page_title: Laplace's equation
hierarchical_section_title: Laplace's equation
caption_attribution_description: Français&#160;: Pierre-Simon de Laplace, mathématicien et astronome, en habit de chancelier du Sénat sous l'Empire
context_page_description: 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
  where is the Laplace operator, is the divergence operator, is the gradient operator, and is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.
  If the right-hand side is specified as a given function, we have
  This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace’s equation is also a special case of the Helmholtz equation.
  The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.
context_section_description: 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
  where  is the Laplace operator,  is the divergence operator (also symbolized "div"),  is the gradient operator (also symbolized "grad"),  and  is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.
  If the right-hand side is specified as a given function, , we have
  This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace’s equation is also a special case of the Helmholtz equation. 
  The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.
caption: Pierre-Simon Laplace
exif: {}