In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a number of concepts employ the word harmonic. The similarity of this terminology to that of music

Harmonic

A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

 is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues corresponding to their modes of vibration. Thus, the term "harmonic" is applied when one is considering functions with sinusoidal variations, or solutions of Laplace's equation

Laplace's equation

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:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

 and related concepts.

See

• harmonic analysis
  Harmonic analysis
  Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
  
  
• harmonic division
• harmonic form
• harmonic function
  Harmonic function
  In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
  
  
• harmonic mean
  Harmonic mean
  In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
  
  
• harmonic mode
• harmonic number
• harmonic series
  Harmonic series (mathematics)
  In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...