I would just like to clarify something about the Divergence of a vector field at a point. Here's what I know
- At any particular point in a volume, the divergence of the vector field is the outgoing flux per unit volume.
Now what troubles me about that statement is the word point. I usually think of flux associated with a volume. Do they mean by point that we have a infinitesimal volume centered on that point?
A flux $\Phi$ is associated with a (flow) vector field ${\bf v}$ and a piece of surface $S$. It is then written as $$\Phi=\int_S{\bf v}\cdot{\bf n}\>{\rm d}\omega$$ (or similar), where ${\rm d}\omega$ denotes the (scalar) surface element.
The divergence of the field ${\bf v}$ at a point ${\bf p}$ can be understood as (the limit of) the outgoing flux through the surface of small balls $B_r$ with center ${\bf p}$, divided by the volume of $B_r$: $${\rm div}\,{\bf v}({\bf p})=\lim_{r\to0+}{\int_{\partial B_r}{\bf v}\cdot{\bf n}\>{\rm d}\omega\over{\rm vol}(B_r)}\ .$$ If the divergence of ${\bf v}$ at ${\bf p}$ has a positive value $\rho>0$ this means that all the while the fluid is flowing through ${\bf p}$ the amount $\rho$ of new fluid is created around ${\bf p}$ per unit of volume and time.