### Poisson's equation

**(Poisson equation)**
(relates one function's Laplacian to another function)

**Poisson's equation** (**Poisson equation**)
is an equation template that fits a number of
physical phenomena, including those governed by an inverse square law,
such as gravitational fields and electric fields.
It asserts a particular relationship between two mathematical fields,
showing that each of them models a particular physical field
in a different manner:

∇²f = g
or
∇·∇f = g
or
Δf = g

- ∇² or Δ -
**Laplacian** operator, which stands for ∇·∇ i.e., the **divergence** of the **gradient**.
- f and g - functions, often of three scalars (representing coordinates in space, i.e., a mathematical field), both functions always yielding scalars. (Sometimes illustrations of how they work use just two coordinates so graph illustrations are clearer.) In relativity where space and time are embedded in a four-dimensional space, it may be four (scalar) parameters or more in phase space.

Through such an equation, given either f or g, the other can be
determined, given some boundary conditions.
The function ∇f (gradient of f) is termed a **potential**
(e.g., gravitational potential)
associated with the function f and has a zero **curl**.
Multiple masses or charges can be modeled by summing the two sides
of the corresponding Poisson equations associated with each
mass/charge.

**Laplace's equation** amounts to a special case of Poisson's equation,
when function g is zero, i.e., evaluates to zero for all parameters:

∇²f = 0

It is of note that given Poisson's equation, throughout any volume
in which g happens to be zero, Laplace's equation is holding over
that volume. If Poisson's equation is used to model the field
surrounding a single point (or single spherically symmetric) charge
or mass, then any region beyond the charge/mass is adhering
to Laplace's equation.

(*mathematics,physics*)
**Further reading:**

http://en.wikipedia.org/wiki/Poisson's_equation

http://en.wikipedia.org/wiki/Laplace%27s_equation

http://en.wikipedia.org/wiki/Vector_calculus_identities

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/laplace.html

http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter2.pdf

**Referenced by pages:**

Legendre polynomials

Schrödinger-Poisson equation

Vlasov-Poisson equation

Index