1. (10+5 points) Let F = P i + Qj be a vector field on a domain D in the plane.

(a) Give two different characterizations for the vector field F to be conservative

which are valid on ANY domain D.

Solution. There are three different possible characterizations:

– F = ∇f for some scalar function f.

– F ds = 0 for all closed curves C in the domain of F.

– For any curve C between points p and q,

C

F · ds depends only on the

endpoints (p and q), not on the curve itself. (Path independence).

(b) Give a condition for F to be conservative on a simply connected domain D

in terms of the derivatives of P and Q.

Solution. ∇ × F = 0. Let C be a closed curve in the domain D, and let S

be the region in D bounded by C. Then,since D (the domain of F) issimply

connected, F will be defined on all of S. So, by Green’s Theorem,

∫∫

S

∇ × F · kdA =

I

F · ds.

If ∇ × F = 0, then for any closed curve C in the domain, C

F · ds = 0, so F

is conservative.

(c) Give an example to show that the condition in (b) is not sufficient if D is not

simply connected.

Solution. Let

F =

−yi + xj

,

x

2 + y

2

with the domain of F being R2 − {(0, 0)}, i.e., the plane minus the origin (F

is not defined at the origin). Then, ∇ × F = 0, but F is not conservative