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{{distinguish|Harmonic form}} |
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In mathematics, a real [[differential form|differential one-form]] ω is called a '''harmonic differential''' if ω and its conjugate one-form, written as ω |
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]]. |
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== Explanation == |
== Explanation == |
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Consider the case of real one-forms defined on a two dimensional [[real manifold]]. Moreover, consider real one-forms |
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'' d''x'' + ''B'' d''y''}}, and formally define the '''conjugate''' one-form to be {{nowrap|1=''ω''<sup>∗</sup> = ''A'' d''y'' − ''B'' d''x''}}. |
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== Motivation == |
== Motivation == |
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There is a clear connection with [[complex analysis]]. Let us write a [[complex number]] ''z'' in terms of its [[real part|real]] and [[imaginary part|imaginary]] parts, say ''x'' and ''y'' respectively, i.e. {{nowrap |
There is a clear connection with [[complex analysis]]. Let us write a [[complex number]] ''z'' in terms of its [[real part|real]] and [[imaginary part|imaginary]] parts, say ''x'' and ''y'' respectively, i.e. {{nowrap|1=''z'' = ''x'' + ''iy''}}. Since {{nowrap|1=''ω'' + ''iω''<sup>∗</sup> = (''A'' − ''iB'')(d''x'' + ''i'' d''y'')}}, from the point of view of [[complex analysis]], the [[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> = −''ω''}} (just as {{nowrap|1=''i''<sup>2</sup> = −1}}). |
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For a given [[function (mathematics)|function]] |
For a given [[function (mathematics)|function]] ''f'', let us write {{nowrap|1=''ω'' = d''f''}}, i.e. {{nowrap|1=''ω'' = {{sfrac|∂''f''|∂''x''}} d''x'' + {{sfrac|∂''f''|∂''y''}} d''y''}}, where ∂ denotes the [[partial derivative]]. Then {{nowrap|1=(d''f'')<sup>∗</sup> = {{sfrac|∂''f''|∂''x''}} d''y'' − {{sfrac|∂''f''|∂''y''}} d''x''}}. Now d((d''f'')<sup>∗</sup>) is not always zero, indeed {{nowrap|1=d((d''f'')<sup>∗</sup>) = Δ''f'' d''x'' 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>}}}}. |
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== Cauchy–Riemann equations == |
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As we have seen above: we call the one-form ω '''harmonic''' if both ω and ω |
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''}} = −{{sfrac|∂''A''|∂''x''}}}} (''ω''<sup>∗</sup> is closed). These are called the [[Cauchy–Riemann equations]] on {{nowrap|''A'' − ''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''}} = −{{sfrac|∂''u''|∂''y''}}}}. |
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== Notable results == |
== Notable results == |
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*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> To prove this one shows that {{nowrap|''u'' + ''iv''}} satisfies the Cauchy–Riemann equations exactly when {{nowrap|''u'' + ''iv''}} is ''[[neighbourhood (topology)|locally]]'' an analytic function of {{nowrap|''x'' + ''iy''}}. Of course an analytic function {{nowrap |
*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–Riemann equations exactly when {{nowrap|''u'' + ''iv''}} is ''[[neighbourhood (topology)|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'') d''z''). |
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==See also== |
==See also== |
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Latest revision as of 20:15, 25 August 2018
In mathematics, a real differential one-form ω on a surface is called a harmonic differential if ω and its conjugate one-form, written as ω∗, are both closed.
Explanation
[edit]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 differentials. Let ω = A dx + B dy, and formally define the conjugate one-form to be ω∗ = A dy − B dx.
Motivation
[edit]There is a clear connection with complex analysis. Let us write a complex number z in terms of its real and imaginary parts, say x and y respectively, i.e. z = x + iy. Since ω + iω∗ = (A − iB)(dx + i dy), from the point of view of complex analysis, the quotient (ω + iω∗)/dz tends to a limit as dz tends to 0. In other words, the definition of ω∗ was chosen for its connection with the concept of a derivative (analyticity). Another connection with the complex unit is that (ω∗)∗ = −ω (just as i2 = −1).
For a given function f, let us write ω = df, i.e. ω = ∂f/∂x dx + ∂f/∂y dy, where ∂ denotes the partial derivative. Then (df)∗ = ∂f/∂x dy − ∂f/∂y dx. Now d((df)∗) is not always zero, indeed d((df)∗) = Δf dx dy, where Δf = ∂2f/∂x2 + ∂2f/∂y2.
Cauchy–Riemann equations
[edit]As we have seen above: we call the one-form ω harmonic if both ω and ω∗ are closed. This means that ∂A/∂y = ∂B/∂x (ω is closed) and ∂B/∂y = −∂A/∂x (ω∗ is closed). These are called the Cauchy–Riemann equations on A − iB. Usually they are expressed in terms of u(x, y) + iv(x, y) as ∂u/∂x = ∂v/∂y and ∂v/∂x = −∂u/∂y.
Notable results
[edit]- A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential.[1]: 172 To prove this one shows that u + iv satisfies the Cauchy–Riemann equations exactly when u + iv is locally an analytic function of x + iy. Of course an analytic function w(z) = u + iv is the local derivative of something (namely ∫w(z) dz).
- The harmonic differentials ω are (locally) precisely the differentials df of solutions f to Laplace's equation Δf = 0.[1]: 172
- If ω is a harmonic differential, so is ω∗.[1]: 172