Electric Field cont Diff
Electromagnetism.ElectromagneticPotential.electricField_contDiff
Project documentation
The electric field of the electromagnetic potential created from the electric field E and the magnetic field B is E, as long as Gauss's law for magnetism and Faraday's law are satisfied. -/ lemma ofElectromagneticField_electricField {c : SpeedOfLight} (E : Time → Space 3 → EuclideanSpace ℝ (Fin 3)) (B : Time → Space 3 → EuclideanSpace ℝ (Fin 3)) (E_...
Exact Lean statement
lemma electricField_contDiff {n} {c : SpeedOfLight} {A : ElectromagneticPotential d}
(hA : ContDiff ℝ (n + 1) A) : ContDiff ℝ n ↿(A.electricField c)Formal artifact
Lean source
lemma electricField_contDiff {n} {c : SpeedOfLight} {A : ElectromagneticPotential d} (hA : ContDiff ℝ (n + 1) A) : ContDiff ℝ n ↿(A.electricField c) := by rw [@contDiff_euclidean] intro i conv => enter [3, x]; change A.electricField c x.1 x.2 i rw [electricField_eq_fieldStrengthMatrix (A) x.1 x.2 i (hA.differentiable (by simp))] change - c * A.fieldStrengthMatrix ((toTimeAndSpace c).symm (x.1, x.2)) (Sum.inl 0, Sum.inr i) apply ContDiff.mul · fun_prop exact (fieldStrengthMatrix_contDiff hA).comp (ContinuousLinearEquiv.contDiff (toTimeAndSpace c).symm)- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Electromagnetism/Kinematics/ElectricField.lean:210-222
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