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, $\Pi$ be the plane perpendicular to $\n$, and consider the reflection $\R$ in the plane $\Pi$.

Quantities such as the position $\r$ and velocity $\v$ are regular vectors and transform in the obvious way under the reflection. Specifically, we write $$ \begin{align} \r &= \r_n + \r_p, \\ \v &= \v_n + \v_p, \end{align} $$ decomposing each vector into components parallel to and perpendicular to $\n$ respectively.

Under the reflection $\R$ we have

$$ \begin{align} \r' &= \R \r = -\r_n + \r_p, \\ \v' &= \R \v = -\v_n + \v_p. \end{align} $$

Now consider the angular momentum (per unit mass) $$ \L = \r \times \v, $$ or alternatively the angular velocity, $\frac{\L}{|\r|^2}$.

We can expand $\L$ as follows: $$ \begin{align} \L &= \r \times \v \\ &= (\r_n + \r_p) \times (\v_n + \v_p) \\ &= \r_n \times \v_n + \r_p \times \v_p + \r_n \times \v_p + \r_p \times \v_n \\ &= 0 + \r_p \times \v_p + (\r_n \times \v_p + \r_p \times \v_n) \\ &= \L_n + \L_p \end{align} $$

where $\L_n = \r_p \times \v_p$ is parallel to $\n$ since both $\r_p$ and $\v_p$ are in the plane $\Pi$ perpendicular to $\n$. And likewise $\L_p = \r_n \times \v_p + \r_p \times \v_n$ is perpendicular to $\n$ since each term involves something parallel to $\n$.

How does $\L$ transform under reflections? We have $$ \begin{align} \L' &= \r' \times \v' \\ &= (\R \r) \times (\R \v) \\ &= (-\r_n + \r_p) \times (-\v_n + \v_p) \\ &= \dots \\ &= \r_p \times \v_p - (\r_n \times \v_p + \r_p \times \v_n) \\ &= \L_n - \L_p \\ &= \L'_n + \L'_p \end{align} $$

Thus the component of $\L$ that is parallel to $\n$ (perpendicular to $\Pi$) stays the same, while it is the component perpendicular to $\n$ (parallel to $\Pi$) that is flipped. Thus the components of $\L$ have the opposite transformation behaviour under reflections – for it is a pseudovector.

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

• When a vector is reflected in a plane perpendicular to n...
  • its component parallel to n is flipped;
  • its component perpendicular to n stays the same.
• When a pseudovector is reflected in a plane perpendicular to n...
  • its component parallel to n stays the same;
  • its component perpendicular to n is flipped.

Pseudovectors usually involve some sort of cross product (including the curl operator) and represent rotational motion in some sense. You can see this for yourself by holding a screw and screwdriver against a mirror: the direction in which the screw moves is the direction of its angular velocity, and is a pseudovector.