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

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Line-integral estimates by arc length and the supremum of the field

Statement

Let $γ$ be a piecewise- $C^{1}$ path of length $L (γ)$ , let $f$ be a continuous scalar field and $F$ a continuous vector field on its trace, and let $M \geq 0$ .

If $∣ f (x) ∣ \leq M$ on the trace of $γ$ , then $∣ \int_{γ} f d s ∣ \leq M L (γ) .$
If $∥ F (x) ∥_{2} \leq M$ on the trace of $γ$ , then $∣ \int_{γ} F \cdot d r ∣ \leq M L (γ) .$

Facts & Assumptions

Given: The path, fields, and bound in the Statement.

[L1]

Line integrals are sums over smooth pieces of $f (γ (t)) ∥ v_{i} (t) ∥_{2}$ or $⟨ F (γ (t)), v_{i} (t) ⟩$ (Scalar line integrals with respect to arc length and vector-field line integrals).

[L2]

The Euclidean Cauchy-Schwarz inequality is $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[L3]

The Riemann integral is linear, and pointwise order between integrable functions is preserved by integration (Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ , If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ).

[L4]

For an admissible partition, $L (γ)$ is the sum of the integrals of the speeds $∥ v_{i} ∥_{2}$ (A continuous piecewise- $C^{1}$ path is rectifiable and its length is the sum of the speed integrals over its pieces).

Proof

technique · direct

1.1

On each smooth piece, $- M ∥ v_{i} (t) ∥_{2} \leq f (γ (t)) ∥ v_{i} (t) ∥_{2} \leq M ∥ v_{i} (t) ∥_{2} .$

givenalgebra

1.2

By [L2] and the bound on $F$ , $- M ∥ v_{i} (t) ∥_{2} \leq ⟨ F (γ (t)), v_{i} (t) ⟩ \leq M ∥ v_{i} (t) ∥_{2} .$

givenL2algebra

2.1

Integrate the two inequalities in step 1.1 using [L3], sum them using [L1], and identify the speed sum with [L4]. This gives $- M L (γ) \leq \int_{γ} f d s \leq M L (γ),$ hence the scalar estimate.

step 1.1L1L3L4algebra

3.1

Repeating step 2.1 with step 1.2 gives the vector estimate.

step 2.1step 1.2L1L3L4

4.1

If $M = 0$ or $L (γ) = 0$ , either two-sided bound has both endpoints equal to zero, so the corresponding integral is zero and the asserted estimate still holds.

step 2.1step 3.1algebra∎

Depends on

Used by

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Sources

J. Lebl, Basic Analysis II, Proposition 9.2.10 (standard reference, not scraped)