= Solution
The <Euler equations for an inviscid fluid> are $\partial_t\mathbf v+(\mathbf v\cdot\nabla)\mathbf v=-\nabla p/\rho+\mathbf f$. Here the <vorticity> is $2\Omega\mathbf e_z$, and direct differentiation gives $(\mathbf v\cdot\nabla)\mathbf v=-\Omega^2(x,y,0)$, whose curl vanishes. Taking the curl therefore gives
$$
2\dot\Omega\mathbf e_z=\nabla\times\mathbf f=(\gamma-\beta)\mathbf e_z,
$$
so
$$
\boxed{\dot\Omega=\frac{\gamma-\beta}{2}.}
$$
For constant force coefficients and the initial rest condition, $\Omega(t)=(\gamma-\beta)t/2$. Only the antisymmetric part of the horizontal force gradient generates rotation; its symmetric part is balanced by pressure.
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