About “Detecting Holes”

I made mention in class that

1) We always have .  In other words, if is conservative, then it has zero curl.

2) When you have a field which is defined on a simply connected domain, then if , it must be that .  That is, on simply connected domains, all fields with zero curl come from the gradient of a function (i.e. are conservative).

A few notes to ward off common confusions:

A) If is defined on a domain with a hole in it, and , then we don’t know anything about whether it is conservative.  The gravitational field is an example that has a hole in it, but it’s still conservative.

B) If is defined on a domain with a hole in it, and , then we could consider just part of that domain that doesn’t have a hole in it, and on that restricted domain, must be conservative.  So being conservative depends on the domain!  A field can be conservative on one part of its domain without being conservative on all of it.  The pdf exam conceptual review packet “Important Example” number one works like this.

Finally [this paragraph is extra-curricular material not necessary for the exam, but for general interest], in class I said that we can detect holes in a domain by using the fact that the implication “zero curl implies conservative” only works on simply connected domains.  More precisely, what I mean is that if the domain has a hole in it, then there will be some field that has zero curl but is not conservative.  If there is no hole, then there will be no such field.  (Of course, it’s not obvious how to check whether some field with a given property exists among all infinitely many possible fields, so I don’t mean this as a practical detection tool, at least not without doing more theoretical work.)

Advertisement