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

1. $\grad\times\FF=\zero$;
2. $\int\FF\cdot d\rr$ is independent of path;
3. $\oint\limits\FF\cdot d\rr=0$ for any closed path;
4. $\FF$ is the gradient of some scalar field, that is, $\FF=\grad f$ for some $f$.

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