Page 101 - Vector Analysis

§4.1 The Line Integrals 97

Definition 4.6 (Rectifiable curves). A curve C Ď Rn with parametrization γ : I Ñ Rn is
called rectifiable if there is an homeomorphism φ : Ir Ñ I, where Ir is again an interval,
such that the map γ ˝ φ : Ir Ñ Rn is Lipschitz.

Remark 4.7. 1. By an homeomorphism it means a continuous bijection whose inverse is
also continuous.

2. We can think of a curve as an equivalence class of continuous maps γ : I Ñ Rn, where

two parametrization γ : I Ñ Rn and γ : Ir Ñ Rn are equivalent if and only if there is
r

an homeomorphism φ: Ir Ñ I such that γ = γ ˝ φ. Each element of the equivalence
r

class is a parametrization of the curve and thus a rectifiable curve is a curve which

has a Lipschitz continuous parametrization.

3. The length of a rectifiable curve parameterized by γ : [a, b] Ñ Rn is finite since by

choosing a Lipschitz parametrization γ : [c, d] Ñ Rn, the number
r

! k ˇ )
ˇk d
ÿ ››γr(ti) γr(ti´1)››Rn ˇ N and

´ P c = t0 ă t1 ă ¨¨ ¨ ă tk =

i=1

is bounded from above by M (d ´ c), where M is the Lipschitz constant of γ.
r

Example 4.8 (Non-rectifiable curves). Let C Ď R2 be a curve parameterized by

# ( π) if t P (0, 1] ,
t, t sin
γ(t) = t

(0, 0) if t = 0 .

Since

( 1 , 1 ) ě ››γ( 1 ) ´ γ ( 1 )››R2 + ››γ( 1 ) ´ γ ( 1 )››R2 ě 2
ℓ γ([ ]) + 1/2 n + 1/2
n+1 n n+ 1 n n + 1/2 n

and 8 2 = 8, by the remark above we conclude that γ([0, 1]) is not a rectifiable

ř

n=1 n + 1/2
curve.

Definition 4.9. A curve C Ď Rn is said to be of class C k or a C k-curve if there exists
a parametrization γ : I Ñ Rn such that γ is k-times continuously differentiable. Such a
parametrization is called a C k-parametrization of the curve. If there exists a parametrization
γ : I Ñ R which is of class C k for all k P N, then the curve is said to be smooth. A curve
C Ď Rn is said to be regular if there exists a C 1-parametrization γ : I Ñ Rn such that
γ 1(t) ‰ 0 for all t P I.