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Laplace–Beltrami operator, generalization to submanifolds in Euclidean space and Riemannian and pseudo-Riemannian manifold. The Laplacian in differential geometry. The discrete Laplace operator is a finite-difference analog of the continuous Laplacian, defined on graphs and grids.
The Hodge Laplacian, also known as the Laplace–de Rham operator, is a differential operator acting on differential forms. (Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemannian- or pseudo-Riemannian metric.
Gradient, divergence, Laplace–Beltrami operator [ edit ] The gradient of a function ϕ {\displaystyle \phi } is obtained by raising the index of the differential ∂ i ϕ d x i {\displaystyle \partial _{i}\phi dx^{i}} , whose components are given by:
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds.
Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in differential geometry ...
In differential geometry, Liouville's equation, named after Joseph Liouville, [1] [2] is the nonlinear partial differential equation satisfied by the conformal factor f of a metric f 2 (dx 2 + dy 2) on a surface of constant Gaussian curvature K: =, where ∆ 0 is the flat Laplace operator
In mathematics, the infinity Laplace (or -Laplace) operator is a 2nd-order partial differential operator, commonly abbreviated .It is alternately defined, for a function : of the variables = (, …,), by
In mathematics, the p-Laplacian, or the p-Laplace operator, is a quasilinear elliptic partial differential operator of 2nd order. It is a nonlinear generalization of the Laplace operator , where p {\displaystyle p} is allowed to range over 1 < p < ∞ {\displaystyle 1<p<\infty } .