Euler equations for ideal fluids, approximations

[Niles]

Homework Statement

The Euler equations for ideal compressible flow are given by
$$\partial_t v + (v\cdot \nabla)v = g-\frac{1}{\rho}\nabla p \\<br /> \partial_t \rho + \nabla \cdot(\rho v) = 0$$
In my book these are written in terms of the small-value expansions $\rho = \rho_0 + \delta \rho$, $p = p_0 + \delta p$ and the equations become
$$\partial_t v = -\frac{1}{\rho_0}\nabla \delta \rho \\<br /> \partial_t (\delta \rho) = -\rho_0 \nabla \cdot v$$

In the second equation, I don't understand why the RHS becomes $\rho_0 \nabla \cdot v$ instead of $(\rho_0+\delta \rho) \nabla \cdot v$?

Thanks in advance.

 

[bigfooted]

$\delta \rho \nabla \cdot v$ is negligible compared to $\rho_0 \nabla \cdot v$ because $\delta \rho << \rho_0$