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Green's function for the three-variable Laplace equation — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Green%27s_function_for_the_three-variable_Laplace_equation
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-31T09:58:08.734Z
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Green's function for the three-variable Laplace equation — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Green%27s_function_for_the_three-variable_Laplace_equation (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_ebcb54cc41ac (Metraipolis archive dc_ebcb54cc41ac, sha256 443c91bc70f56c9fd7eef077ccaf81f5a70d702dcf37525990e2f2a0ad73434d)

BibTeX

@misc{dcebcb54cc41ac,
  title = {Green's function for the three-variable Laplace equation — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Green%27s_function_for_the_three-variable_Laplace_equation},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_ebcb54cc41ac; sha256 443c91bc70f56c9fd7eef077ccaf81f5a70d702dcf37525990e2f2a0ad73434d},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Green%27s_function_for_the_three-variable_Laplace_equation. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In physics, the Green's function (or fundamental solution) for the Laplacian (or Laplace operator) in three variables is used to describe the response of a particular type of physical system to a point source. In particular, this Green's function arises in systems that can be described by Poisson's equation, a partial differential equation (PDE) of the form ∇ 2 u ( x ) = f ( x ) {\displaystyle \nabla ^{2}u(\mathbf {x} )=f(\mathbf {x} )} where ∇ 2 {\displaystyle \nabla ^{2}} is the Laplace operator in R 3 {\displaystyle \mathbb {R} ^{3}} , f ( x ) {\displaystyle f(\mathbf {x} )} is the source term of the system, and u ( x ) {\displaystyle u(\mathbf {x} )} is the solution to the equation. Because ∇ 2 {\displaystyle \nabla ^{2}} is a linear differential operator, the solution u ( x ) {\displaystyle u(\mathbf {x} )} to a general system of this type can be written as an integral over a distribution of source given by f ( x ) {\displaystyle f(\mathbf {x} )} : u ( x ) = ∫ G ( x , x ′ ) f ( x ′ ) d x ′ {\displaystyle u(\mathbf {x} )=\int G(\mathbf {x} ,\mathbf {x'} )f(\mathbf {x'} )d\mathbf {x} '} where the Green's function for Laplacian in three variables G ( x , x ′ ) {\displaystyle G(\mathbf {x} ,\mathbf {x'} )} describes the response of the system at the point x {\displaystyle \mathbf {x} } to a point source located at x ′ {\displaystyle \mathbf {x'} } : ∇ 2 G ( x , x ′ ) = δ ( x − x ′ ) {\displaystyle \nabla ^{2}G(\mathbf {x} ,\mathbf {x'} )=\delta (\mathbf {x} -\mathbf {x'} )} and the point source is given by δ ( x − x ′ ) {\displaystyle \delta (\mathbf {x} -\mathbf {x'} )} , the Dirac delta function.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «differential equation»).
Fonte: Wikipedia in inglese, voce «Green's function for the three-variable Laplace equation» — https://en.wikipedia.org/wiki/Green%27s_function_for_the_three-variable_Laplace_equation
Wikidata Q5602166: «partial differential equations» (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q5602166
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Green%27s_function_for_the_three-variable_Laplace_equation?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1372638675, consultata il 2026-10-31T09:42:48.925Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 9d2aeb9a9e727a6bd3d28386d289ca2949ac20a0127959ec8102bc867f297252