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The Laplace transform's key property is that it converts differentiation and integration in the time domain into multiplication and division by s in the Laplace domain. Thus, the Laplace variable s is also known as an operator variable in the Laplace domain: either the derivative operator or (for s −1) the integration operator.
Discrete Laplace operator is often used in image processing e.g. in edge detection and motion estimation applications. [4] The discrete Laplacian is defined as the sum of the second derivatives and calculated as sum of differences over the nearest neighbours of the central pixel. Since derivative filters are often sensitive to noise in an image ...
The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields, by a similar formula. Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exterior derivative , in terms of which the "geometer's Laplacian" is ...
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 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.
Second derivative; Implicit differentiation ... where ∆ ≡ ∇ 2 is the Laplace operator, ∂U is the boundary of ... The second term in the integral above can be ...
The tensor derivative of a vector field (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as , where represents the dyadic product. This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space.
For any twice-differentiable real-valued function f defined on Euclidean space R n, the Laplace operator (also known as the Laplacian) takes f to the divergence of its gradient vector field, which is the sum of the n pure second derivatives of f with respect to each vector of an orthonormal basis for R n.