Recall our main theorem about vector fields.
Theorem. Let R be an open region in E2 and let F be a C1 vector field on R. The following statements about F are equivalent:
(1) There is a differentiable function f : R → R such that ∇f = F.
(2) If C is a piecewise C1 path in R, then
C F · dx depends only on the endpoints of
C.
(3)
C F · dx = 0 for every piecewise C1 simple, closed curve in R.
Furthermore, statements (1)–(3) imply
(4) curl F = 0, and (4) implies (1)–(3) when R is simply connected (so all four are equivalent when R is simply connected).
The purpose of this handout is to explore what happens when R is not simply connected, that is, when there exist non-gradient vector fields with zero curl.
An important example
Consider the vector field
F = −y x2 + y2 i + x x2 + y2 j defined on R = R2\{(0, 0)}, that is, on all of R2 except the origin. Letting P = −y/(x2+y2) and Q = x/(x2 + y2), it is a simple matter to show that ∂Q
∂x = ∂P
∂y , and so curlF = 0.
We can show that (1)–(3) are false for F by finding a simple, closed curve C for which

C F · dx = 0. Let C be the unit circle parameterized counterclockwise by x = cost, y = sin t, 0 ≤ t ≤ 2π. We have

C
F · dx =

C
−y
x2 + y2 dx + x x2 + y2 dy
=
 2π
0
− sin t
1 (− sin t) dt + cost 1 cost dt =
 2π
0
dt = 2π.
It follows from the theorem that F is not the gradient of any f