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In mathematics, a real [[Differential form|differential one-form]] ''ω'' on a surface is called a '''harmonic differential''' if ''ω'' and its conjugate one-form, written as ''ω''<sup>∗</sup>, are both [[Closed and exact differential forms|closed]].

== Explanation ==

Consider the case of real one-forms defined on a two dimensional real manifold. Moreover, consider real one-forms that are the real parts of [[Complex number|complex]] differentials. Let {{nowrap|1=''ω'' = ''A''&thinsp;d''x'' + ''B''&thinsp;d''y''}}, and formally define the '''conjugate''' one-form to be {{nowrap|1=''ω''<sup>∗</sup> = ''A''&thinsp;d''y'' &minus; ''B''&thinsp;d''x''}}.

== Motivation ==

There is a clear connection with [[Complex analysis|complex analysis]]. Let us write a [[Complex number|complex number]] ''z'' in terms of its real and imaginary parts, say ''x'' and ''y'' respectively, i.e. {{nowrap|1=''z'' = ''x'' + ''iy''}}. Since {{nowrap|1=''ω'' + ''iω''<sup>∗</sup> = (''A'' &minus; ''iB'')(d''x'' + ''i''&thinsp;d''y'')}}, from the point of view of [[Complex analysis|complex analysis]], the [[Quotient|quotient]] {{nowrap|(''ω'' + ''iω''<sup>∗</sup>)/d''z''}} tends to a [[Limit (mathematics)|limit]] as d''z'' tends to 0. In other words, the definition of ''ω''<sup>∗</sup> was chosen for its connection with the concept of a derivative ([[Analytic function|analyticity]]). Another connection with the [[Imaginary unit|complex unit]] is that {{nowrap|1=(''ω''<sup>∗</sup>)<sup>∗</sup> = &minus;''ω''}} (just as {{nowrap|1=''i''<sup>2</sup> = &minus;1}}).

For a given [[Function (mathematics)|function]] ''f'', let us write {{nowrap|1=''ω'' = d''f''}}, i.e. {{nowrap|1=''ω'' = {{sfrac|∂''f''|∂''x''}}&thinsp;d''x'' + {{sfrac|∂''f''|∂''y''}}&thinsp;d''y''}}, where ∂ denotes the [[Partial derivative|partial derivative]]. Then {{nowrap|1=(d''f'')<sup>∗</sup> = {{sfrac|∂''f''|∂''x''}}&thinsp;d''y'' &minus; {{sfrac|∂''f''|∂''y''}}&thinsp;d''x''}}. Now d((d''f'')<sup>∗</sup>) is not always zero, indeed {{nowrap|1=d((d''f'')<sup>∗</sup>) = Δ''f''&thinsp;d''x''&thinsp;d''y''}}, where {{nowrap|1=Δ''f'' = {{sfrac|∂<sup>2</sup>''f''|∂''x''<sup>2</sup>}} + {{sfrac|∂<sup>2</sup>''f''|∂''y''<sup>2</sup>}}}}.

== Cauchy–Riemann equations ==

As we have seen above: we call the one-form ''ω'' '''harmonic''' if both ''ω'' and ''ω''<sup>∗</sup> are closed. This means that {{nowrap|1={{sfrac|∂''A''|∂''y''}} = {{sfrac|∂''B''|∂''x''}}}} (''ω'' is closed) and {{nowrap|1={{sfrac|∂''B''|∂''y''}} = &minus;{{sfrac|∂''A''|∂''x''}}}} (''ω''<sup>∗</sup> is closed). These are called the Cauchy&ndash;Riemann equations on {{nowrap|''A'' &minus; ''iB''}}. Usually they are expressed in terms of {{nowrap|''u''(''x'', ''y'') + ''iv''(''x'', ''y'')}} as {{nowrap|1={{sfrac|∂''u''|∂''x''}} = {{sfrac|∂''v''|∂''y''}}}} and {{nowrap|1={{sfrac|∂''v''|∂''x''}} = &minus;{{sfrac|∂''u''|∂''y''}}}}.

== Notable results ==

*A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential.<ref name="CMRS">{{Citation|first=Harvey|last=Cohn|title=Conformal Mapping on Riemann Surfaces|publisher=McGraw-Hill Book Company|year=1967}}</ref>{{rp|172}} To prove this one shows that {{nowrap|''u'' + ''iv''}} satisfies the Cauchy&ndash;Riemann equations exactly when {{nowrap|''u'' + ''iv''}} is ''locally'' an analytic function of {{nowrap|''x'' + ''iy''}}. Of course an analytic function {{nowrap|1=''w''(''z'') = ''u'' + ''iv''}} is the local derivative of something (namely ∫''w''(''z'')&thinsp;d''z'').
*The harmonic differentials ''ω'' are (locally) precisely the differentials d''f'' of solutions ''f'' to [[Laplace's equation]] {{nowrap|1=Δ''f'' = 0}}.<ref name="CMRS"/>{{rp|172}}
*If ''ω'' is a harmonic differential, so is ''ω''<sup>∗</sup>.<ref name="CMRS"/>{{rp|172}}

==See also==
* [[De Rham cohomology]]

== References ==

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[[Category:Mathematical analysis]]

{{Sourceattribution|Harmonic differential}}