### 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)