Fderiv space eq fderiv curry
Space.fderiv_space_eq_fderiv_curry
Project documentation
The spatial projection is nonexpanding for the product metric. -/ lemma dist_space_le (tx ty : TimeAndSpace d) : dist (space tx) (space ty) ≤ dist tx ty := by rw [Prod.dist_eq] exact le_max_right _ _ end TimeAndSpace namespace Space /-! ### A.2. Space and time derivatives in terms of curried functions
Exact Lean statement
lemma fderiv_space_eq_fderiv_curry {M} [NormedAddCommGroup M] [NormedSpace ℝ M]
(f : Time → Space d → M) (t : Time) (x dx : Space d)
(hf : Differentiable ℝ ↿f) :
fderiv ℝ (fun x' => f t x') x dx = fderiv ℝ ↿f (t, x) (0, dx)Formal artifact
Lean source
lemma fderiv_space_eq_fderiv_curry {M} [NormedAddCommGroup M] [NormedSpace ℝ M] (f : Time → Space d → M) (t : Time) (x dx : Space d) (hf : Differentiable ℝ ↿f) : fderiv ℝ (fun x' => f t x') x dx = fderiv ℝ ↿f (t, x) (0, dx) := by change fderiv ℝ (↿f ∘ fun x' => (t, x')) x dx = _ rw [fderiv_comp, DifferentiableAt.fderiv_prodMk] simp only [ContinuousLinearMap.coe_comp, Function.comp_apply, fderiv_fun_const, Pi.zero_apply, fderiv_fun_id, ContinuousLinearMap.prod_apply, _root_.zero_apply, ContinuousLinearMap.coe_id', id_eq] repeat' fun_prop- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/TimeAndSpace/Basic.lean:119-128
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