Vector density (idea)

More formally a vector density is an object that is represented by coordinates v₁, .., 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_ijv_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_ijv_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 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 L^j = det(L) * ε_kijL^k:

(a' x b')_i = (L^ja_j x L^kb_k)_i = a_jb_k(L^j x L^k)_i = det(L) * a_jb_k(ε_mjkL^m)_i = det(L) * a_jb_kε_mjkL_mi = det(L) * L_im(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.