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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
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