The Hodge decomposition theorem states any smooth vector field on a Riemannian manifold splits uniquely as v = ∇φ + ∇×A + h, where h is harmonic (Δh=0). The curl-free part derives from a potential (irrotational), the div-free part is incompressible, and the harmonic part satisfies both div=0 and curl=0 — it encodes topology.