In mathematics, the **Newtonian potential** or **Newton potential** is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study in potential theory. In its general nature, it is a singular integral operator, defined by convolution with a function having a mathematical singularity at the origin, the Newtonian kernel Γ which is the fundamental solution of the Laplace equation. It is named for Isaac Newton, who first discovered it and proved that it was a harmonic function in the special case of three variables, where it served as the fundamental gravitational potential in Newton's law of universal gravitation. In modern potential theory, the Newtonian potential is instead thought of as an electrostatic potential.

The Newtonian potential of a compactly supported integrable function *ƒ* is defined as the convolution

where the Newtonian kernel Γ in dimension *d* is defined by

Here ω_{d} is the volume of the unit *d*-ball (sometimes sign conventions may vary; compare ( Evans 1998 ) and ( Gilbarg & Trudinger 1983 )). For example, for we have

The Newtonian potential *w* of *ƒ* is a solution of the Poisson equation

which is to say that the operation of taking the Newtonian potential of a function is a partial inverse to the Laplace operator. *w* will be a classical solution, that is twice differentiable, if *f* is bounded and locally Hölder continuous as shown by Otto Hölder. It was an open question whether continuity alone is also sufficient. This was shown to be wrong by Henrik Petrini who gave an example of a continuous *f* for which *w* is not twice differentiable. The solution is not unique, since addition of any harmonic function to *w* will not affect the equation. This fact can be used to prove existence and uniqueness of solutions to the Dirichlet problem for the Poisson equation in suitably regular domains, and for suitably well-behaved functions ƒ: one first applies a Newtonian potential to obtain a solution, and then adjusts by adding a harmonic function to get the correct boundary data.

The Newtonian potential is defined more broadly as the convolution

when *μ* is a compactly supported Radon measure. It satisfies the Poisson equation

in the sense of distributions. Moreover, when the measure is positive, the Newtonian potential is subharmonic on **R**^{d}.

If *ƒ* is a compactly supported continuous function (or, more generally, a finite measure) that is rotationally invariant, then the convolution of *ƒ* with Γ satisfies for *x* outside the support of *ƒ*

In dimension *d* = 3, this reduces to Newton's theorem that the potential energy of a small mass outside a much larger spherically symmetric mass distribution is the same as if all of the mass of the larger object were concentrated at its center.

When the measure *μ* is associated to a mass distribution on a sufficiently smooth hypersurface *S* (a Lyapunov surface of Hölder class *C*^{1,α}) that divides **R**^{d} into two regions *D*_{+} and *D*_{−}, then the Newtonian potential of *μ* is referred to as a **simple layer potential**. Simple layer potentials are continuous and solve the Laplace equation except on *S*. They appear naturally in the study of electrostatics in the context of the electrostatic potential associated to a charge distribution on a closed surface. If d*μ* = *ƒ* d*H* is the product of a continuous function on *S* with the (*d* − 1)-dimensional Hausdorff measure, then at a point *y* of *S*, the normal derivative undergoes a jump discontinuity *ƒ*(*y*) when crossing the layer. Furthermore, the normal derivative is of *w* a well-defined continuous function on *S*. This makes simple layers particularly suited to the study of the Neumann problem for the Laplace equation.

In mathematics and physics, **Laplace's equation** is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

Baron **Siméon Denis Poisson** FRS FRSE was a French mathematician, engineer, and physicist who made many scientific advances.

In mathematics, mathematical physics and the theory of stochastic processes, a **harmonic function** is a twice continuously differentiable function *f* : *U* → **R**, where *U* is an open subset of **R**^{n}, that satisfies Laplace's equation, that is,

In mathematics, the **Laplace operator** or **Laplacian** is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols , , or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δ*f*(*p*) of a function *f* at a point *p* measures by how much the average value of *f* over small spheres or balls centered at *p* deviates from *f*(*p*).

**Poisson's equation** is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson.

In physical science and mathematics, the **Legendre functions***P*_{λ}, *Q*_{λ} and **associated Legendre functions***P*^{μ}_{λ}, *Q*^{μ}_{λ}, and **Legendre functions of the second kind**, *Q _{n}*, are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia articles.

A **Newtonian fluid** is a fluid in which the viscous stresses arising from its flow, at every point, are linearly correlated to the local strain rate—the rate of change of its deformation over time. That is equivalent to saying those forces are proportional to the rates of change of the fluid's velocity vector as one moves away from the point in question in various directions.

**Scalar potential**, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.

In mathematics, a **Dirichlet problem** is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.

In mathematics, the **discrete Laplace operator** is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph, the discrete Laplace operator is more commonly called the Laplacian matrix.

In mathematics, and specifically in potential theory, the **Poisson kernel** is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.

In general relativity and many alternatives to it, the **post-Newtonian formalism** is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields, it may be preferable to solve the complete equations numerically. Some of these post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of the matter forming the gravitational field to the speed of light, which in this case is better called the speed of gravity. In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to Newton's law of gravity.

In mathematics, the **Riesz potential** is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to several variables the Riemann–Liouville integrals of one variable.

In physics, the **Green's function for Laplace's equation in three variables** is used to describe the response of a particular type of physical system to a point source. In particular, this Green's function arises in systems that can be described by Poisson's equation, a partial differential equation (PDE) of the form

There are various **mathematical descriptions of the electromagnetic field** that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

**Newton–Cartan theory** is a geometrical re-formulation, as well as a generalization, of Newtonian gravity first introduced by Élie Cartan and Kurt Friedrichs and later developed by Dautcourt, Dixon, Dombrowski and Horneffer, Ehlers, Havas, Künzle, Lottermoser, Trautman, and others. In this re-formulation, the structural similarities between Newton's theory and Albert Einstein's general theory of relativity are readily seen, and it has been used by Cartan and Friedrichs to give a rigorous formulation of the way in which Newtonian gravity can be seen as a specific limit of general relativity, and by Jürgen Ehlers to extend this correspondence to specific solutions of general relativity.

In the mathematical theory of harmonic analysis, the **Riesz transforms** are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension *d* > 1. They are a type of singular integral operator, meaning that they are given by a convolution of one function with another function having a singularity at the origin. Specifically, the Riesz transforms of a complex-valued function ƒ on **R**^{d} are defined by

In potential theory, an area of mathematics, a **double layer potential** is a solution of Laplace's equation corresponding to the electrostatic or magnetic potential associated to a dipole distribution on a closed surface *S* in three-dimensions. Thus a double layer potential *u*(**x**) is a scalar-valued function of **x** ∈ **R**^{3} given by

In mathematics, the **walk-on-spheres method (WoS)** is a numerical probabilistic algorithm, or Monte-Carlo method, used mainly in order to approximate the solutions of some specific boundary value problem for partial differential equations (PDEs). The WoS method was first introduced by Mervin E. Muller in 1956 to solve Laplace's equation, and was since then generalized to other problems.

In mathematics, the **Poisson boundary** is a measure space associated to a random walk. It is an object designed to encode the asymptotic behaviour of the random walk, i.e. how trajectories diverge when the number of steps goes to infinity. Despite being called a boundary it is in general a purely measure-theoretical object and not a boundary in the topological sense. However, in the case where the random walk is on a topological space the Poisson boundary can be related to the **Martin boundary** which is an analytic construction yielding a genuine topological boundary. Both boundaries are related to harmonic functions on the space via generalisations of the Poisson formula.

- Evans, L.C. (1998),
*Partial Differential Equations*, Providence: American Mathematical Society, ISBN 0-8218-0772-2 . - Gilbarg, D.; Trudinger, Neil (1983),
*Elliptic Partial Differential Equations of Second Order*, New York: Springer, ISBN 3-540-41160-7 . - Solomentsev, E.D. (2001) [1994], "Newton potential",
*Encyclopedia of Mathematics*, EMS Press - Solomentsev, E.D. (2001) [1994], "Simple-layer potential",
*Encyclopedia of Mathematics*, EMS Press - Solomentsev, E.D. (2001) [1994], "Surface potential",
*Encyclopedia of Mathematics*, EMS Press

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