

8.2 What Does the Curl Signify? Why is it Important?


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 nonzero 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 nonzero.
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.