American Journal of Physics · 1986 · 147 citations · 0 references
Natural SciencesMathematicsPhysicsElectromagneticsElectromagnetismMathematical PhysicsElectric FieldsElectromagnetic PotentialsMaxwell's EquationsPartial Differential EquationsDifferential Form FormulationDifferential Forms In ElectromagnetismElectromagnetic Field TensorHelmholtz EquationMaxwell Equations In Covariant FormMaxwell's Equations In Covariant FormMaxwell's Equations In Differential FormSymmetry Arguments In Electromagnetism
Helmholtz’s theorem for three-vectors is applicable to electrostatics and magnetostatics, but it must be generalized to antisymmetric second-rank tensors to be applicable to electromagnetism. A proof of Helmholtz’s theorem for three-vectors, which is shorter than the usual one, is given. Helmholtz’s theorem is shown to be a special case of the Hodge decomposition theorem in the theory of differential forms.