The fundamental theorem implies that vector fields of the form $\FF=\grad{f}$ are special; the corresponding line integrals are always independent of path. One way to think of this is to imagine the level curves of $f$; the change in $f$ depends only on where you start and end, not on how you get there. These special vector fields have a name: A vector field $\FF$ is said to be conservative if there exists a potential function $f$ such that $\FF=\grad{f}$.

If $\FF$ is conservative, then $\DS\Lint\FF\cdot d\rr$ is independent of path; the converse is also true. But how do you know if a given vector field $\FF$ is conservative? That's the next lesson.