page_title: Cauchy–Riemann equations
section_title: Physical interpretation
hierarchical_section_title: Cauchy–Riemann equations / Interpretation and reformulation / Physical interpretation
caption_attribution_description: Contours of the real and imaginary parts of the holomorphic function z sin(z^2)
context_page_description: In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic. This system of equations first appeared in the work of Jean le Rond d'Alembert. Later, Leonhard Euler connected this system to the analytic functions. Cauchy then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.
  The Cauchy–Riemann equations on a pair of real-valued functions of two real variables u and v are the two equations:
  Typically u and v are taken to be the real and imaginary parts respectively of a complex-valued function of a single complex variable z = x + iy, f = u + iv. Suppose that u and v are real-differentiable at a point in an open subset of ℂ, which can be considered as functions from ℝ² to ℝ. This implies that the partial derivatives of u and v exist and we can approximate small variations of f linearly.
context_section_description: A standard physical interpretation of the Cauchy–Riemann equations going back to Riemann's work on function theory (see Klein 1893) is that u represents a velocity potential of an incompressible steady fluid flow in the plane, and v is its stream function.  Suppose that the pair of (twice continuously differentiable) functions  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
  By differentiating the Cauchy–Riemann equations a second time, one shows that u solves Laplace's equation:
  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 .  This implies that the gradient of u must point along the  curves; so these are the streamlines of the flow.  The  curves are the equipotential curves of the flow.
  A holomorphic function can therefore be visualized by plotting the two families of level curves  and .  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 , the stationary points of the flow, the equipotential curves of  intersect.  The streamlines also intersect at the same point, bisecting the angles formed by the equipotential curves.
caption: Contour plot of a pair u and v satisfying the Cauchy–Riemann equations.  Streamlines (v = const, red) are perpendicular to equipotentials (u = const, blue).  The point (0,0) is a stationary point of the potential flow, with six streamlines meeting, and six equipotentials also meeting and bisecting the angles formed by the streamlines.
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