Poisson's equation
The equation is named after French mathematician and physicist Siméon Denis Poisson who published it in 1823.
When the manifold is Euclidean space, the Laplace operator is often denoted as ∇2, and so Poisson's equation is frequently written as
In three-dimensional Cartesian coordinates, it takes the form
In the case of a gravitational field g due to an attracting massive object of density ρ, Gauss's law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity.
Since the gravitational field is conservative (and irrotational), it can be expressed in terms of a scalar potential ϕ:
The corresponding Green's function can be used to calculate the potential at distance r from a central point mass m (i.e., the fundamental solution).
Many problems in electrostatics are governed by the Poisson equation, which relates the electric potential φ to the free charge density
The mathematical details of Poisson's equation, commonly expressed in SI units (as opposed to Gaussian units), describe how the distribution of free charges generates the electrostatic potential in a given region.
Starting with Gauss's law for electricity (also one of Maxwell's equations) in differential form, one has
is the divergence operator, D is the electric displacement field, and ρf is the free-charge density (describing charges brought from outside).
Assuming the medium is linear, isotropic, and homogeneous (see polarization density), we have the constitutive equation
Substituting this into Gauss's law and assuming that ε is spatially constant in the region of interest yields
This equation means that we can write the electric field as the gradient of a scalar function φ (called the electric potential), since the curl of any gradient is zero.
where the minus sign is introduced so that φ is identified as the electric potential energy per unit charge.
Specifying the Poisson's equation for the potential requires knowing the charge density distribution.
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 theory of dilute electrolyte solutions.
Using a Green's function, the potential at distance r from a central point charge Q (i.e., the fundamental solution) is
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 class of cases, computing φ is no longer sufficient to calculate E, since E also depends on the magnetic vector potential A, which must be independently computed.
If there is a static spherically symmetric Gaussian charge density
where Q is the total charge, then the solution φ(r) of Poisson's equation
Furthermore, the error function approaches 1 extremely quickly as its argument increases; in practice, for r > 3σ the relative error is smaller than one part in a thousand.
[8] The goal of this technique is to reconstruct an implicit function f whose value is zero at the points pi and whose gradient at the points pi equals the normal vectors ni.
The set of (pi, ni) is thus modeled as a continuous vector field V. The implicit function f is found by 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 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. In order to effectively apply Poisson's equation to the problem of surface reconstruction, it is necessary to find a good discretization of the vector field V. The basic approach is to bound the data with a finite-difference grid.
It is convenient to define three staggered grids, each shifted in one and only one direction corresponding to the components of the normal data.
On each staggered grid we perform trilinear interpolation on the set of points.
The interpolation weights are then used to distribute the magnitude of the associated component of ni onto the nodes of the particular staggered grid cell containing pi.
