I am currently reading a book titled Notes on Computational Fluid Dynamics: General Principles, which is provided by OpenFOAM at https://doc.cfd.direct/notes/cfd-general-principles/contents. In chapter 2.4 of this book, they define the divergence of a point as follows:
$\nabla \cdot \mathbf{u} = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \int_{S} (\mathbf{dS} \cdot \mathbf{u})$
I understand that Gauss's divergence theorem relates the flux of a vector field through a closed surface to the divergence of the vector field within the volume enclosed by the surface and that this allows us to define $\int_V (\nabla \cdot \mathbf{u}) \, dV = \oint_S (\mathbf{u} \cdot \hat{\mathbf{n}}) \, dS$ for the divergence of a velocity field within a volume V, where V is the volume enclosed by the surface S, $\hat{\mathbf{n}}$ is the outward-pointing unit normal vector at each point on S, and dS is the infinitesimal element of area on S.
However, I don't quite understand how we go from $\int_V (\nabla \cdot \mathbf{u}) \, dV = \oint_S (\mathbf{u} \cdot \hat{\mathbf{n}}) \, dS$ to $\nabla \cdot \mathbf{u} = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \int_{S} (\mathbf{dS} \cdot \mathbf{u})$. Why are we normalizing the flux of the velocity field through the surface with the infinitesimal volume? Could someone please explain the process step by step or point me to a resource that does so?