Page 105 - Vector Analysis

§4.1 The Line Integrals 101

Definition 4.17. Let RC be the collection of all piecewise regular curves. The line element
is a set function s : RC Ñ R that satisfies the following properties:

1. s(C) ą 0 for all C P RC.

2. If C P RC is the union of finitely many regular curves C1, ¨ ¨ ¨ , Ck that do not overlap
except at their end-points, then

s(C) = s(C1) + ¨ ¨ ¨ + s(Ck) .

3. The value of s agrees with the length on straight line segments; that is,
s(L) = ℓ(L) for all line segaments L .

Line integrals of scalar functions

Definition 4.18. Let C Ď Rn be a simple rectifiable curve with an injective Lipschitz

parametrization γ : [a, b] Ñ Rn, and f : C Ñ R be a real-valued function. The line

ż

integral of f along C, denoted by f ds, is the number

C

k ( )( ) ˇ k P N, a = t0 ă t1 ă ¨ ¨¨ ă tk = )
inf f (ξ) ℓ γ([ti´1, ti]) ˇ b
sup ! ÿ ˇ
ξPγ([ti´1,ti])
i=1

provided that it is identical to

! k ( ) ( )ˇ )

inf ÿ sup N,

f (ξ) ℓ γ([ti´1, ti]) ˇ k P a = t0 ă t1 ă ¨ ¨ ¨ ă tk = b .
ˇ

i=1 ξPγ([ti´1,ti])

¿

When C is a closed curve, we also use f ds to denote the line integral of f along C to

C

emphasize that the curve C is a closed loop.

Remark 4.19. Since the parametrization γ is required to be injective, the line integral of
f along C is independent of the choice of the parametrization.

żż

Remark 4.20. In particular, if f ” 1, then ℓ(C) = 1 ds ” ds.

CC

Remark 4.21. If the curve C is a line segment ␣(x, 0) ˇ a ď x ď b(, then the line integral
ˇ

of f along C is simply the Riemann integral of f over [a, b] (by treating f as a function of

x).