Theorems · Theorem · global analysis

MDifferentiableOn.smul_section_of_tsupport

∀ {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [inst : TopologicalSpace B]
  [inst_1 : TopologicalSpace (Bundle.TotalSpace F E)] [inst_2 : (x : B) → TopologicalSpace (E x)]
  [inst_3 : NormedAddCommGroup F] [inst_4 : NontriviallyNormedField 𝕜] [inst_5 : NormedSpace 𝕜 F]
  [inst_6 : FiberBundle F E] [inst_7 : (x : B) → AddCommGroup (E x)] [inst_8 : (x : B) → Module 𝕜 (E x)]
  [VectorBundle 𝕜 F E] {HB : Type u_7} [inst_10 : TopologicalSpace HB] [inst_11 : ChartedSpace HB B] {EB : Type u_8}
  [inst_12 : NormedAddCommGroup EB] [inst_13 : NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {u : Set B}
  {s : (x : B) → E x} {ψ : B → 𝕜},
  MDiff[u] ψ → IsOpen u → tsupport ψ ⊆ u → MDiff[u] (T% s) → MDiff fun x => ⟨x, (ψ • s) x⟩

The scalar product ψ • s of a differentiable function ψ : M → 𝕜 and a section s of a vector bundle V → M is differentiable once s is differentiable on an open set containing tsupport ψ. See ContMDiffOn.smul_section_of_tsupport for the analogous result about C^n sections.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Cited by

0 results in Mathlib

Foundations

Depth 211 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNontriviallyNormedFieldNormedSpaceFiberBundleAddCommGroupModuleVectorBundleTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpace