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) * ajbk(εmjkLm)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.