| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| namespace WeakMTWTransport |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
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| def directionalDerivative (f : E → F) (v : E) (x : E) : F := fderiv ℝ f x v |
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| lemma hasFDerivAt_directionalDerivative {f : E → F} {x : E} |
| (hf : ContDiffAt ℝ ∞ f x) (v : E) : |
| HasFDerivAt (directionalDerivative f v) |
| ((ContinuousLinearMap.apply ℝ F v) ∘L fderiv ℝ (fderiv ℝ f) x) x := by |
| exact (ContinuousLinearMap.apply ℝ F v).hasFDerivAt.comp x |
| ((hf.fderiv_right (m := (∞ : ℕ∞ω)) (by simp)).differentiableAt (by simp)).hasFDerivAt |
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| lemma directionalDerivative_commute {f : E → F} {x : E} |
| (hf : ContDiffAt ℝ ∞ f x) (u v : E) : |
| directionalDerivative (directionalDerivative f u) v x = |
| directionalDerivative (directionalDerivative f v) u x := by |
| change (fderiv ℝ (directionalDerivative f u) x) v = |
| (fderiv ℝ (directionalDerivative f v) x) u |
| rw [(hasFDerivAt_directionalDerivative hf u).fderiv, |
| (hasFDerivAt_directionalDerivative hf v).fderiv] |
| exact (hf.isSymmSndFDerivAt (by simp)).eq v u |
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| lemma contDiffAt_directionalDerivative {f : E → F} {x : E} |
| (hf : ContDiffAt ℝ ∞ f x) (v : E) : |
| ContDiffAt ℝ ∞ (directionalDerivative f v) x := |
| (ContinuousLinearMap.apply ℝ F v).contDiff.contDiffAt.comp x |
| (hf.fderiv_right (m := (∞ : ℕ∞ω)) (by simp)) |
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| lemma hasDerivAt_slice_left {f : ℝ × ℝ → F} {t s : ℝ} |
| (hf : DifferentiableAt ℝ f (t,s)) : |
| HasDerivAt (fun q => f (q,s)) (directionalDerivative f (1,0) (t,s)) t := by |
| exact hf.hasFDerivAt.comp_hasDerivAt t ((hasDerivAt_id t).prodMk (hasDerivAt_const t s)) |
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| lemma hasDerivAt_slice_right {f : ℝ × ℝ → F} {t s : ℝ} |
| (hf : DifferentiableAt ℝ f (t,s)) : |
| HasDerivAt (fun q => f (t,q)) (directionalDerivative f (0,1) (t,s)) s := by |
| exact hf.hasFDerivAt.comp_hasDerivAt s ((hasDerivAt_const s t).prodMk (hasDerivAt_id s)) |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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