EDIT
Can divergence of a vector field be positive everywhere or almost everywhere?
Can it be positive (but finite) at just one point? Is it the case when the point is the source of gas flow, from which a constant (in $\frac{m^3}s$) stream of gas flows? Or is the divergence actually infinite at that point?
Below are my thoughts, which may contain errors, so if you see any, please explain.
If I understand correctly, in a gas which expands due to heating, the divergence of the velocity field is positive at all points (or at least it can be so). But near any point the contribution to the expansion is infinitesimal. Does it mean that in the case of flow with a single point as the source, the divergence at it would be infinite? I think so, because as you decrease the diameter of the neighbourhood around that point, the flux of velocity through its boundary doesn't tend to $0$.
If so, is there a generalization of divergence that can be formulated using measure theory? Or do physicists generally relegate such considerations to pure mathematics, believing that there are no ideal, single point sources in reality?
