Apologies & correction

I went home yesterday afternoon after classes, seminars and meetings and got pukey-sick, which is hopefully the only cause of my stupidity on Friday going over the quiz, for which I apologize and which I'm going to correct here.  If you still are confused on this issue, I'll be in my office tomorrow 11:30-12:30pm and Tuesday 12:30-1:30pm to discuss with you

The second problem on the quiz was to sketch out the regions of positive and negative divergence, which are as follows (blue is positive divergence and red is negative divergence)--I have imposed a coordinate system on the graph to give a language to describe parts of the graph:

Here's the correct reasoning. Calling the vector field F=Pi +Qj , the divergence is ∂P/∂x+∂Q/∂y; we can see that the Q is positive above the line y=1 and also below the line y=-1, and increasing as |y|>1 is more positive; you can see this in the way the vectors point upward-ish and are increasingly upward on each vertical line for x big enough or small enough.   On the other hand Q is negative between y=1 and y=-1, as you can see from the fact that the vectors point slightly downward.  Likewise, for P is positive for x bigger than 1 and also for x less than -1, and is more positive the more |x| is greater than 1 and is negative while x is  between x=-1 and x=1; you can see this from the way that the vectors point somewhat to the left when |x|<1 and to the right when |x|>1, and point increasingly to the right along horizontal lines as |x| is large enough.  

Now, since div(F)=∂P/∂x+∂Q/∂y, div(F) will be positive if the sum of ∂P/∂x and ∂Q/∂y is positive, that is if the rate of increase of P in the positive x-direction is greater than the rate of decrease of Q in the positive y-direction OR vice versa.  In the first quadrant, we can see that both ∂P/∂x and ∂Q/∂y are positive, in that they both P and Q increase from negative to positive values going from left to right (P) and from down to up (Q): this means div(F) must be positive.  The reverse is true in the third quadrant, so  div(F) must be negative.  In the half of the second quadrant above the line y=-x, Q increases a lot with increasing y while P decreases slightly with increasing x so ∂P/∂x is less negative than ∂Q/∂y is positive so div(F) must be positive; the opposite is true in the second quadrant below the line.  In the same manner, in the fourth quadrant above the line P is increasing quite a bit moving from left to right, while Q decreases not so much--it follows that ∂P/∂x is more positive than ∂Q/∂y is negative so div(F) must be positive, and again in the same manner the opposite is true in the fourth quadrant below the line.