Editing Laplace's equation (section)

===Fluid flow===
{{Main|Laplace equation for irrotational flow}}
Let the quantities {{math|''u''}} and {{math|''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 
<math display="block">u_x + v_y=0,</math>
and the condition that the flow be irrotational is that
<math display="block">\nabla \times \mathbf{V} = v_x - u_y = 0.</math>
If we define the differential of a function {{math|''ψ''}} by
<math display="block">d \psi = u \, dy - v \, dx,</math>
then the continuity condition is the integrability condition for this differential: the resulting function is called the [[stream function]] because it is constant along [[Streamlines, streaklines and pathlines|flow lines]]. The first derivatives of {{math|''ψ''}} are given by
<math display="block">\psi_x = -v, \quad \psi_y=u,</math>
and the irrotationality condition implies that {{math|''ψ''}} satisfies the Laplace equation. The harmonic function {{math|''φ''}} that is conjugate to {{math|''ψ''}} is called the [[velocity potential]]. The Cauchy–Riemann equations imply that
<math display="block">\varphi_x=\psi_y=u, \quad \varphi_y=-\psi_x=v.</math>
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.