Negative Divergence of Radial Vector Fields? I'm having trouble understanding the concept of divergence with respect to a set of cases: radial vector fields (in 2d, for the sake of simplicity.)  If we write the vector field as (x,y)/r^n, where r = sqrt(x^2 + y^2), the divergence varies in sign with respect to n.  As n increases to 2, the divergence passes through zero and becomes negative everywhere. 
But I don't understand how a vector field that is always pointing out from the origin can have negative divergence.  Doesn't a negative divergence indicate, in the case of fluid flow, that the fluid is "entering" a point rather than "leaving" it?  But clearly this is not the case near the origin, for example, where in every direction the vectors are pointing away from the origin?  I don't understand what is special about the magnitude of the vectors decreasing more quickly away from the origin (which is what happens as n increases).
Note: I understand the case where n < 0.  I am only considering the cases where n > 0.
 A: 
Doesn't a negative divergence indicate, in the case of fluid flow, that the fluid is "entering" a point rather than "leaving" it?

Yes, or perhaps saying the fluid is compressed at that point is better.

But clearly this is not the case near the origin, for example, where in every direction the vectors are pointing away from the origin?

You need to distinguish between the origin itself, and points near the origin.
The origin is a singularity of the field (for $n \geqslant 1$, the field isn't even continuous at the origin, and for $n > 1$ not even bounded), it is a point source, fluid is created ex nihilo there.
For other points than the origin, in the case of a radial field (whose magnitude only depends on the distance from the origin), it is perhaps best to consider a small annulus (or, in dimensions $> 2$, a spherical shell) centered at the origin containing the point. A negative divergence in that annulus says that the fluid flows into the annulus (shell) faster through the inner bounding circle (sphere) than it flows out through the outer bounding circle (shell).
That that happens when the speed decreases fast enough with increasing distance from the source is, I think, understandable.
