Pseudovectors and transformations

Here are some notes for my Fluid Dynamics II students about vectors, pseudovectors and how they transform under reflections.

Let n be a unit vector and consider the reflection R in the plane perpendicular to n.

Quantities such as the position r and velocity v are regular vectors and transform in the obvious way under the reflection. Specifically, we write

r = rn + rp
v = vn + vp

where rn is the component of r parallel to n, rp is the perpendicular component; and similar for vn and vp. Then under the transformation R we have

r' = R r = -rn + rp
v' = R v = -vn + vp

However, now consider the angular momentum per unit mass am = r * v, using * to denote the cross product. Or indeed the angular velocity, which is the above divided by |r|^2.

We can expand out am explicitly in terms of rn, etc.:

am = (rn + rp) * (vn + vp)
   = rn * vn + rp * vp + rn * vp + rp * vn
   = 0       + rp * vp + (rn * vp + rp * vn)
   =           amn     + amp

Here, the term amn = rp * vp is parallel to n since it is the cross product of two vectors that are perpendicular to n. And the term amp = rn * vp + rp * vn is perpendicular to n since each product involves a vector that is parallel to n.

When r and v are transformed, then am transforms as:

am' = r' * v'
    = (R r) * (R v)
    = (-rn + rp) * (-vn * vp)
    = ...
    = rp * vp - (rn * vp + rp * vn) 
    = amn     - amp
    = amn'    + amp'

Thus the component of am that is parallel to n stays the same, while the component of am that is perpendicular to n is flipped. Thus the components of am transform in the opposite direction to how those of r and v transform – for it is a pseudovector.

In summary, under a reflection in a plane perpendicular to n...

    of a vector...
        the component parallel      to n is flipped
                      perpendicular      stays the same
    of a pseudovector...
        the component parallel      to n stays the same
                      perpendicular      is flipped

(If you don't believe me, take a screw and a screwdriver and hold it up against a mirror.)