Pseudotensor
Two quite different mathematical objects are called a pseudotensor in different contexts. The first context is essentially a tensor multiplied by an extra sign factor, such that the pseudotensor changes sign under reflections when a normal tensor does not. According to one definition, a pseudotensor P of the type formula_1 is a geometric object whose components in an arbitrary basis are enumerated by formula_2indices and obey the transformation rule
formula_3 
under a change of basis. Here formula_4 are the components of the pseudotensor in the new and old bases, respectively, formula_5 is the transition matrix for the contravariant indices, formula_6 is the transition matrix for the covariant indices, and formula_7 
This transformation rule differs from the rule for an ordinary tensor only by the presence of the factor formula_8
The second context where the word "pseudotensor" is used is general relativity. In that theory, one cannot describe the energy and momentum of the gravitational field by an energy–momentum tensor. Instead, one introduces objects that behave as tensors only with respect to restricted coordinate transformations. Strictly speaking, such objects are not tensors at all. A famous example of such a pseudotensor is the Landau–Lifshitz pseudotensor. On 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, 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 change of variables in multi-dimensional integration may be achieved through the incorporation of a factor of the absolute value of the determinant of the 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.
Definition
This is a generalization of a pseudovector. There is a second meaning for pseudotensor (and likewise for pseudovector), restricted to general relativity. Tensors obey strict transformation laws, but pseudotensors are not so constrained.