B.2.a)  The meaning of the sign of
           

Given a vector field
       ,
you calculate
       .
How does the sign of   
       
tell you whether is a source of new fluid or a sink (drain) for old fluid?

Answer:

If , then the point is a source of new fluid.
If , then the point is a sink for old fluid.
Here's why:
Take a small circle centered at .  Calculate the flow of across by calculating
       
       .
Here's the kicker:  
If
       ,
then it is positive for all 's close to . So if is so small that
       
at all 's inside , then you see that
       
       .
This means that if , then the net flow of across small circles centered at is from inside to outside.  
The upshot:
If , then the point is a source of new fluid.
Similarly, if , then the net flow of across small circles centered at is from outside to inside.  
So:
If , then the point is a sink for old fluid.
Check this out:
Go with and look at :

See whether the point is a source or a sink:

Positive.  This tells you that the point is a source.
Take a look at this vector field on a small circle centered at to see whether this calculation agrees with reality:

Confirm by looking at the normal components of the field vectors on the curve:

Yessiree, Bob.  The plot shows a lot more flow from inside to outside than from outside to inside.  
Just as you would expect on a small circle centered at a source.

B.2.b) Sources and sinks

Here's a vector field:

Give a sample plot of some of the sources and sinks in this vector field.

Answer:

Here's :

A point is a source if , and is a sink if .
Here comes the plot:

The larger points are sinks; the smaller points are sources.

Alternate squares of sources and sinks.
Think of the sources as little individual springs feeding the flow.  
Think of the sinks as tiny little holes through which fluid seeps out as the flow goes by.

B.2.c.i) All sources inside

If every point inside a closed curve (like a deformed circle) is a source of a given vector field, and if the vector field has no singularities inside , then how do you know that the net flow of the given vector field across is automatically from inside to outside?

Answer:

If every point inside a closed curve is a source of a given vector field, then
-> new fluid is oozing out of each point inside , and
-> there there is no place within to absorb excess outside-to-inside flow.
The result:
If every point inside a closed curve is a source of a given vector field, then the flow of this vector field across is automatically from inside to outside.
For example, look at:

Unless , .
This tells you that all points except are sources for and the lone exception is not a sink.
This vector field has no singularities.
So:
On the basis of this information you can say with confidence and authority that the flow of this vector field across any closed curve is from inside to outside.

B.2.c.ii) All sinks inside

If every point inside a closed curve (like a deformed circle) is a sink of a given vector field, and if the vector field has no singularities inside , then how do you know that the net flow of the given vector field across is automatically from outside to inside?

Answer:

If every point inside a closed curve is a sink of a given vector field, then
-> old fluid is soaking into each point inside , and
-> there there is no place within to generate excess inside-to-outside flow.
The result:
If every point inside a closed curve is a sink of a given vector field, then the flow of this vector field across is automatically from outside to inside.
For example, look at:

Unless , .
This tells you that all points except are sinks for and the lone exception is not a source.
This vector field has no singularities.
So:
On the basis of this information you can say with confidence and authority that the flow of this vector field across any closed curve is from outside to inside.
The region inside is like a big vacuum sucking up fluid.

B.2.c.iii) No sources or sinks inside

If there are no sinks and there are no sources of a given vector field inside a closed curve (like a deformed circle) , and if the vector field has no singularities inside , then how do you know that the net flow of the given vector field across is ?

Answer:

If there are no sinks and there are no sources of a given vector field inside , and there are no singularities inside , then no new fluid is injected and no old fluid is sucked up inside .
So:
-> What flows from inside to outside must be replaced by equal outside to inside flow.
-> What flows from outside to inside must be replaced by equal inside to outside flow.
The result:
If there are no sinks and there are no sources of a given vector field inside , and there are no singularities inside , then the net flow of the vector field across is .
For example, look at:

This tells you that this vector field has no sources or sinks.
This vector field has no singularities, so:
On the basis if this information you can say with confidence and authority that the net flow of this vector field across any closed curve is .

B.2.d) Divergence

Why do most folks call the divergence of a vector field ?

Answer:

The name fits.  
If , then new fluid is oozing out of the point
and diverges elsewhere.
If , then old fluid is sucked into the point
and converges onto this point.
If , then no new fluid is introduced, and no old fluid is sucked off as the flow passes by .