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# Vetors Essay

279 words - 2 pages

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 ...view middle of the document...

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
=

0
− sin t
1 (− sin t) dt +
cost
1
cost dt =

0
dt = 2π.
It follows from the theorem that F is not the gradient of any f

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