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, and formally define the conjugate one-form to be.

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.. Since, from the point of view of complex analysis, the quotient 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. Another connection with the complex unit is that .
For a given function f, let us write, i.e., where ∂ denotes the partial derivative. Then. Now d is not always zero, indeed, where.

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 and . These are called the Cauchy-Riemann equations on. Usually they are expressed in terms of as and.

Notable results

A harmonic differential is precisely the real part of an complex differential. 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.
The harmonic differentials ω are precisely the differentials df of solutions f to Laplace's equation.
If ω is a harmonic differential, so is ω^∗.