TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
and
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, Laplace's equation is a second-order
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
named after
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar A scholar is a person who pursues academic and intellectual activities, particularly those that develop expertise in an area of Studying, study. A ...

, who first studied its properties. This is often written as :$\nabla^2\! f = 0 \qquad$ or $\qquad \Delta f = 0,$ where $\Delta = \nabla \cdot \nabla = \nabla^2$ is the
Laplace operator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, $\Delta x = x_1 - x_2$. Its use to represent the Laplacian should not be confused with this use. $\nabla \cdot$ is the
divergence In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Produ ...

operator (also symbolized "div"), $\nabla$ is the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

operator (also symbolized "grad"), and $f \left(x, y, z\right)$ is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, $h\left(x, y, z\right)$, we have :$\Delta f = h.$ This is called
Poisson's equation Poisson's equation is an elliptic partial differential equation Second-order linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key i ...
, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of
elliptic partial differential equation Second-order linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be pr ...
s. Laplace's equation is also a special case of the
Helmholtz equation In mathematics, the eigenvalue In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their r ...
. The general theory of solutions to Laplace's equation is known as
potential theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. The solutions of Laplace's equation are the
harmonic function 300px, A harmonic function defined on an annulus. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and cal ...
s, which are important in multiple branches of physics, notably electrostatics, gravitation, and
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
. In the study of
heat conduction Thermal conduction is the transfer of internal energy The internal energy of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes u ...

, the Laplace equation is the
steady-state In systems theory Systems theory is the interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from s ...
heat equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.

# Forms in different coordinate systems

In
rectangular coordinates A Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of number, numerical coordinates, which are the positive and negative numbers, signed distances ...

,Griffiths, David J.
Introduction to Electrodynamics
'. 4th ed., Pearson, 2013. Inner front cover. .
: $\nabla^2 f = \frac + \frac + \frac = 0.$ In
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of th ...

, :$\nabla^2 f=\frac \frac \left\left( r \frac \right\right) + \frac \frac + \frac =0.$ In
spherical coordinates In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, using the $\left(r, \theta, \varphi\right)$ convention, :$\nabla^2 f = \frac\frac \left\left(r^2 \frac\right\right) + \frac \frac \left\left(\sin\theta \frac\right\right) + \frac \frac =0.$ More generally, in
curvilinear coordinates In , curvilinear coordinates are a for in which the s may be curved. These coordinates may be derived from a set of s by using a transformation that is (a one-to-one map) at each point. This means that one can convert a point given in a C ...
, : $\nabla^2 f =\frac\left\left(\fracg^\right\right) + \frac g^\Gamma^n_ =0,$ or : $\nabla^2 f = \frac \frac\!\left\left(\sqrtg^ \frac\right\right) =0, \qquad \left(g=\det\\right).$

# Boundary conditions

The
Dirichlet problem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
for Laplace's equation consists of finding a solution ''φ'' on some domain ''D'' such that ''φ'' on the boundary of ''D'' is equal to some given function. Since the Laplace operator appears in the
heat equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, one physical interpretation of this problem is as follows: fix the temperature on the boundary of the domain according to the given specification of the boundary condition. Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain doesn't change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem. The
Neumann boundary condition In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary differential equation, ordinary or a partial differential equation, the condition specifies the v ...
s for Laplace's equation specify not the function ''φ'' itself on the boundary of ''D'', but its
normal derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of ''D'' alone. For the example of the heat equation it amounts to prescribing the heat flux through the boundary. In particular, at an adiabatic boundary, the normal derivative of ''φ'' is zero. Solutions of Laplace's equation are called
harmonic function 300px, A harmonic function defined on an annulus. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and cal ...
s; they are all analytic within the domain where the equation is satisfied. If any two functions are solutions to Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. This property, called the principle of superposition, is very useful. For example, solutions to complex problems can be constructed by summing simple solutions.

# In two dimensions

Laplace's equation in two independent variables in rectangular coordinates has the form :$\frac + \frac \equiv \psi_ + \psi_ = 0.$

## Analytic functions

The real and imaginary parts of a complex
analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
both satisfy the Laplace equation. That is, if , and if :$f\left(z\right) = u\left(x,y\right) + iv\left(x,y\right),$ then the necessary condition that ''f''(''z'') be analytic is that ''u'' and ''v'' be differentiable and that the
Cauchy–Riemann equations In the field of complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), li ...
be satisfied: :$u_x = v_y, \quad v_x = -u_y.$ where ''ux'' is the first partial derivative of ''u'' with respect to ''x''. It follows that :$u_ = \left(-v_x\right)_y = -\left(v_y\right)_x = -\left(u_x\right)_x.$ Therefore ''u'' satisfies the Laplace equation. A similar calculation shows that ''v'' also satisfies the Laplace equation. Conversely, given a harmonic function, it is the real part of an analytic function, ''f''(''z'') (at least locally). If a trial form is :$f\left(z\right) = \varphi\left(x,y\right) + i \psi\left(x,y\right),$ then the Cauchy–Riemann equations will be satisfied if we set :$\psi_x = -\varphi_y, \quad \psi_y = \varphi_x.$ This relation does not determine ''ψ'', but only its increments: :$d \psi = -\varphi_y\, dx + \varphi_x\, dy.$ The Laplace equation for ''φ'' implies that the integrability condition for ''ψ'' is satisfied: :$\psi_ = \psi_,$ and thus ''ψ'' may be defined by a line integral. The integrability condition and
Stokes' theorem Stokes' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Ba ...
implies that the value of the line integral connecting two points is independent of the path. The resulting pair of solutions of the Laplace equation are called conjugate harmonic functions. This construction is only valid locally, or provided that the path does not loop around a singularity. For example, if ''r'' and ''θ'' are polar coordinates and :$\varphi = \log r,$ then a corresponding analytic function is :$f\left(z\right) = \log z = \log r + i\theta.$ However, the angle ''θ'' is single-valued only in a region that does not enclose the origin. The close connection between the Laplace equation and analytic functions implies that any solution of the Laplace equation has derivatives of all orders, and can be expanded in a power series, at least inside a circle that does not enclose a singularity. This is in sharp contrast to solutions of the
wave equation The wave equation is a second-order linear for the description of s—as they occur in —such as (e.g. waves, and ) or waves. It arises in fields like , , and . Historically, the problem of a such as that of a was studied by , , , and ...
, which generally have less regularity. There is an intimate connection between power series and
Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. If we expand a function ''f'' in a power series inside a circle of radius ''R'', this means that :$f\left(z\right) = \sum_^\infty c_n z^n,$ with suitably defined coefficients whose real and imaginary parts are given by :$c_n = a_n + i b_n.$ Therefore : which is a Fourier series for ''f''. These trigonometric functions can themselves be expanded, using multiple angle formulae.

## Fluid flow

Let the quantities ''u'' and ''v'' be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions. The continuity condition for an incompressible flow is that :$u_x + v_y=0,$ and the condition that the flow be irrotational is that :$\nabla \times \mathbf=v_x - u_y =0.$ If we define the differential of a function ''ψ'' by :$d \psi = v dx - u dy,$ then the continuity condition is the integrability condition for this differential: the resulting function is called the
stream function The stream function is defined for incompressible In fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physica ...
because it is constant along flow lines. The first derivatives of ''ψ'' are given by :$\psi_x = v, \quad \psi_y=-u,$ and the irrotationality condition implies that ''ψ'' satisfies the Laplace equation. The harmonic function ''φ'' that is conjugate to ''ψ'' is called the
velocity potential A velocity potential is a scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energy, potential energies of an object in two different positions depends only on ...
. The Cauchy–Riemann equations imply that :$\varphi_x=-u, \quad \varphi_y=-v.$ Thus every analytic function corresponds to a steady incompressible, irrotational, inviscid fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function.

## Electrostatics

According to
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
, an electric field in two space dimensions that is independent of time satisfies :$\nabla \times \left(u,v,0\right) = \left(v_x -u_y\right)\hat = \mathbf,$ and :$\nabla \cdot \left(u,v\right) = \rho,$ where ''ρ'' is the charge density. The first Maxwell equation is the integrability condition for the differential :$d \varphi = -u\, dx -v\, dy,$ so the electric potential ''φ'' may be constructed to satisfy :$\varphi_x = -u, \quad \varphi_y = -v.$ The second of Maxwell's equations then implies that :$\varphi_ + \varphi_ = -\rho,$ which is the
Poisson equation Poisson's equation is an elliptic partial differential equation Second-order linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key i ...
. The Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions.

# In three dimensions

## Fundamental solution

A
fundamental solution In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of Laplace's equation satisfies :$\Delta u = u_ + u_ + u_ = -\delta\left(x-x\text{'},y-y\text{'},z-z\text{'}\right),$ where the
Dirac delta function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
''δ'' denotes a unit source concentrated at the point . No function has this property: in fact it is a
distributionDistribution may refer to: Mathematics *Distribution (mathematics) Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distr ...
rather than a function; but it can be thought of as a limit of functions whose integrals over space are unity, and whose support (the region where the function is non-zero) shrinks to a point (see
weak solution In mathematics, a weak solution (also called a generalized solution) to an ordinary differential equation, ordinary or partial differential equation is a function (mathematics), function for which the derivatives may not all exist but which is nonet ...
). It is common to take a different sign convention for this equation than one typically does when defining fundamental solutions. This choice of sign is often convenient to work with because −Δ is a
positive operator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. The definition of the fundamental solution thus implies that, if the Laplacian of ''u'' is integrated over any volume that encloses the source point, then :$\iiint_V \nabla \cdot \nabla u \, dV =-1.$ The Laplace equation is unchanged under a rotation of coordinates, and hence we can expect that a fundamental solution may be obtained among solutions that only depend upon the distance ''r'' from the source point. If we choose the volume to be a ball of radius ''a'' around the source point, then Gauss' divergence theorem implies that :$-1= \iiint_V \nabla \cdot \nabla u \, dV = \iint_S \frac \, dS = \left.4\pi a^2 \frac\_.$ It follows that :$\frac = -\frac,$ on a sphere of radius ''r'' that is centered on the source point, and hence :$u = \frac.$ Note that, with the opposite sign convention (used in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

), this is the
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (ph ...

generated by a
point particle A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, ...
, for an
inverse-square law 420px, S represents the light source, while r represents the measured points. The lines represent the flux emanating from the sources and fluxes. The total number of flux lines depends on the strength of the light source and is constant with in ...

force, arising in the solution of
Poisson equation Poisson's equation is an elliptic partial differential equation Second-order linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key i ...
. A similar argument shows that in two dimensions :$u = -\frac.$ where log(''r'') denotes the
natural logarithm The natural logarithm of a number is its logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
. Note that, with the opposite sign convention, this is the
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (ph ...

generated by a pointlike
sink A sink – also known by other names including sinker, washbowl, hand basin, wash basin, and simply basin – is a bowl-shaped plumbing fixture A plumbing fixture is an exchangeable device which can be connected to a plumbing system to deli ...

(see
point particle A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, ...
), which is the solution of the
Euler equations 200px, Leonhard Euler (1707–1783) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
in two-dimensional
incompressible flow In fluid mechanics Fluid mechanics is the branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the o ...

.

## Green's function

A
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous ordinary differential equation, inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means th ...
is a fundamental solution that also satisfies a suitable condition on the boundary ''S'' of a volume ''V''. For instance, :$G\left(x,y,z;x\text{'},y\text{'},z\text{'}\right)$ may satisfy :$\nabla \cdot \nabla G = -\delta\left(x-x\text{'},y-y\text{'},z-z\text{'}\right) \qquad \hbox V,$ :$G = 0 \quad \hbox \quad \left(x,y,z\right) \qquad \hbox S.$ Now if ''u'' is any solution of the Poisson equation in ''V'': :$\nabla \cdot \nabla u = -f,$ and ''u'' assumes the boundary values ''g'' on ''S'', then we may apply Green's identity, (a consequence of the divergence theorem) which states that : The notations ''un'' and ''Gn'' denote normal derivatives on ''S''. In view of the conditions satisfied by ''u'' and ''G'', this result simplifies to :$u\left(x\text{'},y\text{'},z\text{'}\right) = \iiint_V G f \, dV + \iint_S G_n g \, dS. \,$ Thus the Green's function describes the influence at of the data ''f'' and ''g''. For the case of the interior of a sphere of radius ''a'', the Green's function may be obtained by means of a reflection : the source point ''P'' at distance ''ρ'' from the center of the sphere is reflected along its radial line to a point ''P that is at a distance :$\rho\text{'} = \frac. \,$ Note that if ''P'' is inside the sphere, then ''P will be outside the sphere. The Green's function is then given by :$\frac - \frac, \,$ where ''R'' denotes the distance to the source point ''P'' and ''R''′ denotes the distance to the reflected point ''P''′. A consequence of this expression for the Green's function is the Poisson integral formula. Let ''ρ'', ''θ'', and ''φ'' be
spherical coordinates In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

for the source point ''P''. Here ''θ'' denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice. Then the solution of the Laplace equation with Dirichlet boundary values ''g'' inside the sphere is given by :$u\left(P\right) =\frac a^3\left\left(1-\frac\right\right) \int_0^\int_0^ \frac d\theta\text{'} \, d\varphi\text{'}$ where :$\cos \Theta = \cos \theta \cos \theta\text{'} + \sin\theta \sin\theta\text{'}\cos\left(\varphi -\varphi\text{'}\right)$ is the cosine of the angle between and . A simple consequence of this formula is that if ''u'' is a harmonic function, then the value of ''u'' at the center of the sphere is the mean value of its values on the sphere. This mean value property immediately implies that a non-constant harmonic function cannot assume its maximum value at an interior point.

## Laplace's spherical harmonics

Laplace's equation in
spherical coordinates In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

is: : $\nabla^2 f = \frac \frac\left\left(r^2 \frac\right\right) + \frac \frac\left\left(\sin\theta \frac\right\right) + \frac \frac = 0.$ Consider the problem of finding solutions of the form . By
separation of variables In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

, two differential equations result by imposing Laplace's equation: : $\frac\frac\left\left(r^2\frac\right\right) = \lambda,\qquad \frac\frac\frac\left\left(\sin\theta \frac\right\right) + \frac\frac\frac = -\lambda.$ The second equation can be simplified under the assumption that has the form . Applying separation of variables again to the second equation gives way to the pair of differential equations : $\frac \frac = -m^2$ : $\lambda\sin^2\theta + \frac \frac \left\left(\sin\theta \frac\right\right) = m^2$ for some number . A priori, is a complex constant, but because must be a
periodic function A periodic function is a Function (mathematics), function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used th ...

whose period evenly divides , is necessarily an integer and is a linear combination of the complex exponentials . The solution function is regular at the poles of the sphere, where . Imposing this regularity in the solution of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter to be of the form for some non-negative integer with ; this is also explained
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
in terms of the orbital angular momentum. Furthermore, a change of variables transforms this equation into the Legendre equation, whose solution is a multiple of the
associated Legendre polynomial In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation :(1 - x^2) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, or equivalently :\fra ...
. Finally, the equation for has solutions of the form ; requiring the solution to be regular throughout forces . Here the solution was assumed to have the special form . For a given value of , there are independent solutions of this form, one for each integer with . These angular solutions are a product of
trigonometric function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s, here represented as a
complex exponential Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler ...

, and associated Legendre polynomials: : $Y_\ell^m \left(\theta, \varphi \right) = N e^ P_\ell^m \left(\cos \right)$ which fulfill : $r^2\nabla^2 Y_\ell^m \left(\theta, \varphi \right) = -\ell \left(\ell + 1 \right) Y_\ell^m \left(\theta, \varphi \right).$ Here is called a spherical harmonic function of degree and order , is an
associated Legendre polynomial In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation :(1 - x^2) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, or equivalently :\fra ...
, is a normalization constant, and and represent colatitude and longitude, respectively. In particular, the
colatitudeIn a spherical coordinate system File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convent ...
, or polar angle, ranges from at the North Pole, to at the Equator, to at the South Pole, and the
longitude Longitude (, ) is a geographic coordinate A geographic coordinate system (GCS) is a coordinate system associated with position (geometry), positions on Earth (geographic position). A GCS can give positions: *as Geodetic coordinates, ...

, or
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Al ...

, may assume all values with . For a fixed integer , every solution of the eigenvalue problem : $r^2\nabla^2 Y = -\ell \left(\ell + 1 \right) Y$ is a
linear combination In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...
of . In fact, for any such solution, is the expression in spherical coordinates of a
homogeneous polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
that is harmonic (see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
), and so counting dimensions shows that there are linearly independent such polynomials. The general solution to Laplace's equation in a ball centered at the origin is a
linear combination In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...
of the spherical harmonic functions multiplied by the appropriate scale factor , : $f\left(r, \theta, \varphi\right) = \sum_^\infty \sum_^\ell f_\ell^m r^\ell Y_\ell^m \left(\theta, \varphi \right),$ where the are constants and the factors are known as
solid harmonics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...
. Such an expansion is valid in the
ball A ball is a round object (usually spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional s ...
: $r < R = \frac.$ For $r > R$, the solid harmonics with negative powers of $r$ are chosen instead. In that case, one needs to expand the solution of known regions in
Laurent series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

(about $r=\infty$), instead of
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
(about $r=0$), to match the terms and find $f^m_\ell$.

## Electrostatics

Let $\mathbf$ be the electric field, $\rho$ be the electric charge density, and $\varepsilon_0$ be the permittivity of free space. Then
Gauss's law In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
for electricity (Maxwell's first equation) in differential form statesGriffiths, David J. ''Introduction to Electrodynamics''. Fourth ed., Pearson, 2013. Chapter 2: Electrostatics. p. 83-4. . :$\nabla \cdot \mathbf = \frac.$ Now, the electric field can be expressed as the negative gradient of the electric potential $V$, :$\mathbf E=-\nabla V,$ if the field is irrotational, $\nabla \times \mathbf = \mathbf$. The irrotationality of $\mathbf$ is also known as the electrostatic condition. :$\nabla\cdot\mathbf E=\nabla\cdot\left(-\nabla V\right)=-\nabla^2 V$ :$\nabla^2V=-\nabla\cdot\mathbf E$ Plugging this relation into Gauss's law, we obtain Poisson's equation for electricity, :$\nabla^2V = -\frac.$ In the particular case of a source-free region, $\rho = 0$ and Poisson's equation reduces to Laplace's equation for the electric potential. If the electrostatic potential $V$ is specified on the boundary of a region $\mathcal$, then it is uniquely determined. If $\mathcal$ is surrounded by a conducting material with a specified charge density $\rho$, and if the total charge $Q$ is known, then $V$ is also unique.Griffiths, David J. ''Introduction to Electrodynamics''. Fourth ed., Pearson, 2013. Chapter 3: Potentials. p. 119-121. . A potential that doesn't satisfy Laplace's equation together with the boundary condition is an invalid electrostatic potential.

# Gravitation

Let $\mathbf$ be the gravitational field, $\rho$ the mass density, and $G$ the gravitational constant. Then Gauss's law for gravitation in differential form is :$\nabla\cdot\mathbf g=-4\pi G\rho.$ The gravitational field is conservative and can therefore be expressed as the negative gradient of the gravitational potential: :$\mathbf g=-\nabla V,$ :$\nabla\cdot\mathbf g=\nabla\cdot\left(-\nabla V\right)=-\nabla^2V,$ :$\implies\nabla^2V=-\nabla\cdot\mathbf g.$ Using the differential form of Gauss's law of gravitation, we have :$\nabla^2V=4\pi G\rho,$ which is Poisson's equation for gravitational fields. In empty space, $\rho=0$ and we have :$\nabla^2V=0,$ which is Laplace's equation for gravitational fields.

# In the Schwarzschild metric

S. Persides solved the Laplace equation in Schwarzschild spacetime on hypersurfaces of constant ''t''. Using the canonical variables ''r'', ''θ'', ''φ'' the solution is :$\Psi\left(r,\theta,\varphi\right)=R\left(r\right)Y_l\left(\theta,\varphi\right),$ where is a , and :$R\left(r\right)=\left(-1\right)^l\fracP_l\left\left(1-\frac\right\right)+\left(-1\right)^\fracQ_l\left\left(1-\frac\right\right).$ Here ''Pl'' and ''Ql'' are
Legendre functions In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated L ...
of the first and second kind, respectively, while ''rs'' is the
Schwarzschild radius The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius In classical geometry Geometry ...
. The parameter ''l'' is an arbitrary non-negative integer.

* 6-sphere coordinates, a coordinate system under which Laplace's equation becomes ''R''-separable *
Helmholtz equation In mathematics, the eigenvalue In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their r ...
, a general case of Laplace's equation. *
Spherical harmonic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
Potential theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
*
Potential flow In fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other g ...

* Bateman transform *
Earnshaw's theorem Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary mechanical equilibrium, equilibrium configuration solely by the electrostatic interaction of the charges. This was first proven by British math ...
uses the Laplace equation to show that stable static ferromagnetic suspension is impossible *
Vector Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where ...
*
Fundamental solution In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...