density in the coordinate system; and we use the Jacobian determinant. Because the determinant can be negative, which it is for an orientation-reversing coordinate transformation, this formula is applicable only when is an integer. (However, see even and odd tensor densities below.) We say that a tensor density is a pseudotensor density when there is an additional sign flip under an orientation-reversing coordinate transformation. A mixed rank-two pseudotensor density of weight transforms as (pseudotensor density of (integer) weight W) where sgn( ) is a function that returns +1 when its argument is positive or −1 when its argument is negative. Even and odd tensor densities The transformations for even and odd tensor densities have the benefit of being well defined even when is not an integer. Thus one can speak of, say, an odd tensor density of weight +2 or an even tensor density of weight −1/2. When is an even integer the above formula for an (authentic) tensor density can be rewritten as (even tensor density of weight W) Similarly, when is an odd integer the formula for an (authentic) tensor density can be rewritten as (odd tensor density of weight W) Weights of zero and one A tensor density of any type that has weight zero is also called an absolute tensor. An (even) authentic tensor density of weight zero is also called an ordinary tensor. If a weight is not specified but the word "relative" or "density" is used in a context where a specific weight is needed, it is usually assumed that the weight is +1. Algebraic properties A linear combination (also known as a weighted sum) of tensor densities of the same type and weight is again a tensor density of that type and weight. A product of two tensor densities of any types, and with weights and , is a tensor density of weight A product of authentic tensor densities and pseudotensor densities will be an authentic tensor density when an even number of the factors are pseudotensor densities; it will be a pseudotensor density when an odd number of the factors are pseudotensor densities. Similarly, a product of even tensor densities and odd tensor densities will be an even tensor density when an even number of the factors are odd tensor densities; it will be an odd tensor density when an odd number of the factors are odd tensor densities. The contraction of