Page 106 - Vector Analysis

102 CHAPTER 4. Vector Calculus

Remark 4.22 (The interpretation of the line integrals). Let C be a piecewise smooth curve,
and f (x) denote the density of the curve C at position x. Suppose that f is continuous on
C and x = γ(t). Then f (x) is computed by

f (x) = f (γ(t)) = lim m(γ([t, t + ∆t]))
,
∆tÑ0 ℓ(γ([t, t + ∆t]))

where m(¨) denotes the mass. Let ε ą 0 be given. Then by the continuity of f ˝ γ and the
definition of limit, there exists δ ą 0 such that

ˇˇ(f ˝ γ)(t) ´ (f ˝ γ)(s)ˇˇ ă ε if t, s P [a, b], |t ´ s| ă δ
4ℓ(C )

and

ˇˇf (γ(t))ℓ(γ([t, t + ∆t])) ´ m(γ([t, t + ∆t]))ˇˇ ď ℓ(γ([t, t + ε if |∆t| ă δ ;
∆t]))

4ℓ(C )

thus if P = ta = t0 ă t1 ă ¨ ¨ ¨ ă tk = bu is a partition of [a, b] with }P} ă δ, the total mass

k

of the curve m(C), given by m(C) = ř m(γ([ti´1, ti])), validates the following estimate:

i=1

ˇ k ˇ ε
ˇm(C ) f (γ(si´1))ℓ(γ([ti´1, ti]))ˇˇ 2
ˇ ÿ

´ ď .

i=1

As a consequence,

kk

m(C) ´ ε ă ÿ inf ÿ sup f (ξ)ℓ(γ([ti´1, ti])) ă m(C) + ε
f (ξ)ℓ(γ([ti´1, ti])) ď
i=1 ξPγ([ti´1,ti])
i=1 ξPγ([ti´1,ti])

which implies that the line integral of f along C is exactly the mass of the curve.

Theorem 4.23. Let C Ď Rn be a simple curve with C 1-parametrization γ : [a, b] Ñ Rn,
and f : C Ñ R be a real-valued continuous function. Then

ż żb ( ) (4.1)
f ds = f γ(t) }γ 1(t)}Rn dt .

Ca

Proof. Let ε ą 0 be given. Since f ˝ γ and γ 1 are continuous on [a, b], |f ˝ γ| + }γ 1}Rn ď M
on [a, b] for some M ą 0, and there exists δ ą 0 such that

ˇˇ(f ˝ γ)(s) ´ (f ˝ γ)(t)ˇˇ ă 8(M + ε ´ a) whenever s, t P [a, b], |s ´ t| ă δ
1)(b