DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13

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

Definition

Let $γ : [a, b] \to R^{n}$ be piecewise- $C^{1}$ . If $a = b$ , define both line integrals below to be $0$ . If $a < b$ , choose an admissible partition $a = t_{0} < \dots < t_{m} = b$ and a continuous derivative extension $v_{i}$ on each piece. Let $f$ be a continuous scalar field and $F$ a continuous vector field on a set containing the trace of $γ$ . The scalar line integral with respect to arc length and the vector-field line integral are

$\int_{γ} f d s : = \sum_{i < m} \int_{t_{i}}^{t_{i + 1}} f (γ (t)) ∥ v_{i} (t) ∥_{2} d t,$

$\int_{γ} F \cdot d r : = \sum_{i < m} \int_{t_{i}}^{t_{i + 1}} ⟨ F (γ (t)), v_{i} (t) ⟩ d t,$

where the inner product is The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . The summands exist by A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. Independence of the admissible partition is proved in The piecewise-C1 line-integral sums do not depend on the admissible partition ↗.

Depends on

Dependency tree · 2 levels

Sources

J. Lebl, Basic Analysis II, section 9.2