DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13

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The arc-length function $s_{γ} (t) = L (γ ∣_{[a, t]})$ of a rectifiable path

Definition

Let $γ : [a, b] \to R^{n}$ be rectifiable. Its arc-length function is

$s_{γ} : [a, b] \to R, s_{γ} (t) : = L_{[a, t]} (γ ∣_{[a, t]}) .$

The value is finite because every restriction of a rectifiable path is rectifiable by length additivity. In particular

$s_{γ} (a) = 0, s_{γ} (b) = L_{[a, b]} (γ),$

and for $a \leq u \leq v \leq b$ ,

$s_{γ} (v) - s_{γ} (u) = L_{[u, v]} (γ ∣_{[u, v]}) .$

The last identity is the additive length theorem applied at $u$ and $v$ , not an additional convention.

Depends on

Used by

Dependency tree · two levels

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Sources

U. Lang, Differential Geometry I, Section 1.1 (standard reference, not scraped)

The arc-length function sγ(t)=L(γ∣[a,t])sγ​(t)=L(γ∣[a,t]​) of a rectifiable path