Question

How would you explain the physical meaning of vector field divergence?

Answer 1

The divergence of a vector field at a point is the tendency for the field to "flee from", or diverge, from that point. If the magnitude of the vectors gets larger as they pass through a point, the divergence of the vector field at that point is positive and that point is called a source. If, on the other hand, the magnitude of the vectors get smaller as they pass through a point, the divergence of the vector field at that point is negative and that point is called a sink.

In terms of fluid flow, the divergence represents the net rate of change of the mass of fluid flowing from a point per unit volume. Notice the language of derivatives (rate of change). This is the essence of divergence.

If the divergence of F is zero everywhere, F is said to be incompressible.

Answer 2

http://en.wikipedia.org/wiki/Divergence_theorem

If the divergence is zero means, the surface integral on the right hand side is zero -- which is net flux though the surface.
Net flux zero means the total inward going flux is same as total coming outward -- Fair and square.

If the net flux is > 0, means somebody is producing extra fluxes.
If the net flux is < 0, means somebody is eating the fluxes.

Answer 3

http://en.wikipedia.org/wiki/Divergence_theorem

If the divergence is zero means, the surface integral on the right hand side is zero --

The surface integral of a Field over a closed surface gives the net flux of that field through closed surface.

Net flux zero means the total inward going flux is same as total coming outward -- Fair and square.

If the net flux is > 0, means somebody is producing extra fluxes.
If the net flux is < 0, means somebody is eating the fluxes.