Theorems · Theorem · global analysis

IsMIntegralCurveOn.hasDerivWithinAt

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {H : Type u_2} [inst_2 : TopologicalSpace H]
  {I : ModelWithCorners ℝ E H} {M : Type u_3} [inst_3 : TopologicalSpace M] [inst_4 : ChartedSpace H M] {γ : ℝ → M}
  {v : (x : M) → TangentSpace I x} {s : Set ℝ} {t₀ : ℝ} [inst_5 : IsManifold I 1 M],
  IsMIntegralCurveOn γ v s →
    ∀ {t : ℝ},
      t ∈ s →
        γ t ∈ (extChartAt I (γ t₀)).source →
          HasDerivWithinAt (↑(extChartAt I (γ t₀)) ∘ γ) ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t))) s t

If γ is an integral curve of a vector field v, then γ t is tangent to v (γ t) when expressed in the local chart around the initial point γ t₀.

Defined in

Mathlib.Geometry.Manifold.IntegralCurve.Basic

Cited by

0 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifold