### 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 to make the graph easier to understand.) 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 the above f or g
and some boundary conditions, the other can be determined.
The function f is termed a **potential** (e.g., gravitational potential),
∇f (gradient of f) is a vector field indicating the
force-per-whatever (for gravity, the force-per-mass,
equivalent to the resulting acceleration, given f=ma) and g
is the (gravitational) mass, or (for the *electric field* example) the charge.
The vector field ∇f has a zero **curl**, as does any gradient field.
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 for fields f and g described by 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