# Poisson's Equation - Statement of The Equation

Statement of The Equation

The Poisson Equation is

where is the Laplace operator, and f and φ are real or complex-valued functions on a manifold. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇2 and so Poisson's equation is frequently written as

In three-dimensional Cartesian coordinates, it takes the form

$left( frac{partial^2}{partial x^2} + frac{partial^2}{partial y^2} + frac{partial^2}{partial z^2} right)varphi(x,y,z) = f(x,y,z).$

For vanishing f, this equation becomes Laplace's equation

The Poisson equation may be solved using a Green's function; a general exposition of the Green's function for the Poisson equation is given in the article on the screened Poisson equation. There are various methods for numerical solution. The relaxation method, an iterative algorithm, is one example.

### Other articles related to "statement of the equation, equation":

Screened Poisson Equation - Statement of The Equation
... In the homogenous case (f=0), the screened Poisson equation is the same as the time-independent Klein–Gordon equation ... In the inhomogeneous case, the screened Poisson equation is very similar to the inhomogeneous Helmholtz equation, the only difference being the sign within the brackets ...

### Famous quotes containing the words statement of, equation and/or statement:

Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.
Mario Vargas Llosa (b. 1936)

Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.
Anna Quindlen (b. 1952)

The most distinct and beautiful statement of any truth must take at last the mathematical form.
Henry David Thoreau (1817–1862)