Laplace operator

E139493

differential operator elliptic differential operator second-order differential operator

The Laplace operator is a second-order differential operator widely used in mathematics and physics to describe phenomena such as diffusion, heat flow, and wave propagation.

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All labels observed (4)

Label	Occurrences
Laplacian	3
Laplace–Beltrami operator	2
Laplace operator canonical	1
d’Alembert operator	1

Statements (49)

Predicate	Object
instanceOf	differential operator ⓘ elliptic differential operator ⓘ second-order differential operator ⓘ
actsOn	scalar fields ⓘ vector fields ⓘ
alsoKnownAs	Laplace operator ⓘ surface form: Laplacian
appearsIn	Schrödinger equation ⓘ heat equation ∂u/∂t = κΔu ⓘ wave equation ⓘ
definitionInCartesianCoordinates	sum of second partial derivatives with respect to spatial coordinates ⓘ
definitionInRn	Δf = ∑_{i=1}^n ∂²f/∂x_i² ⓘ
discreteAnalogue	discrete Laplacian ⓘ
domain	functions on Euclidean space ⓘ functions on Riemannian manifolds ⓘ
equation	Laplace equation Δu = 0 ⓘ Poisson equation Δu = f ⓘ
field	mathematics ⓘ physics ⓘ
generalization	Hodge Laplacian ⓘ Laplace operator self-linksurface differs ⓘ surface form: Laplace–Beltrami operator
graphAnalogue	graph Laplacian ⓘ
historicalPeriod	18th century ⓘ
invariance	invariant under Euclidean isometries ⓘ rotation invariant in Euclidean space ⓘ
linearity	linear ⓘ
namedAfter	Pierre-Simon Laplace ⓘ
order	2 ⓘ
property	negative semi-definite on appropriate function spaces ⓘ self-adjoint under suitable boundary conditions ⓘ
relatedConcept	Dirichlet boundary conditions ⓘ surface form: Dirichlet boundary condition Green's function ⓘ Neumann boundary conditions in potential theory ⓘ surface form: Neumann boundary condition harmonic function ⓘ
relatedProcess	Brownian motion ⓘ
spectralTheory	eigenvalues form the Laplacian spectrum ⓘ
symbol	Δ ⓘ ∇² ⓘ
type	local operator ⓘ
usedFor	electrostatics ⓘ fluid dynamics ⓘ gravitation ⓘ modeling diffusion ⓘ modeling heat flow ⓘ modeling wave propagation ⓘ potential theory ⓘ quantum mechanics ⓘ
usedIn	Fourier analysis ⓘ partial differential equations ⓘ stochastic processes ⓘ

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Pierre-Simon Laplace → developedConcept → Laplace operator ⓘ

Jean d’Alembert → knownFor → Laplace operator ⓘ

this entity surface form: d’Alembert operator

Laplace equation → involvesOperator → Laplace operator ⓘ

this entity surface form: Laplacian

Laplace operator → alsoKnownAs → Laplace operator ⓘ

this entity surface form: Laplacian

Laplace operator → generalization → Laplace operator self-linksurface differs ⓘ

this entity surface form: Laplace–Beltrami operator

Selberg trace formula → coreConcept → Laplace operator ⓘ

this entity surface form: Laplace–Beltrami operator

Laplacian spectrum → hasOperator → Laplace operator ⓘ

this entity surface form: Laplacian