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 `v`_{1}, ..,
`v`_{n} in any orthonormal frame,
and which satisies the follwing transformation law:
if the components of the vector density in one
coordinate frame are `v`_{i} 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`)
* `L`_{i}_{j}v_{i}

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

`v'`_{i} =
`L`_{i}_{j}v_{i}

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
**R**^{3}. To show that it is we let
`L`^{i} denote the `i`th column
of `L` (regarded as a vector in R^{3})
and use the fact that `L`^{i} x
`L`^{j} = det(`L`) *
`ε`_{k}_{i}_{j}L^{k}:

(`a`' x `b`')_{i} =
(`L`^{j}a_{j} x `L`^{k}b_{k})_{i}
= `a`_{j}b_{k}(`L`^{j} x
`L`^{k})_{i} =
det(`L`) * `a`_{j}b_{k}(ε_{m}_{j}_{k}L^{m})_{i}
= det(`L`) * `a`_{j}b_{k}ε_{m}_{j}_{k}L_{m}_{i} =
det(`L`) * `L`_{i}_{m}(`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.