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
where is the Laplace operator in, is the source term of the system, and is the solution to the equation. Because is a linear differential operator, the solution to a general system of this type can be written as an integral over a distribution of source given by :
where the Green's function for Laplace's equation in three variables describes the response of the system at the point to a point source located at :
and the point source is given by, the Dirac delta function.
Read more about Green's Function For The Three-variable Laplace Equation: Motivation, Mathematical Exposition, Rotationally Invariant Green's Functions For The Three-variable Laplace Equation
Famous quotes containing the words green, function, laplace and/or equation:
“Chaucer is fresh and modern still, and no dust settles on his true passages. It lightens along the line, and we are reminded that flowers have bloomed, and birds sung, and hearts beaten in England. Before the earnest gaze of the reader, the rust and moss of time gradually drop off, and the original green life is revealed. He was a homely and domestic man, and did breathe quite as modern men do.”
—Henry David Thoreau (1817–1862)
“The uses of travel are occasional, and short; but the best fruit it finds, when it finds it, is conversation; and this is a main function of life.”
—Ralph Waldo Emerson (1803–1882)
“Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes.”
—Pierre Simon De Laplace (1749–1827)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)