Editing Cauchy–Riemann equations (section)

=== Physical interpretation ===
[[File:Contours of holomorphic function.png|right|thumb|[[Contour plot]] of a pair ''u'' and ''v'' satisfying the Cauchy–Riemann equations. Streamlines (''v''&nbsp;=&nbsp;const, red) are perpendicular to equipotentials (''u''&nbsp;=&nbsp;const, blue). The point (0,0) is a [[Critical point (mathematics)|stationary point]] of the potential flow, with six streamlines meeting, and six equipotentials also meeting and bisecting the angles formed by the streamlines.]]
A standard physical interpretation of the Cauchy–Riemann equations going back to Riemann's work on function theory{{r|Klein1893_see}} is that ''u'' represents a [[velocity potential]] of an incompressible [[potential flow|steady fluid flow]] in the plane, and ''v'' is its [[stream function]]. Suppose that the pair of (twice [[Differentiable function|continuously differentiable]]) functions ''u'' and ''v'' satisfies the Cauchy–Riemann equations. We will take ''u'' to be a velocity potential, meaning that we imagine a flow of fluid in the plane such that the [[velocity vector]] of the fluid at each point of the plane is equal to the [[gradient]] of ''u'', defined by
<math display="block">\nabla u = \frac{\partial u}{\partial x}\mathbf i + \frac{\partial u}{\partial y}\mathbf j.</math>

By differentiating the Cauchy–Riemann equations for the functions ''u'' and ''v'', with the [[symmetry of second derivatives]], one shows that ''u'' solves [[Laplace's equation]]:
<math display="block">\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0.</math>
That is, ''u'' is a [[harmonic function]]. This means that the [[divergence]] of the gradient is zero, and so the fluid is incompressible.

The function ''v'' also satisfies the Laplace equation, by a similar analysis. Also, the Cauchy–Riemann equations imply that the [[dot product]] <math display="inline">\nabla u\cdot\nabla v = 0</math> (<math display="inline">\nabla u\cdot\nabla v = \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \cdot \frac{\partial v}{\partial y} = \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x} - \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x}  = 0</math>), i.e., the direction of the maximum slope of ''u'' and that of ''v'' are orthogonal to each other. This implies that the gradient of ''u'' must point along the <math display="inline">v = \text{const}</math> curves; so these are the [[Streamlines, streaklines, and pathlines|streamlines]] of the flow. The <math display="inline">u = \text{const}</math> curves are the [[equipotential curve]]s of the flow.

A holomorphic function can therefore be visualized by plotting the two families of [[level curve]]s <math display="inline">u=\text{const}</math> and <math display="inline">v=\text{const}</math>. Near points where the gradient of ''u'' (or, equivalently, ''v'') is not zero, these families form an [[orthogonal]] family of curves. At the points where <math display="inline">\nabla u=0</math>, the stationary points of the flow, the equipotential curves of <math display="inline">u=\text{const}</math> intersect. The streamlines also intersect at the same point, bisecting the angles formed by the equipotential curves.