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In spherical coordinates in N dimensions, with the parametrization x = rθ ∈ R N with r representing a positive real radius and θ an element of the unit sphere S N−1, = + + where Δ S N−1 is the Laplace–Beltrami operator on the (N − 1)-sphere, known as the spherical Laplacian.
The spherical Laplacian is the Laplace–Beltrami operator on the (n − 1)-sphere with its canonical metric of constant sectional curvature 1. It is convenient to regard the sphere as isometrically embedded into R n as the unit sphere centred at the origin. Then for a function f on S n−1, the spherical Laplacian is defined by
The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator.
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.
In Cartesian coordinates, the Laplacian of a function (,,) is = = = + +. The Laplacian is a measure of how much a function is changing over a small sphere centered at the point. When the Laplacian is equal to 0, the function is called a harmonic function .
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.
Because is a linear differential operator, the solution () to a general system of this type can be written as an integral over a distribution of source given by (): = (, ′) (′) ′ where the Green's function for Laplacian in three variables (, ′) describes the response of the system at the point to a point source located at ...
The symmetrically normalized Laplacian is a symmetric matrix if and only if the adjacency matrix A is symmetric and the diagonal entries of D are nonnegative, in which case we can use the term the symmetric normalized Laplacian. The symmetric normalized Laplacian matrix can be also written as