= Solution
For the <Boussinesq approximation>, subtract the resting background <buoyancy> $N^2z$ and denote its perturbation by $b$. With constant nonzero $f$ and constant $N>0$, the leading <geostrophic flow> and hydrostatic balance define the <quasi-geostrophic streamfunction>
$$
p'=f\psi,\qquad\mathbf u_g=(-\psi_y,\psi_x),\qquad b=f\psi_z.
$$
The leading horizontal flow is nondivergent. At the next order, its relative vertical <vorticity> $q_g=\nabla_H^2\psi$ is stretched by the small vertical <velocity>, while the background <buoyancy> is advected vertically. Thus the curl and <buoyancy> equations give
$$
D_g\nabla_H^2\psi=f w_z,\qquad D_gb+N^2w=0,
\qquad D_g=\partial_t-\psi_y\partial_x+\psi_x\partial_y.
$$
These follow from $Dq_a/Dt=-q_a\nabla_H\cdot\mathbf u$ with $q_a\simeq f$ and $\nabla_H\cdot\mathbf u_a=-w_z$, retaining the geostrophic advection of the first-order anomalies. Vertical advection of those anomalies is of higher quasi-geostrophic order.
Differentiate the <buoyancy> equation in $z$. It is important not to assume without justification that $\partial_z$ commutes with $D_g$. Their commutator on $b$ is
$$
[\partial_z,D_g]b=\mathbf u_{g,z}\cdot\nabla_Hb
=\frac1f(-b_y,b_x)\cdot(b_x,b_y)=0,
$$
by hydrostatic balance and <thermal wind>. Hence $D_gb_z=-N^2w_z$. Adding $f/N^2$ times this equation to the <vorticity> equation eliminates $w_z$, yielding
$$
\boxed{D_gQ=0,\qquad Q=f+\nabla_H^2\psi+\frac{f^2}{N^2}\psi_{zz}.}
$$
This is the <quasi-geostrophic potential-vorticity equation> on an $f$-plane. The additive $f$ makes $Q$ equal to the first-order normalized Ertel PV of total <buoyancy> $N^2z+b$; using a PV anomaly instead removes that constant and does not change the dynamics.
For <stretched coordinates for quasi-geostrophic inversion>, put
$$
X=x,\qquad Y=y,\qquad Z=\frac N{|f|}z.
$$
Then $(f^2/N^2)\partial_z^2=\partial_Z^2$, and the potential-vorticity relation becomes
$$
\boxed{Q=f+\nabla_*^2\psi,\qquad\nabla_*^2=\partial_X^2+\partial_Y^2+\partial_Z^2.}
$$
Stable <stratification> makes this operator elliptic. Uniform $N$ and nonzero $f$ are essential to this simple constant-coefficient stretching.
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