Theorems · Theorem · global analysis

extDeriv_pullback

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {n : ℕ} {r : WithTop ℕ∞} {x : E}
  {ω : F → F [⋀^Fin n]→L[𝕜] G} {f : E → F},
  DifferentiableAt 𝕜 ω (f x) →
    ContDiffAt 𝕜 r f x →
      minSmoothness 𝕜 2 ≤ r →
        extDeriv (fun x => (ω (f x)).compContinuousLinearMap (fderiv 𝕜 f x)) x =
          (extDeriv ω (f x)).compContinuousLinearMap (fderiv 𝕜 f x)

Exterior derivative commutes with pullback.

Defined in

Mathlib.Analysis.Calculus.DifferentialForm.Basic

Cited by

0 results in Mathlib

Foundations

Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace