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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

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

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