CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A continuous piecewise- $C^{1}$ path is rectifiable and its length is the sum of the speed integrals over its pieces

Statement

Let $γ : [a, b] \to R^{n}$ be continuous. Suppose there is a partition $a = t_{0} < \dots < t_{m} = b$ such that on each $[t_{i}, t_{i + 1}]$ the restriction is differentiable in the interior and its derivative has a continuous extension $v_{i}$ to that closed subinterval. Then $γ$ is rectifiable and

$L_{[a, b]} (γ) = \sum_{i < m} \int_{t_{i}}^{t_{i + 1}} ∥ v_{i} (t) ∥_{2} d t .$

No agreement between $v_{i - 1} (t_{i})$ and $v_{i} (t_{i})$ is required, so corners are allowed. For a singleton interval the empty sum and the length are zero.

Facts & Assumptions

Given: The piecewise $C^{1}$ path and subdivision.

[L1]

Each $C^{1}$ piece is rectifiable with length equal to its speed integral (If $γ : [a, b] \to R^{n}$ is continuous, differentiable on $(a, b)$ , and $γ^{'}$ extends continuously to $[a, b]$ , then $L (γ) = \int_{a}^{b} ∥ γ^{'} (t) ∥_{2} d t$ ).

[L2]

Arc length is additive over adjacent parameter subintervals (Arc length is additive across every subdivision point and decreases under restriction).

Proof

technique · decomposition

1.1

By [L1], the $i$ -th restriction is rectifiable and has length $\int_{t_{i}}^{t_{i + 1}} ∥ v_{i} ∥_{2}$ .

givenL1

2.1

Repeated application of [L2] expresses the total length as the sum of the finitely many piece lengths.

step 1.1L2

3.1

Substituting step 1.1 into step 2.1 proves the formula and finiteness. Endpoint derivative extensions are local to each piece, so no matching condition is used.

step 1.1step 2.1

4.1

On a singleton there are no nondegenerate pieces, and both the empty sum and the defined length are zero.

givenalgebra∎

Depends on

Used by

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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

A continuous piecewise-C1C1 path is rectifiable and its length is the sum of the speed integrals over its pieces