Poisson's equation

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson.

Poisson's equation is

where ${\displaystyle \Delta }$ is the Laplace operator, and ${\displaystyle f}$ and ${\displaystyle \varphi }$ are real or complex-valued functions on a manifold. Usually, ${\displaystyle f}$ is given and ${\displaystyle \varphi }$ is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇² and so Poisson's equation is frequently written as

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