Harmonic differential

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 ω = Adx + Bdy, and formally define the conjugate one-form to be ω = Ady Bdx.

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. z = x + iy. Since ω + = (A iB)(dx + idy), from the point of view of complex analysis, the quotient (ω + )/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/xdx + f/ydy, where ∂ denotes the partial derivative. Then (df) = f/xdy f/ydx. Now d((df)) is not always zero, indeed d((df)) = Δfdxdy, where Δf = 2f/x2 + 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 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

  • 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 CauchyRiemann 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

See also

References

  1. Cohn, Harvey (1967), Conformal Mapping on Riemann Surfaces, McGraw-Hill Book Company
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