A vector field $\FF$ is said to be divergence free if any one of the following conditions holds:

1. $\grad\cdot\FF=0$;
2. $\int\FF\cdot d\AA$ is independent of surface;
3. $\oint\FF\cdot d\AA=0$ for any closed surface;
4. $\FF$ is the curl of some other vector field, that is, $\FF=\grad\times\GG$ for some $\GG$.

Each of these conditions implies the others. Do you see why? Spend some time thinking about these equivalences and why they hold.