A vector density is a vampire vector: it looks like a vector, has components like a vector and rotates like a vector, but it does not reflect like a vector.

More formally a vector density is an object that is represented by coordinates v1, .., vn in any orthonormal frame, and which satisies the follwing transformation law: if the components of the vector density in one coordinate frame are vi and make an orthogonal transformation given by a matrix L to a new frame then the components of the vector density in the new frame are

v'i = det(L) * Lijvi

Here the summation convention is used, so a repeated index implies summation. This transformation law can compared with that for vectors:

v'i = Lijvi

So the only difference between a vector density and a vector is the appearance of det(L) in the transformation law. This means that a vector density will transform precisely like a vector under rotations (which have det(L) = 1) while they will undergo an additional sign-change under reflections (which have det(L) = -1).

The best known example of a vector density (although it may not be so well known that it is a vector density) is the vector product of two vectors in R3. To show that it is we let Li denote the ith column of L (regarded as a vector in R3) and use the fact that Li x Lj = det(L) * εkijLk:

(a' x b')i = (Ljaj x Lkbk)i = ajbk(Lj x Lk)i = det(L) * ajbkmjkLm)i = det(L) * ajbkεmjkLmi = det(L) * Lim(a x b)m

The differential operator curl of a vector field can be regarded as a vector product, so it too is a vector density.