Curl magnetic Field Matrix eq electric Field field Strength Matrix
Electromagnetism.ElectromagneticPotential.curl_magneticFieldMatrix_eq_electricField_fieldStrengthMatrix
Project documentation
The matrix corresponding to the magnetic field in general dimensions. In 3 space-dimensions this reduces to a vector. -/ noncomputable def magneticFieldMatrix (c : SpeedOfLight := 1) (A : ElectromagneticPotential d) : Time → Space d → (Fin d × Fin d) → ℝ := timeSlice c <| fun x ij => A.fieldStrengthMatrix x (Sum.inr ij.1, Sum.inr ij.2) lemma magneticFie...
Exact Lean statement
lemma curl_magneticFieldMatrix_eq_electricField_fieldStrengthMatrix {d : ℕ} {c : SpeedOfLight}
(A : ElectromagneticPotential d)
(hA : ContDiff ℝ 2 A) (t : Time) (x : Space d) (i : Fin d) :
∑ j, Space.deriv j (A.magneticFieldMatrix c t · (j, i)) x =
(1/c^2) * ∂ₜ (fun t => A.electricField c t x) t i +
(∑ (μ : (Fin 1 ⊕ Fin d)), (∂_ μ (A.fieldStrengthMatrix · (μ, Sum.inr i))
((toTimeAndSpace c).symm (t, x))))Formal artifact
Lean source
lemma curl_magneticFieldMatrix_eq_electricField_fieldStrengthMatrix {d : ℕ} {c : SpeedOfLight} (A : ElectromagneticPotential d) (hA : ContDiff ℝ 2 A) (t : Time) (x : Space d) (i : Fin d) : ∑ j, Space.deriv j (A.magneticFieldMatrix c t · (j, i)) x = (1/c^2) * ∂ₜ (fun t => A.electricField c t x) t i + (∑ (μ : (Fin 1 ⊕ Fin d)), (∂_ μ (A.fieldStrengthMatrix · (μ, Sum.inr i)) ((toTimeAndSpace c).symm (t, x)))) := by trans (1/c^2) * ∂ₜ (fun t => A.electricField c t x) t i + (- (1/c^2) * ∂ₜ (fun t => A.electricField c t x) t i + ∑ j, Space.deriv j (A.magneticFieldMatrix c t · (j, i)) x) · ring congr 1 rw [Fintype.sum_sum_type] congr · simp rw [time_deriv_electricField_eq_fieldStrengthMatrix hA t x i] field_simp · funext j rw [SpaceTime.deriv_sum_inr c] simp rfl · apply fieldStrengthMatrix_differentiable hA- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Electromagnetism/Kinematics/MagneticField.lean:459-480
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