Editing Cauchy–Riemann equations (section)

=== Harmonic vector field ===
Another interpretation of the Cauchy–Riemann equations can be found in [[Problems and Theorems in Analysis|Pólya & Szegő]].{{r|PólyaSzegő1978}} Suppose that ''u'' and ''v'' satisfy the Cauchy–Riemann equations in an open subset of '''R'''<sup>2</sup>, and consider the [[vector field]]
<math display="block">\bar{f} = \begin{bmatrix} u\\ -v \end{bmatrix}</math>
regarded as a (real) two-component vector. Then the second Cauchy–Riemann equation ({{EquationNote|1b}}) asserts that <math>\bar{f}</math> is [[irrotational vector field|irrotational]] (its [[Curl (mathematics)|curl]] is 0):
<math display="block">\frac{\partial (-v)}{\partial x} - \frac{\partial u}{\partial y} = 0.</math>

The first Cauchy–Riemann equation ({{EquationNote|1a}}) asserts that the vector field is [[solenoidal vector field|solenoidal]] (or [[divergence]]-free):
<math display="block">\frac{\partial u}{\partial x} + \frac{\partial (-v)}{\partial y}=0.</math>

Owing respectively to [[Green's theorem]] and the [[divergence theorem]], such a field is necessarily a [[conservative vector field|conservative]] one, and it is free from sources or sinks, having net flux equal to zero through any open domain without holes. (These two observations combine as real and imaginary parts in [[Cauchy's integral theorem]].) In [[fluid dynamics]], such a vector field is a [[potential flow]].{{r|Chanson2007}} In [[magnetostatics]], such vector fields model static [[magnetic field]]s on a region of the plane containing no current. In [[electrostatics]], they model static electric fields in a region of the plane containing no electric charge.

This interpretation can equivalently be restated in the language of [[differential form]]s. The pair ''u'' and ''v'' satisfy the Cauchy–Riemann equations if and only if the [[one-form]] <math>v\,dx + u\, dy</math> is both [[Closed and exact differential forms|closed]] and [[codifferential|coclosed]] (a [[Hodge theory|harmonic differential form]]).