Does the numerical value of the divergence of a vector field at a point has any significance?

Question

If the divergence is positive, the point acts like a source.If negative it acts like a sink.If zero, the field is unaffected at the point.But Does the numerical value of the divergence of a vector field at a point has any significance or are we only concerned whether it is positive or negative or zero?

Answer 1

Consider a point ${\bf p}$ within the flow field ${\bf v}$ and a small ball $B_r$ of radius $r>0$ with center ${\bf p}$. An application of Gass' theorem then shows that $${\rm div}\,{\bf v}({\bf p})=\lim_{r\to0+}{1\over{\rm vol}(B_r)}\int_{\partial B_r}{\bf v}\cdot {\bf n}\>{\rm d}\omega\ .$$ There will be fluid flowing into $B_r$ across some parts of $\partial B_r$ and fluid flowing out of $B_r$ across other parts of $B_r$. The flux integral $\int_{\partial B_r}{\bf v}\cdot {\bf n}\>{\rm d}\omega$ is equal to the surplus of fluid coming out of the ball $B_r$ per unit of time. This surplus is somehow generated within $B_r$. We therefore can say the following: ${\rm div}\,{\bf v}({\bf p})$ is the amount of fluid being produced per unit of time and unit of volume in the immediate surroundings of ${\bf p}$.