Coordinate-free definition of pseudotensors How to define pseudotensors (particularly, pseudovectors) in a coordinate-free form? Can it be defined on a manifold (like a tensor field)?
Or may be the objects that physicists model via pseudotensors, can be modeled in more naturally coordinate-free way?
 A: You're gonna need a few concepts.  First, $k$-vectors: objects formed from wedge products of $k$ linearly independent vectors. Such products allow us to talk naturally about planes and volumes and other higher-dimensional geometric objects. So for instance, using two vectors $a$ and $b$, the wedge product $a \wedge b$ is called a 2-vector (or a bivector) and represents the plane spanned by $a$ and $b$.
Linear maps have natural extensions to acting on $k$-vectors.  If $\underline T(a)$ is a linear map on a vector $a$, then the natural extension to a 2-vectot $a \wedge b$ is
$$\underline T(a \wedge b) = \underline T(a) \wedge \underline T(b)$$
You also need to know about Hodge duality: this is the means through which we turn a $k$-vector to an $(n-k)$-vector that is the orthogonal complement of the original $k$-vector.  This is done through multiplication with the pseudoscalar $\epsilon$ (the multiplication I refer to is the geometric product, but really, you don't need to know how that is performed, only that this multiplication with $\epsilon$ does what I have described).
Rotations and reflections obey particular special properties with respect to Hodge duality.  In particular, a rotation $\underline R$ obeys
$$\underline R(B \epsilon) = \underline R(B) \epsilon$$
for any $k$-vector $B$, while a reflection $\underline N$ has an additional sign change:
$$\underline N(B \epsilon) = - \underline N(B) \epsilon$$
Both of these properties can be derived from more general formulas relating inverses and adjoint transformations, for the special cases where the determinant is $\pm 1$.
If $B$ is an $(n-1)$-vector, then $B\epsilon$ is the vector normal to it (which is unique to within a sign).  What these two transformation laws say is that you can rotate $B$ and the normal rotates appropriately, but if you reflect $B$ and find the new normal, it's opposite from reflecting the normal itself.  This is the classic behavior of a pseudovector.
