Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

The paper proves an n-dimensional Helmholtz decomposition by defining densities γ=div f and ρij=∂jf_i−∂if_j, building Newton potentials G=N∫γ and Rij=N∫ρij, and then setting g=grad G and r=ROT R so that f=g+r, with ROT grad G=0 and div ROT R=0. This is stated and shown via operator identities D, D and the key identity DD f=grad div f+ROT ROT f=Δf, ultimately giving DF=f (cf. Theorem 1 and Propositions 2–3, 6–9) . The candidate solution uses the classical fundamental-solution approach: ΔK=δ0, a careful cutoff/dominated-convergence justification to move derivatives and integrate by parts at infinity, and the componentwise identity Δf_k=∂k(div f)−∑m∂m(∂kf_m−∂mf_k), leading to f=grad G+ROT R. Up to notation and a minor sign slip in one sentence (the componentwise derivation is correct), the arguments coincide in substance. The paper’s proof invokes a lemma to commute D with the Newton potential that nominally requires decay faster than 1/|x|^2 (applied to f in Corollary 5/Eq. (48)), whereas the theorem assumes decay faster than 1/|x|; this is a mild assumptions mismatch but can be fixed by either strengthening the hypothesis or using the same cutoff argument the candidate employs. Hence both are correct; the proofs are substantially the same (Newton potential + integration by parts), with minor presentational/assumption gaps to be tightened in the paper.