Page 113 - Vector Analysis

§4.2 Conservative Vector Fields 109

Choose δ ą 0 such that B(x, δ) Ď D. Let C be a piecewise smooth curve joining
P0, and L be the line segment joining x and x + hej, where 0 ă h ă δ and ej =
(0, ¨ ¨ ¨ , 0, 1, 0, ¨ ¨ ¨ , 0) is the unit vector whose j-th component is 1. Then with the
parametrization of L: γ(t) = x + tej for t P [0, h], we have

ϕ(x + hej ) ´ ϕ(x) = 1 ż ¨ dr = 1 żh + tej ) ¨ ej dt ;
h h F h F(x

L 0

thus passing to the limit as h Ñ 0, we find that

Bϕ (x) = F(x) ¨ ej .
B xj

As a consequence, F(x) = (∇ϕ)(x) which implies that F is conservative. ˝

Let D Ď R2, and F = (M, N ) : D Ñ R2. If F is conservative, then M = ϕx and N = ϕy
for some scalar function ϕ : D Ñ R; thus if ϕ is of class C 2, we must have My = Nx. In
other words, if F : D Ñ R2 is a smooth vector field, then it is necessary that My = Nx. The
converse statement is not true in general, and we have the following counter-example.

Example 4.31. Let D Ď (Rx,2yb) e=t(hxe2a+ynnyu2 l,axr2´r+exgyi2o)n. D = ␣(x, y) ˇ 1 ă x2 + y2 ă 4(, and
consider the vector field F ˇ

Then

B y = x2 ´ y2 = B ´x ;
B y x2 + y2 (x2 + y2)2 B x x2 + y2

however, if F = ∇ϕ for some differentiable scalar function ϕ : D Ñ R, we must have

y
ϕx(x, y) = x2 + y2

which further implies that

ϕ(x, y) = arctan x + f (y) .
y

Using that ϕy(x, y) = y, we conclude that f is a constant function; thus

x2 + y2

ϕ(x, y) = arctan x + C .
y

Since ϕ is not differentiable on the positive x-axis, F ‰ ∇ϕ.