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- Laplace_operator abstract "In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or ∆. The Laplacian ∆f(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p, deviates from f(p) as the radius of the sphere grows. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems such as cylindrical and spherical coordinates, the Laplacian also has a useful form.The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics, where the operator gives a constant multiple of the mass density when it is applied to a given gravitational potential. Solutions of the equation ∆f = 0, now called Laplace's equation, are the so-called harmonic functions, and represent the possible gravitational fields in free space.The Laplacian occurs in differential equations that describe many physical phenomena, such as electric and gravitational potentials, the diffusion equation for heat and fluid flow, wave propagation, and quantum mechanics. The Laplacian represents the flux density of the gradient flow of a function. For instance, the net rate at which a chemical dissolved in a fluid moves toward or away from some point is proportional to the Laplacian of the chemical concentration at that point; expressed symbolically, the resulting equation is the diffusion equation. For these reasons, it is extensively used in the sciences for modelling all kinds of physical phenomena. The Laplacian is the simplest elliptic operator, and is at the core of Hodge theory as well as the results of de Rham cohomology. In image processing and computer vision, the Laplacian operator has been used for various tasks such as blob and edge detection.".
- Laplace_operator wikiPageID "174706".
- Laplace_operator wikiPageRevisionID "600330256".
- Laplace_operator hasPhotoCollection Laplace_operator.
- Laplace_operator id "p/l057510".
- Laplace_operator title "Laplace operator".
- Laplace_operator title "Laplacian".
- Laplace_operator urlname "Laplacian".
- Laplace_operator subject Category:Differential_operators.
- Laplace_operator subject Category:Elliptic_partial_differential_equations.
- Laplace_operator subject Category:Fourier_analysis.
- Laplace_operator subject Category:Harmonic_functions.
- Laplace_operator subject Category:Linear_operators_in_calculus.
- Laplace_operator subject Category:Multivariable_calculus.
- Laplace_operator type Abstraction100002137.
- Laplace_operator type Communication100033020.
- Laplace_operator type DifferentialEquation106670521.
- Laplace_operator type DifferentialOperators.
- Laplace_operator type EllipticPartialDifferentialEquations.
- Laplace_operator type Equation106669864.
- Laplace_operator type Function113783816.
- Laplace_operator type HarmonicFunctions.
- Laplace_operator type LinearOperator113786595.
- Laplace_operator type LinearOperatorsInCalculus.
- Laplace_operator type MathematicalRelation113783581.
- Laplace_operator type MathematicalStatement106732169.
- Laplace_operator type Message106598915.
- Laplace_operator type Operator113786413.
- Laplace_operator type PartialDifferentialEquation106670866.
- Laplace_operator type Relation100031921.
- Laplace_operator type Statement106722453.
- Laplace_operator comment "In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or ∆. The Laplacian ∆f(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p, deviates from f(p) as the radius of the sphere grows.".
- Laplace_operator label "Laplace operator".
- Laplace_operator label "Laplace-Operator".
- Laplace_operator label "Laplace-operator".
- Laplace_operator label "Laplaciano".
- Laplace_operator label "Operador laplaciano".
- Laplace_operator label "Operator Laplace’a".
- Laplace_operator label "Operatore di Laplace".
- Laplace_operator label "Opérateur laplacien".
- Laplace_operator label "Оператор Лапласа".
- Laplace_operator label "لابلاسي".
- Laplace_operator label "ラプラス作用素".
- Laplace_operator label "拉普拉斯算子".
- Laplace_operator sameAs Laplaceův_operátor.
- Laplace_operator sameAs Laplace-Operator.
- Laplace_operator sameAs Operador_laplaciano.
- Laplace_operator sameAs Opérateur_laplacien.
- Laplace_operator sameAs Operatore_di_Laplace.
- Laplace_operator sameAs ラプラス作用素.
- Laplace_operator sameAs 라플라스_연산자.
- Laplace_operator sameAs Laplace-operator.
- Laplace_operator sameAs Operator_Laplace’a.
- Laplace_operator sameAs Laplaciano.
- Laplace_operator sameAs m.017lmx.
- Laplace_operator sameAs Q203484.
- Laplace_operator sameAs Q203484.
- Laplace_operator sameAs Laplace_operator.
- Laplace_operator wasDerivedFrom Laplace_operator?oldid=600330256.
- Laplace_operator isPrimaryTopicOf Laplace_operator.