Show that the volume element

\[
d \omega=\prod_{i=1}^{3 N}\left(d q_{i} d p_{i}ight)
\]

of the phase space remains invariant under a canonical transformation of the (generalized) coordinates \((q, p)\) to any other set of (generalized) coordinates \((Q, P)\).

Before considering the most general transformation of this kind, which is referred to as a contact transformation, it may be helpful to consider a point transformation - one in which the new coordinates \(Q_{i}\) and the old coordinates \(q_{i}\) transform only among themselves.