Research proof index

A constructive form of the statement that if the curl of a function is zero, then it is equal to the grad of another function. In the context of e.g. electromagnetism the potential given by this lemma corresponds to that defined by the work done to move a unit charge from the origin to the point x along a straight line.

Curl and time derivative commute. -/ lemma time_deriv_curl_commute (fₜ : Time → Space → EuclideanSpace ℝ (Fin 3)) (t : Time) (x : Space) (hf : ContDiff ℝ 2 ↿fₜ) : ∂ₜ (fun t => (∇ ⨯ fₜ t) x) t = (∇ ⨯ fun x => (∂ₜ (fun t => fₜ t x) t)) x:= by ext i rw [← Time.deriv_euclid] · fin_cases i all_goals simp [curl] rw [Time.deriv_eq, fderiv_fun_sub] simp [← Time.d...

Cross product and fderiv commute.

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

Derivatives along space coordinates and time commute.

The gradient in the direction of the space position.