According to Maxwell's equations, an electric field (u, v) in two space dimensions that is independent of time satisfies ∇ ∇ × × ( u , v , 0 ) = ( v x − − u y ) k ^ ^ = 0 , and ∇ ∇ ⋅ ⋅ ( u , v ) = ρ ρ , where ρ is the charge density. The first Maxwell equation is the integrability condition for the differential d φ φ = − − u d x − − v d y , so the electric potential φ may be constructed to satisfy φ φ x = − − u , φ φ y = − − v . The second of Maxwell's equations then implies that φ φ x x + φ φ y y = − − ρ ρ , which is the Poisson equation. The Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions.