Let the quantities u and v be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions. The continuity condition for an incompressible flow is that u x + v y = 0 , and the condition that the flow be irrotational is that ∇ ∇ × × V = v x − − u y = 0. If we define the differential of a function ψ by d ψ ψ = v d x − − u d y , then the continuity condition is the integrability condition for this differential: the resulting function is called the stream function because it is constant along flow lines. The first derivatives of ψ are given by ψ ψ x = v , ψ ψ y = − − u , and the irrotationality condition implies that ψ satisfies the Laplace equation. The harmonic function φ that is conjugate to ψ is called the velocity potential. The Cauchy–Riemann equations imply that φ φ x = − − u , φ φ y = − − v . Thus every analytic function corresponds to a steady incompressible, irrotational, inviscid fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function.