|
|
Line 17: |
Line 17: |
|
== Notable results == |
|
== 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> 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 begin}}''w''(''z'') = ''u'' + ''iv''{{nowrap end}} is the local derivative of something (namely ∫''w''(''z'') d''z'') |
|
*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 <math>\scriptstyle u \, + \, iv</math> satisfies the Cauchy–Riemann equations exactly when <math>\scriptstyle u \, + \, iv</math> is ''[[neighbourhood (topology)|locally]]'' an analytic function of <math>\scriptstyle x \, + \, iy</math>. Of course an analytic function <math>\scriptstyle w(z) \, = \, u \, + \, iv</math> is the local derivative of something (namely <math>\scriptstyle \int \! w(z)\,\text{d}z</math>) |
|
|
|
|
|
*The harmonic differentials ω are (locally) precisely the differentials dƒ of solutions ƒ to [[Laplace's equation]] {{nowrap begin}}Δƒ = 0{{nowrap end}}.<ref name="CMRS"/> |
|
*The harmonic differentials <math>\scriptstyle \omega</math> are (locally) precisely the differentials <math>\scriptstyle \text{d}\! f</math> of solutions <math>\scriptstyle f</math> to [[Laplace's equation]] <math>\scriptstyle \Delta f \, = \, 0.</math><ref name="CMRS"/> |
|
|
|
|
|
*If ω is a harmonic differential, so is ω*.<ref name="CMRS"/> |
|
*If <math>\scriptstyle \omega</math> is a harmonic differential, so is <math>\scriptstyle \omega^*.</math><ref name="CMRS"/> |
|
|
|
|
|
==See also== |
|
==See also== |
In mathematics, a real differential one-form ω is called a harmonic differential if ω and its conjugate one-form, written as ω*, are both closed.
Explanation
Consider the case of real one-forms defined on a two dimensional real manifold. Moreover, consider real one-forms which are the real parts of complex differentials. Let , and formally define the conjugate one-form to be .
Motivation
There is a clear connection with complex analysis. Let us write a complex number in terms of its real and imaginary parts, say and respectively, i.e. . Since , from the point of view of complex analysis, the quotient tends to a limit as tends to 0. In other words, the definition of was chosen for its connection with the concept of a derivative (analyticity). Another connection with the complex unit is that (just as ).
For a given function , let us write , i.e. where denotes the partial derivative. Then Now is not always zero, indeed where
Cauchy–Riemann equations
As we have seen above: we call the one-form harmonic if both and are closed. This means that ( is closed) and ( is closed). These are called the Cauchy–Riemann equations on Usually they are expressed in terms of as and
Notable results
- A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential.[1] To prove this one shows that satisfies the Cauchy–Riemann equations exactly when is locally an analytic function of . Of course an analytic function is the local derivative of something (namely )
- The harmonic differentials are (locally) precisely the differentials of solutions to Laplace's equation [1]
- If is a harmonic differential, so is [1]
See also
References
- ^ a b c Cohn, Harvey (1967), Conformal Mapping on Riemann Surfaces, McGraw-Hill Book Company