Non-uniform field pressure on fluid

I have been studying fluid mechanics and currently I was trying to understand Euler's equation for fluid flow. For that purpose, I was following this webpage.

My problem is in understanding Eq.(12). The author states that that is the contribution due to an external $\underline{\text{uniform}}$ (gravitational) field, $g$. What I don't understand is why does that definition only applies to an uniform field. What would the contribution due to a, say, central field be?

Can anyone help me?

In general, any external body force would give a contribution $\int \mathbf{f} \ dV$, where $\mathbf{f}$ is the force per unit volume resulting from the external force. For those who didn't read the link, the integral is over a small volume at the point in consideration. Here external means that the source of the force is anything but the fluid itself.
The specific case, which is common in fluids, is that the external force is gravity, and $\mathbf{f} = \rho \mathbf{g}$.
If you didn't want to make a uniform gravity approximation and instead took the Newton's gravitational law for the gravitational force you would get $\mathbf{f} = \frac{G \rho m}{ r^2}$, where $m$ is the mass gravitational field source, and $r$ is the distance to that body (assuming for simplicity a spherically symmetric object).