  Exact analytical expressions for available potential energy density (APE) and
available potential vorticity (APV) are derived from first principles. These
APE and APV expressions align with previously known quantities found using
perturbation expansions in Holliday & McIntyre (1981) and Wagner & Young
(2015), respectively. The key is to recast the equations of motion and their
conservation laws in terms of the coordinate label $z-\eta$, where z is a fluid
parcels height at the current time, and $\eta$ is the isopycnal deviation from
the parcels current height after adiabatic rearrangement of all parcels into
the no-motion state. In addition to their intuitive appeal and simplicity, the
new APE and APV expressions are easily implemented in numerical computations of
Boussinesq dynamics with non-constant stratification.
