|
|
| Line 1: |
Line 1: |
| In [[mathematics]], '''Poisson's equation''' is a [[partial differential equation]] of elliptic type with broad utility in [[electrostatics]], [[mechanical engineering]] and [[theoretical physics]]. Commonly used to model diffusion, it is named after the [[France|French]] [[mathematician]], [[geometer]], and [[physicist]] [[Siméon Denis Poisson]].<ref>{{citation|title=Glossary of Geology|editor1-first=Julia A.|editor1-last=Jackson|editor2-first=James P.|editor2-last=Mehl|editor3-first=Klaus K. E.|editor3-last=Neuendorf|series=American Geological Institute|publisher=Springer|year=2005|isbn=9780922152766|page=503|url=http://books.google.com/books?id=SfnSesBc-RgC&pg=PA503}}.</ref>
| | The ac1st16.dll error is annoying and quite widespread with all sorts of Windows computers. Not just does it make the computer run slower, however it could also prevent we from using a range of programs, including AutoCAD. To fix this problem, you should use a simple way to cure all possible problems which cause it. Here's what we should do...<br><br>If it is very not as big of a issue because you think it's, it will possibly be solved easily by running a Startup Repair or by System Restore Utility. Again it may be because easy as running an anti-virus check or cleaning the registry.<br><br>The error is basically a result of problem with Windows Installer package. The Windows Installer is a tool used to install, uninstall and repair the many programs on the computer. Let you discuss a limited items that helped a great deal of individuals whom facing the synonymous matter.<br><br>Always see with it which you have installed antivirus, anti-spyware and anti-adware programs and have them up-to-date on a regular basis. This can help stop windows XP running slow.<br><br>So to fix this, we simply should be able to make all the registry files non-corrupted again. This can dramatically speed up the loading time of your computer and may allow you to a big number of factors on it again. And fixing these files couldn't be easier - you just should employ a tool called a [http://bestregistrycleanerfix.com/fix-it-utilities fix it utilities].<br><br>Turn It Off: Chances are should you are like me; then we spend a great deal of time on your computer on a daily basis. Try providing a computer some time to do absolutely nothing; this might sound funny however, in the event you have an elder computer you may be asking it to do too much.<br><br>The initial reason the computer could be slow is because it demands more RAM. You'll notice this issue right away, especially if you have lower than a gig of RAM. Most brand-new computers come with a least which much. While Microsoft says Windows XP will run on 128 MB, it plus Vista absolutely need at least a gig to run smoothly plus let you to run multiple programs at when. Fortunately, the price of RAM has dropped greatly, plus there are a gig installed for $100 or less.<br><br>Ally Wood is a expert software reviewer plus has worked in CNET. Then she is functioning for her own review software organization to give feedback to the software creator plus has completed deep test inside registry cleaner software. After reviewing the most popular registry cleaner, she has created complete review on a review website for you that is accessed for free. |
| | |
| ==Statement of the equation==
| |
| | |
| Poisson's equation is
| |
| | |
| :<math>\Delta\varphi=f</math>
| |
| | |
| where <math>\Delta</math> is the [[Laplace operator]], and ''f'' and ''φ'' are [[real number|real]] or [[complex number|complex]]-valued [[function (mathematics)|functions]] on a [[manifold]]. When the manifold is [[Euclidean space]], the Laplace operator is often denoted as ∇<sup>2</sup> and so Poisson's equation is frequently written as
| |
| | |
| :<math>\nabla^2 \varphi = f.</math>
| |
| | |
| In three-dimensional [[Cartesian coordinate]]s, it takes the form
| |
| | |
| :<math>
| |
| \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right)\varphi(x,y,z) = f(x,y,z).
| |
| </math>
| |
| | |
| | |
| Poisson's equation may be solved using a [[Green's function]]; a general exposition of the Green's function for Poisson's equation is given in the article on the [[screened Poisson equation]]. There are various methods for numerical solution. The [[relaxation method]], an iterative algorithm, is one example.
| |
| | |
| ==Newtonian gravity==
| |
| | |
| {{main|gravitational field|Gauss' law for gravity}}
| |
| | |
| In the case of a gravitational field '''g''' due to an attracting massive object, of density ''ρ'', Gauss' law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity. Gauss' law for gravity is:
| |
| | |
| :<math>\nabla\cdot\bold{g} = -4\pi G\rho </math>,
| |
| | |
| and since the gravitational field is conservative, it can be expressed in terms of a scalar potential ''Φ'':
| |
| | |
| :<math>\bold{g} = -\nabla \Phi </math>,
| |
| | |
| substituting into Gauss' law
| |
| | |
| :<math>\nabla\cdot(-\nabla \Phi) = - 4\pi G \rho</math>
| |
| | |
| obtains '''Poisson's equation''' for gravity:
| |
| | |
| :<math>{\nabla}^2 \Phi = 4\pi G \rho.</math>
| |
| | |
| ==Electrostatics==
| |
| | |
| {{main|Electrostatics}}
| |
| | |
| One of the cornerstones of [[electrostatics]] is setting up and solving problems described by the Poisson equation. Finding φ for some given ''f'' is an important practical problem, since this is the usual way to find the [[electric potential]] for a given [[electric charge|charge]] distribution described by the density function.
| |
| | |
| The mathematical details behind Poisson's equation in electrostatics are as follows ([[SI]] units are used rather than [[Gaussian units]], which are also frequently used in [[electromagnetism]]).
| |
| | |
| Starting with [[Gauss' law]] for electricity (also one of [[Maxwell's equations]]) in differential form, we have:
| |
| | |
| :<math>\mathbf{\nabla} \cdot \mathbf{D} = \rho_f</math>
| |
| | |
| where <math>\mathbf{\nabla} \cdot</math> is the [[divergence|divergence operator]], '''D''' = [[electric displacement field]], and ''ρ<sub>f</sub>'' = [[free charge]] [[charge density|density]] (describing charges brought from outside). Assuming the medium is linear, isotropic, and homogeneous (see [[polarization density]]), we have the [[constitutive equation#Electromagnetism|constitutive equation]]:
| |
| | |
| :<math>\mathbf{D} = \varepsilon \mathbf{E}</math>
| |
| | |
| where ''ε'' = [[permittivity]] of the medium and '''E''' = [[electric field]]. Substituting this into Gauss' law and assuming ''ε'' is spatially constant in the region of interest obtains:
| |
| | |
| :<math>\mathbf{\nabla} \cdot \mathbf{E} = \frac{\rho_f}{\varepsilon}</math>
| |
| | |
| In the absence of a changing magnetic field, '''B''', [[Faraday's law of induction]] gives:
| |
| | |
| :<math>\nabla \times \mathbf{E} = -\dfrac{\partial \mathbf{B}} {\partial t} = 0</math>
| |
| | |
| where <math>\nabla \times</math> is the [[curl (mathematics)|curl operator]] and ''t'' is time. Since the [[Curl (mathematics)|curl]] of the electric field is zero, it is defined by a scalar electric potential field, <math>\varphi</math> (see [[Helmholtz decomposition]]).
| |
| | |
| :<math>\mathbf{E} = -\nabla \varphi</math>
| |
| | |
| The derivation of Poisson's equation under these circumstances is straightforward. Substituting the potential gradient for the electric field
| |
| | |
| :<math>\nabla \cdot \bold{E} = \nabla \cdot ( - \nabla \varphi ) = - {\nabla}^2 \varphi = \frac{\rho_f}{\varepsilon},</math>
| |
| | |
| directly obtains '''Poisson's equation''' for electrostatics, which is:
| |
| | |
| :<math>{\nabla}^2 \varphi = -\frac{\rho_f}{\varepsilon}.</math>
| |
| | |
| Solving Poisson's equation for the potential requires knowing the charge density distribution. If the charge density is zero, then [[Laplace's equation]] results. If the charge density follows a [[Boltzmann distribution]], then the [[Poisson-Boltzmann equation]] results. The Poisson–Boltzmann equation plays a role in the development of the [[Debye–Hückel equation|Debye–Hückel theory of dilute electrolyte solutions]].
| |
| | |
| The above discussion assumes that the magnetic field is not varying in time. The same Poisson equation arises even if it does vary in time, as long as the [[Coulomb gauge]] is used. In this more general context, computing ''φ'' is no longer sufficient to calculate '''E''', since '''E''' also depends on the [[magnetic vector potential]] '''A''', which must be independently computed. See [[Mathematical descriptions of the electromagnetic field#Maxwell's equation in potential formulation|Maxwell's equation in potential formulation]] for more on ''φ'' and '''A''' in Maxwell's equations and how Poisson's equation is obtained in this case. | |
| | |
| === Potential of a Gaussian charge density ===
| |
| If there is a static spherically symmetric [[Gaussian distribution|Gaussian]] charge density
| |
| | |
| :<math> \rho_f(r) = \frac{Q}{\sigma^3\sqrt{2\pi}^3}\,e^{-r^2/(2\sigma^2)},</math>
| |
| | |
| where ''Q'' is the total charge, then the solution ''φ''(''r'') of Poisson's equation,
| |
| | |
| :<math>{\nabla}^2 \varphi = - { \rho_f \over \varepsilon } </math>,
| |
| | |
| is given by
| |
| | |
| :<math> \varphi(r) = { 1 \over 4 \pi \varepsilon } \frac{Q}{r}\,\mbox{erf}\left(\frac{r}{\sqrt{2}\sigma}\right)</math>
| |
| | |
| where erf(''x'') is the [[error function]].
| |
| | |
| This solution can be checked explicitly by evaluating <math>{\nabla}^2 \varphi</math>. Note that, for ''r'' much greater than ''σ'', the erf function approaches unity and the potential φ (''r'') approaches the [[electrical potential|point charge]] potential | |
| | |
| :<math> \varphi \approx { 1 \over 4 \pi \varepsilon } {Q \over r} </math>,
| |
| | |
| as one would expect. Furthermore the erf function approaches 1 extremely quickly as its argument increases; in practice for r > 3''σ'' the relative error is smaller than one part in a thousand.
| |
| | |
| ==Surface Reconstruction==
| |
| | |
| Poisson's equation is also used to reconstruct a smooth 2D surface (in the sense of [[curve fitting]]) based on a large number of points ''p<sub>i</sub>'' (a [[point cloud]]) where each point also carries an estimate of the local [[surface normal]] '''n'''<sub>''i''</sub>.<ref>F. Calakli and G. Taubin, [http://mesh.brown.edu/ssd/pdf/Calakli-pg2011.pdf Smooth Signed Distance Surface Reconstruction], Pacific Graphics Vol 30-7, 2011</ref>
| |
| | |
| This technique reconstructs the [[implicit function]] ''f'' whose value is zero at the points ''p<sub>i</sub>'' and whose gradient at the points ''p<sub>i</sub>'' equals the normal vectors '''n'''<sub>''i''</sub>. The set of (''p<sub>i</sub>'', '''n'''<sub>''i''</sub>) is thus a sampling of a continuous [[Euclidean vector|vector]] field '''V'''. The implicit function ''f'' is found by [[Integral|integrating]] the vector field '''V'''. Since not every vector field is the [[gradient]] of a function, the problem may or may not have a solution: the necessary and sufficient condition for a smooth vector field '''V''' to be the gradient of a function ''f'' is that the [[Curl (mathematics)|curl]] of '''V''' must be identically zero. In case this condition is difficult to impose, it is still possible to perform a [[least-squares]] fit to minimize the difference between '''V''' and the gradient of ''f''.
| |
| | |
| ==See also==
| |
| * [[Discrete Poisson equation]]
| |
| * [[Poisson–Boltzmann equation]]
| |
| * [[Uniqueness theorem for Poisson's equation]]
| |
| | |
| ==References==
| |
| <references /> | |
| | |
| <div class="references-small"> | |
| * [http://eqworld.ipmnet.ru/en/solutions/lpde/lpde302.pdf Poisson Equation] at EqWorld: The World of Mathematical Equations.
| |
| * L.C. Evans, ''Partial Differential Equations'', American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
| |
| * A. D. Polyanin, ''Handbook of Linear Partial Differential Equations for Engineers and Scientists'', Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
| |
| </div> | |
| | |
| ==External links==
| |
| *{{springer|title=Poisson equation|id=p/p073290}}
| |
| *[http://planetmath.org/encyclopedia/PoissonsEquation.html Poisson's equation] on [[PlanetMath]].
| |
| *[http://www.youtube.com/watch?v=sMJTWa-Z9Ho Poisson's Equation] Poisson's Equation video
| |
| | |
| [[Category:Potential theory]]
| |
| [[Category:Partial differential equations]]
| |
| [[Category:Electrostatics]]
| |
The ac1st16.dll error is annoying and quite widespread with all sorts of Windows computers. Not just does it make the computer run slower, however it could also prevent we from using a range of programs, including AutoCAD. To fix this problem, you should use a simple way to cure all possible problems which cause it. Here's what we should do...
If it is very not as big of a issue because you think it's, it will possibly be solved easily by running a Startup Repair or by System Restore Utility. Again it may be because easy as running an anti-virus check or cleaning the registry.
The error is basically a result of problem with Windows Installer package. The Windows Installer is a tool used to install, uninstall and repair the many programs on the computer. Let you discuss a limited items that helped a great deal of individuals whom facing the synonymous matter.
Always see with it which you have installed antivirus, anti-spyware and anti-adware programs and have them up-to-date on a regular basis. This can help stop windows XP running slow.
So to fix this, we simply should be able to make all the registry files non-corrupted again. This can dramatically speed up the loading time of your computer and may allow you to a big number of factors on it again. And fixing these files couldn't be easier - you just should employ a tool called a fix it utilities.
Turn It Off: Chances are should you are like me; then we spend a great deal of time on your computer on a daily basis. Try providing a computer some time to do absolutely nothing; this might sound funny however, in the event you have an elder computer you may be asking it to do too much.
The initial reason the computer could be slow is because it demands more RAM. You'll notice this issue right away, especially if you have lower than a gig of RAM. Most brand-new computers come with a least which much. While Microsoft says Windows XP will run on 128 MB, it plus Vista absolutely need at least a gig to run smoothly plus let you to run multiple programs at when. Fortunately, the price of RAM has dropped greatly, plus there are a gig installed for $100 or less.
Ally Wood is a expert software reviewer plus has worked in CNET. Then she is functioning for her own review software organization to give feedback to the software creator plus has completed deep test inside registry cleaner software. After reviewing the most popular registry cleaner, she has created complete review on a review website for you that is accessed for free.