CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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For a C1 path the arc-length accumulation function has derivative equal to speed

Statement

Let $a < b$ , let $γ : [a, b] \to R^{n}$ be $C^{1}$ , and let

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

Then $s_{γ}$ is differentiable on $[a, b]$ in the relative sense and

$s_{γ}^{'} (t) = ∥ γ^{'} (t) ∥_{2} .$

At $a$ and $b$ these are the relative one-sided derivatives.

Facts & Assumptions

Given: The $C^{1}$ path in the Statement.

[L1]

The arc-length function is $s_{γ} (t) = L_{[a, t]} (γ ∣_{[a, t]})$ , with $s_{γ} (a) = 0$ (The arc-length function $s_{γ} (t) = L (γ ∣_{[a, t]})$ of a rectifiable path).

[L2]

A $C^{1}$ path has length equal to the integral of its continuous speed, including on a singleton interval where both values are zero (If $γ : [a, b] \to R^{n}$ is continuous, differentiable on $(a, b)$ , and $γ^{'}$ extends continuously to $[a, b]$ , then $L (γ) = \int_{a}^{b} ∥ γ^{'} (t) ∥_{2} d t$ ).

[L3]

The integral function of an integrable function is differentiable at every point where the integrand is continuous, with derivative equal to the integrand; at endpoints this means the relative one-sided derivative (The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive).

Proof

technique · direct

1.1

The speed $v (t) : = ∥ γ^{'} (t) ∥_{2}$ is continuous because $γ^{'}$ and the Euclidean norm are continuous.

givenalgebra

2.1

By [L1] and [L2], for every $t \in [a, b]$ , $s_{γ} (t) = \int_{a}^{t} v (u) d u,$ including $t = a$ .

L1L2step 1.1

3.1

Apply [L3] to step 2.1. It gives $s_{γ}^{'} (t) = v (t) = ∥ γ^{'} (t) ∥_{2}$ throughout $[a, b]$ , with the asserted endpoint interpretation.

step 1.1step 2.1L3∎

Depends on

Used by

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Sources

J. Lebl, Basic Analysis II, section 9.2 (standard reference, not scraped)