Poisson's equation is Δ Δ φ φ = f , where Δ Δ is the Laplace operator, and f and φ φ are real or complex -valued functions on a manifold. Usually, f is given, and φ φ is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇, and so Poisson's equation is frequently written as ∇ ∇ 2 φ φ = f .