Harmonic differential

Last updated April 06, 2019

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.

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero, and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d.

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

A complex number is a number that can be expressed in the form $a + bi$ , where $a$ and $b$ are real numbers, and $i$ is a solution of the equation $x 2 = -1$ . Because no real number satisfies this equation, $i$ is called an imaginary number. For the complex number $a + bi$ , $a$ is called the real part, and $b$ is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

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−iB)(dx + i 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^∗)^∗ = −ω (just as i² = −1).

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

Quotient Mathematical result of division

In arithmetic, a quotient is the quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as a fraction or a ratio. For example, when dividing twenty by three, the quotient is six and two thirds. In this sense, a quotient is the ratio of a dividend to its divisor.

In mathematics, a limit is the value that a function "approaches" as the input "approaches" some value. Limits are essential to calculus and are used to define continuity, derivatives, and integrals.

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

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable. The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in vector calculus and differential geometry.

Cauchy–Riemann equations

As we have seen above: we call the one-form ωharmonic 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.

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic. This system of equations first appeared in the work of Jean le Rond d'Alembert. Later, Leonhard Euler connected this system to the analytic functions. Cauchy (1814) then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.

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 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) 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

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

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References

1 2 3 Cohn, Harvey (1967), Conformal Mapping on Riemann Surfaces, McGraw-Hill Book Company

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[CMRS-1] 1 2 3 Cohn, Harvey (1967), Conformal Mapping on Riemann Surfaces, McGraw-Hill Book Company