openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /HeatKernelsNormalTwoNeg.lean
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.HeatKernelsContDiffKernel | |
| 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 | |
| @[simp] theorem normal_two_neg (Y : Coord 2) : normal 2 (-Y) = normal 2 Y := by | |
| simp [normal_two] | |
| theorem momentKernel_neg (i : Fin 2) (Y : Coord 2) : momentKernel i (-Y) = -momentKernel i Y := by | |
| simp [momentKernel] | |
| theorem integral_momentKernel (i : Fin 2) : ∫ Y, momentKernel i Y = 0 := by | |
| have hh := integral_neg_eq_self (momentKernel i) volume | |
| simp only [momentKernel_neg, integral_neg] at hh | |
| linarith | |
| theorem integral_partialD_kernel {ν t : ℝ} (hν : 0 < ν) (ht : 0 < t) (i : Fin 2) : | |
| (∫ Y, SpatialCalculus.partialD i (kernel ν t) Y) = 0 := by | |
| rw [partialD_kernel_rescale hν ht, integral_const_mul, | |
| integral_rescale _ (Real.sqrt_pos.mpr (by positivity)), integral_momentKernel, mul_zero] | |
| end VelocityDetection.HeatKernels | |
| end | |
| end OAI | |