integral representation of length of smooth curve

Suppose γ:[0,1]m is a continuously differentiable curve. Then the definition of its length as a rectifiable curve


is equal to its length as computed in differential geometry:


Let the partition {ti} of [0,1] be arbitrary. Then

i=1nγ(ti)-γ(ti-1) =i=1nti-1tiγ(t)𝑑t (fundamental theorem of calculusMathworldPlanetmathPlanetmath)
i=1nti-1tiγ(t)𝑑t (triangle inequalityMathworldMathworldPlanetmath for integrals)

Hence L01γ(t)𝑑t. (By the way, this also shows that γ is rectifiable in the first place.)

The inequalityMathworldPlanetmath in the other direction is more tricky. Given ϵ>0, we know that 01γ(t)𝑑t can be approximated up to ϵ by a Riemann sumMathworldPlanetmath of the form


provided the partition {ti} is fine enough, i.e. has mesh width Δ for some small Δ>0. We want to approximate γ(ti-1) with [γ(ti)-γ(ti-1)]/(ti-ti-1), but this only works if ti-ti-1 is small.

To get the precise estimates, use uniform continuity of γ on [0,1] to obtain a δ>0 such that γ(τ)-γ(t)ϵ whenever |τ-t|δ. Then for all 0<hδ and t[0,1],


Let the partition {ti} have a mesh width less than both δ and Δ. Then setting h=ti-ti-1 successively in each summand, we have

01γ(t)𝑑t i=1nγ(ti-1)(ti-ti-1)+ϵ

Taking ϵ0 yields 01γ(t)𝑑tL. ∎

We remark that L=01γ(t)𝑑t is true for piecewise smooth curves γ also, simply by adding together the results for each smooth segment of γ.

Title integral representation of length of smooth curve
Canonical name IntegralRepresentationOfLengthOfSmoothCurve
Date of creation 2013-03-22 15:39:39
Last modified on 2013-03-22 15:39:39
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 11
Author stevecheng (10074)
Entry type Derivation
Classification msc 51N05
Related topic ArcLength
Related topic Rectifiable
Related topic TotalVariation