Green's function for the three-variable Laplace equation

In physics, the Green's function (or fundamental solution) for Laplace's equation 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 In physics, the Green's function (or fundamental solution) for Laplace's equation 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 where ∇ 2 {displaystyle abla ^{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 abla ^{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} )} : where the Green's function for Laplace's equation 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'} } : and the point source is given by δ ( x − x ′ ) {displaystyle delta (mathbf {x} -mathbf {x'} )} , the Dirac delta function. One physical system of this type is a charge distribution in electrostatics. In such a system, the electric field is expressed as the negative gradient of the electric potential, and Gauss's law in differential form applies: