PEM in the Coulomb Gauge The Helmholtz theorem guarantees that any vector can be decomposed in the form v = v_⊥ + v_ΙΙ where ∇ · v_⊥ = 0 and ∇ × v_ΙΙ = 0. Use this fact, the identity

and the Coulomb gauge constraint to prove that the electromagnetic momentum of a charge distribution ρ(r, t) in the presence of a magnetic field B(r, t) can be written in the form

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fd³r (u. V)w = [ (d$· u)w - [ d³r (V - u)w,