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 = A&thinsp;dx + B&thinsp;dy, and formally define the conjugate one-form to be ω∗ = A&thinsp;dy &minus; B&thinsp;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. z = x + iy. Since ω + iω∗ = (A &minus; iB)(dx + i&thinsp;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∗)∗ = &minus;ω (just as i2 = &minus;1).

For a given function f, let us write ω = df, i.e. ω = $∂f⁄∂x$&thinsp;dx + $∂f⁄∂y$&thinsp;dy, where ∂ denotes the partial derivative. Then (df)∗ = $∂f⁄∂x$&thinsp;dy &minus; $∂f⁄∂y$&thinsp;dx. Now d((df)∗) is not always zero, indeed d((df)∗) = Δf&thinsp;dx&thinsp;dy, where Δf = $∂^{2}f⁄∂x^{2}$ + $∂^{2}f⁄∂y^{2}$.

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 $∂B⁄∂y$ = &minus;$∂A⁄∂x$ (ω∗ is closed). These are called the Cauchy–Riemann equations on A &minus; iB. Usually they are expressed in terms of u(x, y) + iv(x, y) as $∂u⁄∂x$ = $∂v⁄∂y$ and $∂v⁄∂x$ = &minus;$∂u⁄∂y$.

Notable results

 * A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential. 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)&thinsp;dz).
 * The harmonic differentials ω are (locally) precisely the differentials df of solutions f to Laplace's equation Δf = 0.
 * If ω is a harmonic differential, so is ω∗.