Theorems · Theorem · real analysis

ContinuousOn.curveIntegrable_of_contDiffOn

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {a b : E} {ω : E → E →L[𝕜] F}
  {γ : Path a b} [inst_5 : NormedSpace ℝ E] {s : Set E},
  ContinuousOn ω s → ContDiffOn ℝ 1 (⇑γ.extend) unitInterval → (∀ (t : ↑unitInterval), γ t ∈ s) → CurveIntegrable ω γ

If a 1-form ω is continuous on a set s, then it is curve integrable along any $C^1$ path in this set.

Defined in

Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Cited by

0 results in Mathlib

Foundations

Depth 247 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedSpace