Theorems · Theorem · global analysis

inTangentCoordinates_eq

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {E' : Type u_3} [inst_3 : NormedAddCommGroup E'] [inst_4 : NormedSpace 𝕜 E'] {H : Type u_4}
  [inst_5 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [inst_6 : TopologicalSpace H']
  {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_6} [inst_7 : TopologicalSpace M] [inst_8 : ChartedSpace H M]
  {M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] [inst_11 : IsManifold I 1 M]
  [inst_12 : IsManifold I' 1 M'] {N : Type u_9} (f : N → M) (g : N → M') (ϕ : N → E →L[𝕜] E') {x₀ x : N},
  f x ∈ (chartAt H (f x₀)).source →
    g x ∈ (chartAt H' (g x₀)).source →
      inTangentCoordinates I I' f g ϕ x₀ x =
        (tangentBundleCore I' M').coordChange (achart H' (g x)) (achart H' (g x₀)) (g x) ∘SL
          ϕ x ∘SL (tangentBundleCore I M).coordChange (achart H (f x₀)) (achart H (f x)) (f x)

To write a linear map between tangent spaces in coordinates amounts to precomposing and postcomposing it with suitable coordinate changes. For a concrete version expressing the change of coordinates as derivatives of extended charts, see inTangentCoordinates_eq_mfderiv_comp.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.Tangent

Cited by

1 results in Mathlib

Foundations

Depth 218 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifold