“Harmonic differential”的意思、由来-开放百科全书

词条

Harmonic differential

释义

Explanation
Motivation
Cauchy–Riemann equations
Notable results
See also
References

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

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 {{nowrap|1=ω = A dx + B dy}}, and formally define the conjugate one-form to be {{nowrap|1=ω^∗ = A dy − B dx}}.

Motivation

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. {{nowrap|1=z = x + iy}}. Since {{nowrap|1=ω + iω^∗ = (A − iB)(dx + i dy)}}, from the point of view of complex analysis, the quotient {{nowrap|(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 {{nowrap|1=(ω^∗)^∗ = −ω}} (just as {{nowrap|1=i² = −1}}).

For a given function f, let us write {{nowrap|1=ω = df}}, i.e. {{nowrap|1=ω = {{sfrac|∂f|∂x}} dx + {{sfrac|∂f|∂y}} dy}}, where ∂ denotes the partial derivative. Then {{nowrap|1=(df)^∗ = {{sfrac|∂f|∂x}} dy − {{sfrac|∂f|∂y}} dx}}. Now d((df)^∗) is not always zero, indeed {{nowrap|1=d((df)^∗) = Δf dx dy}}, where {{nowrap|1=Δf = {{sfrac|∂²f|∂x²}} + {{sfrac|∂²f|∂y²}}}}.

Cauchy–Riemann equations

As we have seen above: we call the one-form ω harmonic if both ω and ω^∗ 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^∗ 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}}}}.

Notable results

A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential.^[1]{{rp|172}} To prove this one shows that {{nowrap|u + iv}} satisfies the Cauchy–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) dz).
The harmonic differentials ω are (locally) precisely the differentials df of solutions f to Laplace's equation {{nowrap|1=Δf = 0}}.^[1]{{rp|172}}
If ω is a harmonic differential, so is ω^∗.^[1]{{rp|172}}

References

1. ^¹²{{Citation|first=Harvey|last=Cohn|title=Conformal Mapping on Riemann Surfaces|publisher=McGraw-Hill Book Company|year=1967}}

1 : Mathematical analysis

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

Explanation

Motivation

Cauchy–Riemann equations

Notable results

See also

References