine connection itself doesn't depend on the choice of coordinates, its Christoffel symbols do, making them a pseudotensor quantity according to the second definition.