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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 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.
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
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 .
Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it is well suited for the analysis or retrieval of non-rigid shapes, i.e. bendable objects such as humans, animals, plants, etc.
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 ...
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.
6.11 Hodge Laplacian on p-forms. ... These coordinates are also called normal coordinates. In such a frame, the expression for several operators is simpler.
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