Constant EB magnetic Field Matrix
Electromagnetism.ElectromagneticPotential.constantEB_magneticFieldMatrix
Project documentation
An electric potential which gives a given constant E-field and B-field. -/ @[nolint unusedArguments] noncomputable def constantEB {d : ℕ} (c : SpeedOfLight) (E₀ : EuclideanSpace ℝ (Fin d)) (B₀ : Fin d × Fin d → ℝ) (B₀_antisymm : ∀ i j, B₀ (i, j) = - B₀ (j, i)) : ElectromagneticPotential d where val := fun x μ => match μ with | Sum.inl _ => - (1/c) * ⟪E₀,...
Exact Lean statement
@[simp]
lemma constantEB_magneticFieldMatrix {c : SpeedOfLight}
{E₀ : EuclideanSpace ℝ (Fin d)} {B₀ : Fin d × Fin d → ℝ}
{B₀_antisymm : ∀ i j, B₀ (i, j) = - B₀ (j, i)} :
(constantEB c E₀ B₀ B₀_antisymm).magneticFieldMatrix c = fun _ _ => B₀Formal artifact
Lean source
@[simp]lemma constantEB_magneticFieldMatrix {c : SpeedOfLight} {E₀ : EuclideanSpace ℝ (Fin d)} {B₀ : Fin d × Fin d → ℝ} {B₀_antisymm : ∀ i j, B₀ (i, j) = - B₀ (j, i)} : (constantEB c E₀ B₀ B₀_antisymm).magneticFieldMatrix c = fun _ _ => B₀ := by funext t x funext i match i with | (i, j) => rw [magneticFieldMatrix_eq_vectorPotential] rw [constantEB_vectorPotential_space_deriv, constantEB_vectorPotential_space_deriv] conv_lhs => enter [2] rw [B₀_antisymm] ring apply constantEB_smooth.differentiable (by simp)- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Electromagnetism/Vacuum/Constant.lean:214-229
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