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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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 92 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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