Theorems · Theorem · global analysis

mdifferentiableWithinAt_section

∀ {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 F] [inst_3 : TopologicalSpace (Bundle.TotalSpace F E)]
  [inst_4 : (x : B) → TopologicalSpace (E x)] {EB : Type u_7} [inst_5 : NormedAddCommGroup EB]
  [inst_6 : NormedSpace 𝕜 EB] {HB : Type u_8} [inst_7 : TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB)
  [inst_8 : TopologicalSpace B] [inst_9 : ChartedSpace HB B] [inst_10 : FiberBundle F E] (s : (b : B) → E b) {u : Set B}
  {b₀ : B}, MDiffAt[u] (T% s) b₀ ↔ (MDiffAt[u] fun b => (↑(trivializationAt F E b₀) ⟨b, s b⟩).2) b₀

Characterization of differentiable sections of a vector bundle at a point within a set in terms of the preferred trivialization at that point.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Cited by

4 results in Mathlib

Foundations

Depth 207 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceFiberBundle