= Solution
The vertical <relative vorticity> is
$$
\zeta=v_x-u_y.
$$
Differentiate the $v$-momentum equation with respect to $x$ and the $u$-momentum equation with respect to $y$, then subtract. The mixed pressure derivatives cancel, giving
$$
\zeta_t
=(-fu-g\eta_y)_x-(fv-g\eta_x)_y
=-f(u_x+v_y).
$$
The mass-conservation equation says $\eta_t=-h(u_x+v_y)$. Therefore the linearized <shallow-water potential vorticity>
$$
q=\zeta-\frac fh\eta
$$
satisfies
$$
q_t=\zeta_t-\frac fh\eta_t
=-f(u_x+v_y)+f(u_x+v_y)=0.
$$
Thus
$$
\boxed{\frac{\partial q}{\partial t}=0}.
$$
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