Page 112 - Vector Analysis

108 CHAPTER 4. Vector Calculus

Theorem 4.30. Let D be an open, connected domain in Rn, and let F be a smooth vector
field defined on D. Then the following three statements are equivalent:

(1) F is conservative in D.

¿

(2) F ¨ dr = 0 for every piecewise smooth, closed curve C in D.

C

ż

(3) Given any two point P0, P1 P D, F ¨ dr has the same value for all piecewise smooth

C

curves in D starting at P0 and ending at P1.

Proof. (1) ñ (2): Suppose that F = ∇ϕ in D for some scalar function ϕ : D Ñ R. Let
C Ď Rn be a piecewise smooth closed curve parameterized by γ : [a, b] Ñ Rn such
that γ : [ti´1, ti] Ñ Rn is smooth for all 1 ď i ď N , where a = t0 ă t1 ă ¨ ¨ ¨ ă tN = b.
Let Ci = γ([ti´1, ti]). Then the chain rule implies that

¿ Nż N ż ti

F ¨ dr = ÿ ∇ϕ ¨ dr = ÿ (∇ϕ ˝ γ)(t) ¨ γ 1(t) dt

C i=1 Ci i=1 ti´1

N ż ti d N ˇt=ti
ÿ
ÿ

= (ϕ ˝ γ)(t) dt = (ϕ ˝ γ)(t)ˇ = ϕ(γ(b)) ´ ϕ(γ(a)) = 0.
dt ˇt=ti´1
i=1 ti´1 i=1

(2) ñ (3): Let C1 and C2 be two piecewise smooth curves in D starting at P0 and ending
at P1 parameterized by γ1 : [a, b] Ñ Rn and γ2 : [c, d] Ñ Rn, respectively. Define
γ : [a, b + d ´ c] Ñ Rn by

" γ1(t) if t P [a, b] ,
γ(t) = if t P [b, b + d ´ c] .
γ2(b + d ´ t)

Then C = γ([a, b + d ´ c]) is a piecewise smooth closed curve; thus

¿ żb ż b+d´c

0 = F ¨ dr = (F ˝ γ1)(t) ¨ γ11(t) dt ´ (F ˝ γ2)(b + d ´ t)γ21(b + d ´ t) dt

Ca b

ż żd żż

= F ¨ dr ´ (F ˝ γ2)(t)γ21(t)dt = F ¨ dr ´ F ¨ dr .

C1 c C1 C2

ż

(3) ñ (1): Let P0 P D. For x P D, define ϕ(x) = F ¨ dr, where C is any piecewise

C

smooth curve starting at P0 and ending at x. Note that by assumption, ϕ : D Ñ R is

well-defined.