OAI/Analysis/AffineBernstein/GraphParametricArea.lean

GitHub source file

12 declarations, 0 contain sorry, 0 declared as axioms

1. def OAI.AffineBernstein.horizontalVelocity line 14
2. def OAI.AffineBernstein.graphParamVariation line 17
3. def OAI.AffineBernstein.graphParamConormal line 21
4. lemma OAI.AffineBernstein.graphParamConormal_apply line 29
5. lemma OAI.AffineBernstein.graphParamConormal_vertical line 37
6. lemma OAI.AffineBernstein.differentiableAt_horizontalVelocity line 41
7. lemma OAI.AffineBernstein.differentiableAt_graphParamVariation line 46
8. lemma OAI.AffineBernstein.graphParamVariation_fderiv line 53
9. lemma OAI.AffineBernstein.graphParam_frame_determinant line 66
10. lemma OAI.AffineBernstein.hessian_coordinate line 89
11. lemma OAI.AffineBernstein.graphParam_secondForm line 95
12. theorem OAI.AffineBernstein.parametricAreaDensity_graphVariation line 132

    Source comment

    The abstract conormal formula is exactly the literal graph-variation density.

    theorem parametricAreaDensity_graphVariation {n : ℕ} {Ω : Set (Space n)} (hΩ : IsOpen Ω)
        {u β : Space n → ℝ} {a : Fin n → Space n → ℝ}
        (hu : ContDiffOn ℝ ∞ u Ω) (hβ : ContDiffOn ℝ ∞ β Ω)
        (ha : ∀ k, ContDiffOn ℝ ∞ (a k) Ω) {x : Space n} (hx : x ∈ Ω) (t : ℝ) :
        parametricAreaDensity (graphAmbientBasis n) (graphParamVariation u β a t)
          (graphParamConormal u β a x t) ((0 : Space n), 1) x = graphVariationArea u β a x t := by
    …

    Mentions: OAI.AffineBernstein.Space, OAI.AffineBernstein.parametricAreaDensity, OAI.AffineBernstein.graphAmbientBasis, OAI.AffineBernstein.graphParamVariation, OAI.AffineBernstein.graphParamConormal, OAI.AffineBernstein.graphVariationArea