Research proof index

If two lagrangians, L and L', differ by a total time derivative, and L has a variational derivative grad, then so does L'. -/ lemma totalTimeDerivative_hasVarGradientAt_equivalence [CompleteSpace X] (L δL : Time → X → X → ℝ) (hδL : IsTotalTimeDerivative δL) (q : Time → X) (hq : ContDiff ℝ ∞ q) (grad : Time → X) (hgrad : HasVarGradientAt (fun q' t => L t (...

A vector-valued plane wave travelling in the direction of s with propagation speed c. -/ noncomputable def planeWave (f₀ : ℝ → EuclideanSpace ℝ (Fin d)) (c : ℝ) (s : Direction d) : Time → Space d → EuclideanSpace ℝ (Fin d) := fun t x => f₀ (⟪x, s.unit⟫ℝ - c * t) lemma planeWave_eq {d f₀ c x} {s : Direction d} : planeWave f₀ c s t x = f₀ (⟪x, s.unit⟫...

If f₀ is a function of (inner ℝ x s - c * t), the fderiv of its components with respect to spatial coordinates is equal to the corresponding component of the propagation direction s times time derivative.

The transverse harmonic planewave representation is equivalent to the general planewave expression with ‖k‖ = ω/c.

Let $T>0$, $1 \le p < \infty$, and let $f$ belong to $L^p((0,T])$. For every $\varepsilon>0$, there is a smooth $T$-periodic function $f_0 : \mathbb{R}\to\mathbb{C}$ such that

$\lVert f-f_0\rVert_{L^p((0,T])} \le \varepsilon.$

Closed ball centered at z with radius z.im/2 is contained in the upper half plane.