Divergence plays a big role in physics when one studies electrodynamics. Here divergence is introduced when studying positive and negative charges as sources and sinks of electric field. It is said that positive charges mean positive divergence and negative charges negative divergence.
I thought I had a clear intuitive understanding od divergence until I began to take a look at some particular fields.
Here are two simple vector fields:
$\vec{A}=x\hat{x}+y\hat{y}$
$\vec{A}$">$\vec{B}=y\hat{x}+x\hat{y}$
$\vec{B}$">The first one has a divergence of 2 at every point and the second one zero divergence at every point. I can understand this result when observing fields at the origin, but when looking at other points, especially points where for example $x=y>0$, around which fields look very similar, I don't understand how this can be.
So my question is: how can I intuitively see and understand divergence at the arbitrary point in terms of sources and sinks.
Best Answer
To "see" the divergence around a point different from the origin, just take out a constant term (the vector value at that point): a constant field has zero divergence, so you do not alter the situation in subtracting.
You can easily see, for instance in the first case, that upon subtracting you get at $P$ the same configuration as at the origin.