# Talk:Poisson's equation

## merge with Laplace equation (discussion)

I totally and completely disagree with the proposal to merge this page with that for the Laplace equation. Although they appear to be superficially similar, they lead in very different directions. The Laplace equation leads to the study of Harmonic functions, de Rham cohomology, and Hodge theory. On the other hand, Poisson's equation leads to the study of electrostatics, the Green's function, and anything that uses Green's functions or propagators, such as quantum field theory. Laplace's equation has absolutely no sense of a propagator, because it does not need one. Just because this article sucks, doesn't mean that it should be merged with another article that sucks even more. Both articles should be fixed. linas 14:00, 26 Apr 2005 (UTC)

Put another way, the wave equation is just the Laplace equation with signature +---, yet I notice there is no proposal to merge those two articles. That is because the Laplace equation is big enough to deserve multiple articles discussing it. linas 14:11, 26 Apr 2005 (UTC)

There is also the Poison-Boltzman equation concerning the distribution of ions around a macromolecule, that is solved to give the mapping of the epectrostatic potential around the molecule. From what i know it emerges from Poison equation and Boltzman distribution.

## Electrostatics

Recent edits on this article have changed it from being a discussion of Poisson's equation, to its application to the narrow, specific field of electrostatics. I strongly urge that this article be structured so that there is a general introduction, pointing out its general utility, and have the special case of electrostatics treated in a subsection. linas 17:11, 11 December 2005 (UTC)

Never mind, I did it myself. linas 17:33, 11 December 2005 (UTC)

## Solutions ???

For completeness, I think this article needs to have a section describing the major solutions to the equation and some of the methods for finding the solutions. -- Metacomet 18:17, 11 December 2005 (UTC)

## Where is the solution

No Poisson solution in this site?

## less algebra, and more explanation please

I completely agree. I don't even know what it is supposed to be about. Psi is not explained, and there is no indication why Poisson would have dreamed it up in the first place. What does it solve? It's not clear here at all. Telling us that the LHS equals a number on the RHS is no more than trivial. I assume psi is to represent the wave function, but there is no clue from the article. --Centroyd

## more applications

other applications include image processing. e.g. filtering light and shade effects from sequences of pictures. see: http://www.cs.huji.ac.il/course/2006/csip/iccv01.pdf

There should be something about how Poisson's equation is used as del-squared-phi=4(pi)G(rho) to compute gravitational forces in the universe, as an alternative to Newton's inverse-square law of gravitation. Phi is the potential field, and one gets to the force per unit mass (mass of the earth, say) by calculating -del(phi). —The preceding unsigned comment was added by 82.12.27.202 (talk) 12:10, August 23, 2007 (UTC)

## Poisson's equation in Gravity's Rainbow

The Thomas Pynchon novel has a description of Poisson's equation that makes this article look like technical babble. pages 54-56. Fuzzform 22:39, 11 July 2007 (UTC)

## Del vs. Delta

There seems to be some confusion on this page between Del and Delta, that is:

${\displaystyle \nabla }$ (Del) vs. ${\displaystyle \Delta }$ (Delta)

This should be cleared up. I think that all of the Deltas should be turned into Dels. This is vector calculus here, not thermodynamics... but maybe I'm missing something.

askewchan 16:10, 25 October 2007 (UTC)

The explanation of the notation appears to already be in the article. Take a look at the paragraph after the first equation. Note that del squared is short hand for del dot del and both of these are equivalent to delta. Feel free to add expository text to clear that up in the article if you think it is needed. ChrisChiasson 17:02, 26 October 2007 (UTC)

May call Del Nabla? — Preceding unsigned comment added by 188.98.208.59 (talk) 21:39, 26 July 2012 (UTC)

## Thermodynamics?

Maybe i am confusing the values but i think thermodynamics also has Poisson's equation describing the adiabatic process. Am i right and do this article need disambiguation? --159.148.226.100 (talk) 20:01, 13 December 2009 (UTC)

I am surprised i have found no such term in the article in English. In any case there is an eguation pVγ=const called Poisson's equation in the article of Russian Wikipedia, in our books in Russian in the university (Riga Technical Univ.) and we have been taught so. --159.148.226.100 (talk) 20:11, 13 December 2009 (UTC)

## Non-constant coefficients

I don't have time to edit this right now, but the article should really mention the generalization to non-constant coefficients, i.e.:

${\displaystyle \nabla \cdot (c\nabla \phi )=f}$

where c is a function of position (and may in general be a matrix; a positive-definite matrix with some boundedness if one wishes to preserve the basic properties of the Poisson operator). For example, this is the equation in linear electrostatics for an inhomogeneous medium, in which case c is the permittivity at each point in space.

Even more generally, you can have:

${\displaystyle b\nabla \cdot (c\nabla \phi )=f}$

where b is another (positive) function. e.g. in finding the static shape of a stretched membrane b and c are related to the mass density and elasticity, respectively.

— Steven G. Johnson (talk) 07:04, 1 May 2011 (UTC)

Is there any reason to go through everything currently in the derivation? To derive the Poisson equation for electrostatics, all that needs to be done is substitute the electric potential

${\displaystyle {\mathbf {E}}=-\nabla \varphi }$

into gauss' electrostatic law:

${\displaystyle \nabla \cdot {\mathbf {E} }=\rho /\epsilon }$

so the laplacian is obtained:

${\displaystyle \nabla \cdot (-\nabla \varphi )=\rho /\epsilon }$
${\displaystyle -\nabla ^{2}\varphi =\rho /\epsilon }$

It is clear that the field is static and not dynamic - given the title Electrostatics (hence can be desrcibed by a time-independant potential φ, without the ${\displaystyle \scriptstyle \partial {\mathbf {A} }/\partial t}$). Faraday's law, the electric displacement D, all the assupmtions of the euclidean space, linear homogeneous isotropic media and constant permittivity ε etc is completley irrelavent (though if preferred these statements could be encapsulated into the fact that the field is electrostatic - in written paragraph form, not mathematically as then it would still be padded).

I have added the same derivation to the gravity section to illustrate the simplicity. If no-one objects i'll simplify the section myself.-- 12:51, 18 December 2011 (UTC)

On second thought maybe the section could be re-written in a way that presents the conditions under which the Poisson equation for the electric field is true. I'll do that, rather than remove the content. This content is not applicable to the gravity case, so the extra details behind the electrostatic Poisson equation are probably worth it. Also the gravity application will be moved in front of the electrostatics one since there is less detail, readers will see how potential gradients substituted into Gaussian-type field equations are Poisson equations.-- 01:06, 21 December 2011 (UTC)

## Diffusion

If so important in diffusion, why no examples/discussion? [1] —DIV (137.111.13.4 (talk) 03:12, 29 January 2014 (UTC))