ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-13

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Scalar and vector line integrals along an affine line segment

Example

For $p, q \in R^{n}$ , let $γ (t) = (1 - t) p + t q$ on $[0, 1]$ , and let $f : R^{n} \to R$ and $F : R^{n} \to R^{n}$ be continuous on a set containing the segment from $p$ to $q$ . Then

$\int_{γ} f d s = ∥ q - p ∥_{2} \int_{0}^{1} f ((1 - t) p + t q) d t,$

$\int_{γ} F \cdot d r = \int_{0}^{1} ⟨ F ((1 - t) p + t q), q - p ⟩ d t .$

Facts & Assumptions

Given: The affine path and continuous fields in the Example.

[L1]

On a $C^{1}$ path, scalar and vector line integrals use respectively the integrands $f (γ (t)) ∥ γ^{'} (t) ∥_{2}$ and $⟨ F (γ (t)), γ^{'} (t) ⟩$ (Scalar line integrals with respect to arc length and vector-field line integrals).

Verification

technique · direct

1.1

Differentiation gives $γ^{'} (t) = q - p$ , so the speed is the constant $∥ q - p ∥_{2}$ .

givenalgebra

2.1

Substitute step 1.1 and the formula for $γ (t)$ into the scalar clause of [L1]. Pulling out the constant speed gives the first displayed formula.

step 1.1L1algebra

2.2

Substitute step 1.1 into the vector clause of [L1] to obtain the second displayed formula.

step 1.1L1

3.1

If $p = q$ , then $q - p = 0$ and $∥ q - p ∥_{2} = 0$ , so both formulas give zero. Thus the constant-segment case is included.

step 2.1step 2.2algebra∎

Depends on

Scalar line integrals with respect to arc length and vector-field line integrals

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

J. Lebl, Basic Analysis II, Example 9.2.18 (standard reference, not scraped)