Suppose we describe our vector field by arrows as described above; the curl is then a measure of the curliness of the arrows.
If v is zero throughout R and R is simply connected, then v is the gradient of a potential in R. (This is exactly the condition that cross partials are equal previously described.) If it holds v is said to be irrotational.With curl non-zero the arrows describing a vector field can form closed paths like the swirls in a fingerprint.
Line integrals of vector fields along paths are path dependent when the field is not irrotational, that is when the curl is non-zero. The difference of the "circulation" integral of a vector field along two paths with the same endpoints can be described as the integral on the closed path obtained by going up one and down the other.
We will soon describe a formula for such integrals in terms of the flux integral of the curl of v on any surface in R having the closed path as boundary. Thus the curl supplies a quantitative measure of how non conservative a vector field is.
A vector field that is the curl of another is divergence free: its divergence vanishes, which means its arrows have no sources or sinks, and consist only of swirls.