Theorems · Theorem · global analysis

extDerivWithin_constOfIsEmpty

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E} {x : E} (f : E → F),
  UniqueDiffWithinAt 𝕜 s x →
    extDerivWithin (fun x => ContinuousAlternatingMap.constOfIsEmpty 𝕜 E (Fin 0) (f x)) s x =
      (ContinuousAlternatingMap.ofSubsingleton 𝕜 E F 0) (fderivWithin 𝕜 f s x)

The exterior derivative of a 0-form given by a function f within a set is the 1-form given by the derivative of f within the set.

Defined in

Mathlib.Analysis.Calculus.DifferentialForm.Basic

Cited by

1 results in Mathlib

Foundations

Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace