differentiation - What is the physical meaning of divergence?

I want to visualize the concept of divergence of a vector field. I also have searched the web.Some says it is

1.the amount of flux per unit volume in a region around some point

2.Divergence of vector quantity indicates how much the vector spreads out from the certain point.(is a measure of how much a field comes together or flies apart.).
3.The divergence of a vector field is the rate at which"density"exists in a given region of space.
4.Divergence measures the net flow of fluid out of (i.e. diverging from) a given point. If fluid is flowing instead into that point the divergence will be negative.

I am confused with all these definition.Can someone give me a proper visualizable definition which also satisfies in some way all the above definition and descriptions? Thanks for your response.

Answer

You can think of the divergence of a vector field as the number of lines of field getting out from a point in space.

This is a (very) rough explanation in natural language. A more precise explanation is that the divergence is the volume density of flux of the vector field. This can be seen from the Gauss theorem for a volume $V$ inside a closed surface $S$.

Say $\vec{E}$ is the vector field. Then its flux through $S$ is $\Phi (\vec{E}) = \int \int _S \vec{E} \cdot d\vec{S}$. The Gauss theorem states that:

$\int \int _S \vec{E} \cdot d\vec{S} = \int \int \int _V \nabla \cdot \vec{E} dV$.

If you take a very small volume $V$ (going to zero in limit) where the field doesn't vary much, then:

$\Phi (\vec{E}) = \int \int _S \vec{E} \cdot d\vec{S} = \nabla \cdot \vec{E} V$,

thus:

$\nabla \cdot \vec{E} = \frac{\Phi (\vec{E})}{V}$.

Generally

$\nabla \cdot \vec{E} = \frac{d \Phi (\vec{E})}{dV}$.

If $\vec{E}$ comes out of a point in space (say a charge at that point) then $\nabla \cdot \vec{E} > 0$ at that point; if $\vec{E}$ goes into the point then $\nabla \cdot \vec{E} < 0$; if it goes both in and out of the point then the divergence is given by the difference between the flux going out and the (absolute value of) flux going into the point. If equal numbers of lines of field go in and come out of the point then $\nabla \cdot \vec{E} = 0$ and $\vec{E}$ is a rotational field (i.e. lines of field have to return back to the point of origin so they form closed loops). The latter is the case of the magnetic field.

Note that $d\vec{S}$ is always perpendicular to surface $S$ and oriented towards the outside of $S$. Thus field lines coming out of $S$ (or V) give a positive flux, while those entering $S$ (or V) give a negative flux.