# What does it intuitively mean that the divergence of a vector field is 0?

I was going through an Electrodynamics textbook, and as a prerequisite it requires elemenets of Vector calculus and Multivariable calculus. They discussed divergence, and gave examples of fields with positive and negative divergence. And they also gave a graphic example of a vector field where all the vectors are equal and parallel to each other as a field with 0 divergence.

However, to me it seems that this should have a positive divergence, not 0. Can someone explain this to me?

• Check out the divergence theorem on Wikipedia and if you still don't understand look at The Feynman Lectures Vol. 1 I believe. – Tom Copeland Sep 4 '12 at 6:53
• rather The Feynman Lectures on Physics Vol. II. – Tom Copeland Sep 4 '12 at 7:18

The divergence of a vector field at a point is the net flow generated by a vector field into (or out of) a small region around the point. If all the vectors of the field are parallel, then in any small region, there is just as much flow inwards as outwards, so the net flow is 0.

• Pleas also note the "Divergence Theorem" as a special case of Stoke's theorem: en.wikipedia.org/wiki/Divergence_theorem for some intuition. A nice example is found population dynamics. Suppose you want to measure the change of population in your country (immigration, emigration) in a simplified example (without birth/death rates). Then you could either look at the integral of the divergence of your vector field $F$ (it denotes where people move) in the country or you could integrate $F$ along the border (surface). If $F$ has divergence $0$ you have no net in/outflows in the country... – vanguard2k Sep 4 '12 at 6:53