Poisson's equation

Poisson's equation (Poisson equation) is an equation 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.

A special case of Poisson's equation is Laplace's equation, the term for the equation if the 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.