Eq grad integral of curl zero
Space.eq_grad_integral_of_curl_zero
Project documentation
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.
Exact Lean statement
lemma eq_grad_integral_of_curl_zero (f : Space → EuclideanSpace ℝ (Fin 3)) (hf : ContDiff ℝ 1 f)
(hcurl : curl f = 0) :
f = grad (fun x => ∫ t in (0 : ℝ)..1, ⟪f (t • x), basis.repr x⟫_ℝ ∂(volume))Formal artifact
Lean source
lemma eq_grad_integral_of_curl_zero (f : Space → EuclideanSpace ℝ (Fin 3)) (hf : ContDiff ℝ 1 f) (hcurl : curl f = 0) : f = grad (fun x => ∫ t in (0 : ℝ)..1, ⟪f (t • x), basis.repr x⟫_ℝ ∂(volume)) := by obtain ⟨g, ⟨rfl, hg⟩⟩ := exists_grad_of_curl_zero f (hf.differentiable (by simp)) (by simp [hcurl]) suffices h1 : ContDiff ℝ 1 g by nth_rewrite 1 [eq_integral_grad h1] simp rw [contDiff_one_iff_hasFDerivAt] refine ⟨fun x => ((toDual ℝ Space) (basis.repr.symm (∇ g x))), by fun_prop, fun x => ?_⟩ exact hasGradientAt_iff_hasFDerivAt.mpr (DifferentiableAt.hasGradientAt_grad x (hg x))- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/Space/Derivatives/Curl.lean:640-650
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Adiabatic relation log
adiabatic_relation_log
Plain-language statement
Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.
Source project: Physlib
Person-level attribution pending.
Adiabatic relation Ua Ub Va Vb
adiabatic_relation_UaUbVaVb
Plain-language statement
Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.
Source project: Physlib
Person-level attribution pending.
Deriv Within mean Energy Beta eq neg variance
CanonicalEnsemble.derivWithin_meanEnergy_Beta_eq_neg_variance
Plain-language statement
(∂U/∂β) = -Var(E) for finite systems.
Source project: Physlib
Person-level attribution pending.