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Laplace's equation on an annulus

A harmonic function over an annulus

The Laplace operator or Laplacian is a differential operator equal to

\nabla\cdot\nabla f=\nabla^2f=\Delta f

or in other words, the divergence of the gradient of a function. In terms of the del operator, the Laplacian is written as

\nabla\cdot\nabla=\begin{bmatrix}
\dfrac{\part}{\part x_1}\\
\vdots\\
\dfrac{\part}{\part x_n}\end{bmatrix}\cdot\begin{bmatrix}\dfrac{\part}{\part x_1}\\
\vdots\\
\dfrac{\part}{\part x_n}\end{bmatrix}=\dfrac{\part^2}{\part x_1^{\ 2}}+\cdots+\dfrac{\part^2}{\part x_n^{\ 2}}

Intuitively, it represents how fast the average value of f changes for a growing sphere, or how the value of f at a given point compares to the average of the points around it.

Since the Laplacian is a scalar, it can be multiplied by vectors as well to produce the vector Laplacian, a vector triple product equal to the Laplacian of each component of the vector field.

Functions where the Laplacian is equal to zero are called harmonic functions. This particular case is so important it is called Laplace's equation, and appears frequently in partial differential equations. Given an arbitrary boundary in n dimensions, there will be exactly one harmonic function along said boundary. Intuitively, this function can be thought of as the minimum surface area for a function crossing said boundary; for example, a balloon stretched over a cup or similar surface will take on the shape of a harmonic function.

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