in the standard basis by If we now try to express this same expression in a basis other than the standard basis, then the components of the vectors will change, say according to where is some 2 by 2 matrix of real numbers. Given that the area of the spanned parallelogram is a geometric invariant, it cannot have changed under the change of basis, and so the new representation of this matrix must be: which, when expanded is just the original expression but multiplied by the determinant of which is also In fact this representation could be thought of as a two index tensor transformation, but instead, it is computationally easier to think of the tensor transformation rule as multiplication by rather than as 2 matrix multiplications (In fact in higher dimensions, the natural extension of this is matrix multiplications, which for large is completely infeasible). Objects which transform in this way are called tensor densities because they arise naturally when considering problems regarding areas and volumes, and so are frequently used in integration. Definition Some authors classify tensor densities into the two types called (authentic) tensor densities and pseudotensor densities in this article. Other authors classify them differently, into the types called even tensor densities and odd tensor densities. When a tensor density weight is an integer there is an equivalence between these approaches that depends upon whether the integer is even or odd. Note that these classifications elucidate the different ways that tensor densities may transform somewhat pathologically under orientation-reversing coordinate transformations. Regardless of their classifications into these types, there is only one way that tensor densities transform under orientation-preserving coordinate transformations. In this article we have chosen the convention that assigns a weight of +2 to , the determinant of the metric tensor expressed with covariant indices. With this choice, classical densities, like charge density, will be represented by tensor densities of weight +1. Some authors use a sign convention for weights that is the negation of that presented here. In contrast to the meaning used in this article, in general relativity "pseudotensor" sometimes means an object that does not transform like a tensor or relative tensor of any weight. Tensor and pseudotensor densities For example, a mixed rank-two (authentic) tensor density of weight transforms as: ((authentic) tensor density of (integer) weight W) where is the rank-two tensor density in the coordinate system, is the transformed tensor