Page 102 - Vector Analysis

98 CHAPTER 4. Vector Calculus

Theorem 4.10. Let C Ď Rn be a curve with C 1-parametrization γ : [a, b] Ñ Rn. Then

żb
ℓ(C) = }γ 1(t)}Rn dt .

a

Proof. Let ε ą 0 be given. Since γ : [a, b] Ñ Rn is C 1, there exists δ ą 0 such that

}γ 1(t) ´ γ 1(s)››Rn ă ε ´ a) whenever s, t P [a, b], |s ´ t| ă δ .
4?n(b

By the definition of the length of curves, there exists a partition P = ta = t0 ă t1 ă ¨ ¨ ¨ ă

tk = bu of [a, b] such that

ℓ(C ) ´ ε ă k ´ γ(ti´1)››Rn ď ℓ(C) .
4
ÿ ››γ(ti)

i=1

W.L.O.G., we can assume that }P} ă δ. For each component γj of γ, the mean value

theorem implies that for some ξi P [ti´1, ti],

γj(ti) ´ γj(ti´1) = γj1(ξi)(ti ´ ti´1) ;

thus for each i P t1, ¨ ¨ ¨ , ku and si P [ti´1, ti],

ˇˇγj (ti ) ´ γj(ti´1) ´ γj1(si)(ti ´ ti´1)ˇˇ ď ˇˇγj1(ξi) ´ γj1(si)ˇˇ|ti ´ ti´1| ă ε ´ a) |ti ´ ti´1| .
4?n(b

As a consequence, for each i P t1, ¨ ¨ ¨ , ku and si P [ti´1, ti],

ˇ ˇˇ ˇ

ˇ››γ (ti) ´ γ(ti´1)››Rn ´ ››γ 1(si)››Rn |ti ´ ti´1|ˇˇ ă ˇ››γ (ti) ´ γ(ti´1)››Rn ´ ››γ 1(si)(ti ´ ti´1 )››Rn ˇ
ˇ ˇ ˇ

››γ(ti) ti´1)››Rn [ n( ε )2 ] 1
ε ÿ ti´1| 2
γ 1(si)(ti
ď ´ γ(ti´1) ´ ´ ď 4?n(b ´ a) |ti ´

j=1

ă 4(b ´ a) |ti ´ ti´1|

which further implies that

ˇ k k ˇ ε
ˇ ti´1|ˇˇ 4
ˇ ÿ ››γ(ti γ(ti´1)››Rn ÿ ››γ 1(si)››Rn|ti

) ´ ´ ´ ă .

i=1 i=1

Therefore, for a = t0 ď s0 ď t1 ď s1 ¨ ¨ ¨ ď sk ď tk = b,

ε k ››γ 1(si)››Rn|ti ε
2 2
ÿ

ℓ(C ) ´ ă ´ ti´1| ă ℓ(C ) + .

i=1

Since }γ 1} is Riemann integrable over [a, b], we must have

żb ˝
ℓ(C) ´ ε ă L(}γ 1}Rn, P) ď ››γ 1(t)››Rndt ď U (}γ 1}Rn, P) ă ℓ(C) + ε ,

a

and the theorem is concluded because ε ą 0 is given arbitrarily.