openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /HeatKernelsNormalEqKernel.lean
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.HeatKernelsNormalTwoNeg | |
| namespace OAI | |
| noncomputable section | |
| namespace VelocityDetection.HeatKernels | |
| open scoped BigOperators Topology ContDiff | |
| open Set Function Filter | |
| open Set Function Filter MeasureTheory | |
| open scoped Topology BigOperators ContDiff | |
| open scoped Topology ContDiff BigOperators | |
| open scoped Topology ContDiff ZeroAtInfty | |
| open scoped Topology ContDiff ZeroAtInfty BigOperators | |
| open scoped Topology | |
| open SpatialCalculus | |
| theorem normal_eq_kernel : normal 2 = kernel (1 / 2) 1 := by | |
| ext X | |
| rw [normal_two] | |
| simp only [kernel] | |
| congr 1 <;> congr 1 <;> ring | |
| theorem contDiff_normal : ContDiff ℝ ∞ (normal 2) := by | |
| rw [normal_eq_kernel] | |
| exact contDiff_kernel _ _ | |
| theorem partialD_normal (i : Fin 2) (X : Coord 2) : | |
| partialD i (normal 2) X = -momentKernel i X := by | |
| rw [normal_eq_kernel, partialD_kernel] | |
| simp only [momentKernel, normal_eq_kernel] | |
| ring_nf | |
| end VelocityDetection.HeatKernels | |
| end | |
| end OAI | |