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In mathematics and physics, 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 = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient (). Thus if f {\displaystyle f} is a twice-differentiable real-valued function , then the Laplacian of f {\displaystyle f} is the real-valued function defined by:
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid.For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the Laplacian matrix.
On a Riemannian manifold, one can define the conformal Laplacian as an operator on smooth functions; it differs from the Laplace–Beltrami operator by a term involving the scalar curvature of the underlying metric. In dimension n ≥ 3, the conformal Laplacian, denoted L, acts on a smooth function u by
Given a simple graph with vertices , …,, its Laplacian matrix is defined element-wise as [1],:= { = , or equivalently by the matrix =, where D is the degree matrix and A is the adjacency matrix of the graph.
A useful starting point and organizing principle in the study of harmonic functions is a consideration of the symmetries of the Laplace equation. Although it is not a symmetry in the usual sense of the term, we can start with the observation that the Laplace equation is linear. This means that the fundamental object of study in potential theory ...
Considering higher dimensional analogues of the harmonics on the unit n-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" was used to refer to all functions satisfying Laplace's equation. [1]
More generally, one can formulate a similar trick using the normal bundle to define the Laplace–Beltrami operator of any Riemannian manifold isometrically embedded as a hypersurface of Euclidean space. One can also give an intrinsic description of the Laplace–Beltrami operator on the sphere in a normal coordinate system.