Template:Short description Template:For Template:Use American English Template:Complex analysis sidebar In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as <math display="block"> \nabla^2\! f = 0 </math> or <math display="block"> \Delta f = 0,</math> where <math> \Delta = \nabla \cdot \nabla = \nabla^2</math> is the Laplace operator,<ref group="note">The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, <math>\Delta x = x_1 - x_2</math>. Its use to represent the Laplacian should not be confused with this use.</ref> <math>\nabla \cdot</math> is the divergence operator (also symbolized "div"), <math>\nabla</math> is the gradient operator (also symbolized "grad"), and <math>f (x, y, z)</math> 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, <math>h(x, y, z)</math>, we have <math display="block">\Delta f = h</math>
This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation.
The general theory of solutions to Laplace's equation is known as potential theory. The twice continuously differentiable solutions of Laplace's equation are the harmonic functions,<ref>Stewart, James. Calculus : Early Transcendentals. 7th ed., Brooks/Cole, Cengage Learning, 2012. Chapter 14: Partial Derivatives. p. 908. Template:ISBN.</ref> which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation.<ref>Zill, Dennis G, and Michael R Cullen. Differential Equations with Boundary-Value Problems. 8th edition / ed., Brooks/Cole, Cengage Learning, 2013. Chapter 12: Boundary-value Problems in Rectangular Coordinates. p. 462. Template:ISBN.</ref> In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.
Forms in different coordinate systemsEdit
In rectangular coordinates,<ref name="Griffiths">Griffiths, David J. Introduction to Electrodynamics. 4th ed., Pearson, 2013. Inner front cover. Template:ISBN.</ref> <math display="block"> \nabla^2 f = \frac{\partial^2 f}{\partial x^2 } + \frac{\partial^2 f}{\partial y^2 } + \frac{\partial^2 f}{\partial z^2 } = 0.</math>
In cylindrical coordinates,<ref name="Griffiths"/> <math display="block">\nabla^2 f=\frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2} = 0.</math>
In spherical coordinates, using the <math>(r, \theta, \varphi)</math> convention,<ref name="Griffiths"/> <math display="block"> \nabla^2 f = \frac{1}{r^2}\frac{\partial}{\partial r} \left(r^2 \frac{\partial f}{\partial r}\right) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \left(\sin\theta \frac{\partial f}{\partial \theta}\right) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 f}{\partial \varphi^2} =0.</math>
More generally, in arbitrary curvilinear coordinates Template:Math, <math display="block"> \nabla^2 f =\frac{\partial}{\partial \xi^j}\left(\frac{\partial f}{\partial \xi^k}g^{kj}\right) + \frac{\partial f}{\partial \xi^j} g^{jm}\Gamma^n_{mn} =0,</math> or <math display="block"> \nabla^2 f = \frac{1}{\sqrt{|g|}} \frac{\partial}{\partial \xi^i}\!\left(\sqrt{|g|}g^{ij} \frac{\partial f}{\partial \xi^j}\right) =0, \qquad (g=\det\{g_{ij}\})</math> where Template:Math is the Euclidean metric tensor relative to the new coordinates and Template:Math denotes its Christoffel symbols.
Boundary conditionsEdit
Template:See also The Dirichlet problem for Laplace's equation consists of finding a solution Template:Math on some domain Template:Mvar such that Template:Math on the boundary of Template:Mvar is equal to some given function. Since the Laplace operator appears in the heat equation, 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 does not change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem.
The Neumann boundary conditions for Laplace's equation specify not the function Template:Math itself on the boundary of Template:Mvar but its normal derivative. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of Template:Math 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 Template:Math is zero.
Solutions of Laplace's equation are called harmonic functions; 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 dimensionsEdit
Laplace's equation in two independent variables in rectangular coordinates has the form <math display="block">\frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial y^2} \equiv \psi_{xx} + \psi_{yy} = 0.</math>
Analytic functionsEdit
The real and imaginary parts of a complex analytic function both satisfy the Laplace equation. That is, if Template:Math, and if <math display="block">f(z) = u(x,y) + iv(x,y),</math> then the necessary condition that Template:Math be analytic is that Template:Math and Template:Mvar be differentiable and that the Cauchy–Riemann equations be satisfied: <math display="block">u_x = v_y, \quad v_x = -u_y.</math> where Template:Math is the first partial derivative of Template:Math with respect to Template:Mvar. It follows that <math display="block">u_{yy} = (-v_x)_y = -(v_y)_x = -(u_x)_x.</math> Therefore Template:Math satisfies the Laplace equation. A similar calculation shows that Template:Math also satisfies the Laplace equation. Conversely, given a harmonic function, it is the real part of an analytic function, Template:Math (at least locally). If a trial form is <math display="block">f(z) = \varphi(x,y) + i \psi(x,y),</math> then the Cauchy–Riemann equations will be satisfied if we set <math display="block">\psi_x = -\varphi_y, \quad \psi_y = \varphi_x.</math> This relation does not determine Template:Math, but only its increments: <math display="block">d \psi = -\varphi_y\, dx + \varphi_x\, dy.</math> The Laplace equation for Template:Math implies that the integrability condition for Template:Math is satisfied: <math display="block">\psi_{xy} = \psi_{yx},</math> and thus Template:Math may be defined by a line integral. The integrability condition and Stokes' theorem 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 Template:Mvar and Template:Mvar are polar coordinates and <math display="block">\varphi = \log r,</math> then a corresponding analytic function is <math display="block">f(z) = \log z = \log r + i\theta.</math>
However, the angle Template:Mvar 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, which generally have less regularityTemplate:Citation needed.
There is an intimate connection between power series and Fourier series. If we expand a function Template:Math in a power series inside a circle of radius Template:Mvar, this means that <math display="block">f(z) = \sum_{n=0}^\infty c_n z^n,</math> with suitably defined coefficients whose real and imaginary parts are given by <math display="block">c_n = a_n + i b_n.</math> Therefore <math display="block">f(z) = \sum_{n=0}^\infty \left[ a_n r^n \cos n \theta - b_n r^n \sin n \theta\right] + i \sum_{n=1}^\infty \left[ a_n r^n \sin n\theta + b_n r^n \cos n \theta\right],</math> which is a Fourier series for Template:Math. These trigonometric functions can themselves be expanded, using multiple angle formulae.
Fluid flowEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Let the quantities Template:Math and Template:Math 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 <math display="block">u_x + v_y=0,</math> and the condition that the flow be irrotational is that <math display="block">\nabla \times \mathbf{V} = v_x - u_y = 0.</math> If we define the differential of a function Template:Math by <math display="block">d \psi = u \, dy - v \, dx,</math> then the continuity condition is the integrability condition for this differential: the resulting function is called the stream function because it is constant along flow lines. The first derivatives of Template:Math are given by <math display="block">\psi_x = -v, \quad \psi_y=u,</math> and the irrotationality condition implies that Template:Math satisfies the Laplace equation. The harmonic function Template:Math that is conjugate to Template:Math is called the velocity potential. The Cauchy–Riemann equations imply that <math display="block">\varphi_x=\psi_y=u, \quad \varphi_y=-\psi_x=v.</math> 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.
ElectrostaticsEdit
According to Maxwell's equations, an electric field Template:Math in two space dimensions that is independent of time satisfies <math display="block">\nabla \times (u,v,0) = (v_x -u_y)\hat{\mathbf{k}} = \mathbf{0},</math> and <math display="block">\nabla \cdot (u,v) = \rho,</math> where Template:Math is the charge density. The first Maxwell equation is the integrability condition for the differential <math display="block">d \varphi = -u\, dx -v\, dy,</math> so the electric potential Template:Math may be constructed to satisfy <math display="block">\varphi_x = -u, \quad \varphi_y = -v.</math> The second of Maxwell's equations then implies that <math display="block">\varphi_{xx} + \varphi_{yy} = -\rho,</math> which is the Poisson equation. The Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions.
In three dimensionsEdit
Fundamental solutionEdit
A fundamental solution of Laplace's equation satisfies <math display="block"> \Delta u = u_{xx} + u_{yy} + u_{zz} = -\delta(x-x',y-y',z-z'),</math> where the Dirac delta function Template:Math denotes a unit source concentrated at the point Template:Math. No function has this property: in fact it is a distribution 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). 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. The definition of the fundamental solution thus implies that, if the Laplacian of Template:Math is integrated over any volume that encloses the source point, then <math display="block"> \iiint_V \nabla \cdot \nabla u \, dV =-1.</math>
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 Template:Mvar from the source point. If we choose the volume to be a ball of radius Template:Mvar around the source point, then Gauss's divergence theorem implies that <math display="block"> -1= \iiint_V \nabla \cdot \nabla u \, dV = \iint_S \frac{du}{dr} \, dS = \left.4\pi a^2 \frac{du}{dr}\right|_{r=a}.</math>
It follows that <math display="block"> \frac{du}{dr} = -\frac{1}{4\pi r^2},</math> on a sphere of radius Template:Mvar that is centered on the source point, and hence <math display="block"> u = \frac{1}{4\pi r}.</math>
Note that, with the opposite sign convention (used in physics), this is the potential generated by a point particle, for an inverse-square law force, arising in the solution of Poisson equation. A similar argument shows that in two dimensions <math display="block"> u = -\frac{\log(r)}{2\pi}.</math> where Template:Math denotes the natural logarithm. Note that, with the opposite sign convention, this is the potential generated by a pointlike sink (see point particle), which is the solution of the Euler equations in two-dimensional incompressible flow.
Green's functionEdit
A Green's function is a fundamental solution that also satisfies a suitable condition on the boundary Template:Mvar of a volume Template:Mvar. For instance, <math display="block">G(x,y,z;x',y',z')</math> may satisfy <math display="block"> \nabla \cdot \nabla G = -\delta(x-x',y-y',z-z') \qquad \text{in } V,</math> <math display="block"> G = 0 \quad \text{if} \quad (x,y,z) \qquad \text{on } S.</math>
Now if Template:Math is any solution of the Poisson equation in Template:Mvar: <math display="block"> \nabla \cdot \nabla u = -f,</math>
and Template:Math assumes the boundary values Template:Math on Template:Mvar, then we may apply Green's identity, (a consequence of the divergence theorem) which states that
<math display="block"> \iiint_V \left[ G \, \nabla \cdot \nabla u - u \, \nabla \cdot \nabla G \right]\, dV = \iiint_V \nabla \cdot \left[ G \nabla u - u \nabla G \right]\, dV = \iint_S \left[ G u_n -u G_n \right] \, dS. \,</math>
The notations un and Gn denote normal derivatives on Template:Math. In view of the conditions satisfied by Template:Math and Template:Math, this result simplifies to
<math display="block"> u(x',y',z') = \iiint_V G f \, dV - \iint_S G_n g \, dS. \,</math>
Thus the Green's function describes the influence at Template:Math of the data Template:Math and Template:Math. For the case of the interior of a sphere of radius Template:Math, the Green's function may be obtained by means of a reflection Template:Harv: the source point Template:Math at distance Template:Math from the center of the sphere is reflected along its radial line to a point P' that is at a distance
<math display="block"> \rho' = \frac{a^2}{\rho}. \,</math>
Note that if Template:Math is inside the sphere, then P′ will be outside the sphere. The Green's function is then given by <math display="block"> \frac{1}{4 \pi R} - \frac{a}{4 \pi \rho R'}, \,</math> where Template:Mvar denotes the distance to the source point Template:Mvar and Template:Math denotes the distance to the reflected point P′. A consequence of this expression for the Green's function is the Poisson integral formula. Let Template:Mvar, Template:Mvar, and Template:Mvar be spherical coordinates for the source point Template:Math. Here Template:Mvar 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 Template:Math inside the sphere is given by Template:Harv <math display="block">u(P) =\frac{1}{4\pi} a^3\left(1-\frac{\rho^2}{a^2}\right) \int_0^{2\pi}\int_0^{\pi} \frac{g(\theta',\varphi') \sin \theta'}{(a^2 + \rho^2 - 2 a \rho \cos \Theta)^{\frac{3}{2}}} d\theta' \, d\varphi'</math> where <math display="block"> \cos \Theta = \cos \theta \cos \theta' + \sin\theta \sin\theta'\cos(\varphi -\varphi')</math> is the cosine of the angle between Template:Math and Template:Math. A simple consequence of this formula is that if Template:Math is a harmonic function, then the value of Template:Math 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 harmonicsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
Laplace's equation in spherical coordinates is:<ref>The approach to spherical harmonics taken here is found in Template:Harv.</ref>
<math display="block"> \nabla^2 f = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial f}{\partial r}\right)
+ \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta}\left(\sin\theta \frac{\partial f}{\partial \theta}\right)
+ \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 f}{\partial \varphi^2} = 0.</math>
Consider the problem of finding solutions of the form Template:Math. By separation of variables, two differential equations result by imposing Laplace's equation:
<math display="block">\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) = \lambda,\qquad \frac{1}{Y}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta \frac{\partial Y}{\partial\theta}\right) + \frac{1}{Y}\frac{1}{\sin^2\theta}\frac{\partial^2Y}{\partial\varphi^2} = -\lambda.</math>
The second equation can be simplified under the assumption that Template:Math has the form Template:Math. Applying separation of variables again to the second equation gives way to the pair of differential equations
<math display="block">\frac{1}{\Phi} \frac{d^2 \Phi}{d\varphi^2} = -m^2</math> <math display="block">\lambda\sin^2\theta + \frac{\sin\theta}{\Theta} \frac{d}{d\theta} \left(\sin\theta \frac{d\Theta}{d\theta}\right) = m^2</math>
for some number Template:Math. A priori, Template:Math is a complex constant, but because Template:Math must be a periodic function whose period evenly divides Template:Math, Template:Math is necessarily an integer and Template:Math is a linear combination of the complex exponentials Template:Math. The solution function Template:Math is regular at the poles of the sphere, where Template:Math. Imposing this regularity in the solution Template:Math of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter Template:Math to be of the form Template:Math for some non-negative integer with Template:Math; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables Template:Math transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial Template:Math . Finally, the equation for Template:Math has solutions of the form Template:Math; requiring the solution to be regular throughout Template:Math forces Template:Math.<ref group=note>Physical applications often take the solution that vanishes at infinity, making Template:Math. This does not affect the angular portion of the spherical harmonics.</ref>
Here the solution was assumed to have the special form Template:Math. For a given value of Template:Math, there are Template:Math independent solutions of this form, one for each integer Template:Math with Template:Math. These angular solutions are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: <math display="block"> Y_\ell^m (\theta, \varphi ) = N e^{i m \varphi } P_\ell^m (\cos{\theta} )</math> which fulfill <math display="block"> r^2\nabla^2 Y_\ell^m (\theta, \varphi ) = -\ell (\ell + 1 ) Y_\ell^m (\theta, \varphi ).</math>
Here Template:Math is called a spherical harmonic function of degree Template:Mvar and order Template:Mvar, Template:Math is an associated Legendre polynomial, Template:Math is a normalization constant, and Template:Mvar and Template:Mvar represent colatitude and longitude, respectively. In particular, the colatitude Template:Mvar, or polar angle, ranges from Template:Math at the North Pole, to Template:Math at the Equator, to Template:Math at the South Pole, and the longitude Template:Mvar, or azimuth, may assume all values with Template:Math. For a fixed integer Template:Mvar, every solution Template:Math of the eigenvalue problem <math display="block"> r^2\nabla^2 Y = -\ell (\ell + 1 ) Y</math> is a linear combination of Template:Math. In fact, for any such solution, Template:Math is the expression in spherical coordinates of a homogeneous polynomial that is harmonic (see below), and so counting dimensions shows that there are Template:Math linearly independent such polynomials.
The general solution to Laplace's equation in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor Template:Math, <math display="block"> f(r, \theta, \varphi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m r^\ell Y_\ell^m (\theta, \varphi ), </math> where the Template:Math are constants and the factors Template:Math are known as solid harmonics. Such an expansion is valid in the ball <math display="block"> r < R = \frac{1}{\limsup_{\ell\to\infty} |f_\ell^m|^{{1}/{\ell}}}.</math>
For <math> r > R</math>, the solid harmonics with negative powers of <math>r</math> are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about <math>r=\infty</math>), instead of Taylor series (about <math>r = 0</math>), to match the terms and find <math>f^m_\ell</math>.
Electrostatics and magnetostaticsEdit
Let <math>\mathbf{E}</math> be the electric field, <math>\rho</math> be the electric charge density, and <math>\varepsilon_0</math> be the permittivity of free space. Then Gauss's law for electricity (Maxwell's first equation) in differential form states<ref name="Griffiths-2">Griffiths, David J. Introduction to Electrodynamics. 4th ed., Pearson, 2013. Chapter 2: Electrostatics. p. 83-4. Template:ISBN.</ref> <math display="block">\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}.</math>
Now, the electric field can be expressed as the negative gradient of the electric potential <math>V</math>, <math display="block">\mathbf E=-\nabla V,</math> if the field is irrotational, <math>\nabla \times \mathbf{E} = \mathbf{0}</math>. The irrotationality of <math>\mathbf{E}</math> is also known as the electrostatic condition.<ref name="Griffiths-2"/>
<math display="block">\nabla\cdot\mathbf E = \nabla\cdot(-\nabla V)=-\nabla^2 V</math> <math display="block">\nabla^2 V = -\nabla\cdot\mathbf E</math>
Plugging this relation into Gauss's law, we obtain Poisson's equation for electricity,<ref name="Griffiths-2"/> <math display="block">\nabla^2 V = -\frac{\rho}{\varepsilon_0}.</math>
In the particular case of a source-free region, <math>\rho = 0</math> and Poisson's equation reduces to Laplace's equation for the electric potential.<ref name="Griffiths-2"/>
If the electrostatic potential <math>V</math> is specified on the boundary of a region <math>\mathcal{R}</math>, then it is uniquely determined. If <math>\mathcal{R}</math> is surrounded by a conducting material with a specified charge density <math>\rho</math>, and if the total charge <math>Q</math> is known, then <math>V</math> is also unique.<ref name="Griffiths-3">Griffiths, David J. Introduction to Electrodynamics. 4th ed., Pearson, 2013. Chapter 3: Potentials. p. 119-121. Template:ISBN.</ref>
For the magnetic field, when there is no free current, <math display="block">\nabla\times\mathbf{H} = \mathbf{0}.</math>We can thus define a magnetic scalar potential, Template:Math, as <math display="block">\mathbf{H} = -\nabla\psi.</math>With the definition of Template:Math: <math display="block">\nabla\cdot\mathbf{B} = \mu_{0}\nabla\cdot\left(\mathbf{H} + \mathbf{M}\right) = 0,</math> it follows that <math display="block">\nabla^2 \psi = -\nabla\cdot\mathbf{H} = \nabla\cdot\mathbf{M}.</math>
Similar to electrostatics, in a source-free region, <math>\mathbf{M} = 0</math> and Poisson's equation reduces to Laplace's equation for the magnetic scalar potential , <math display="block">\nabla^2 \psi = 0</math>
A potential that does not satisfy Laplace's equation together with the boundary condition is an invalid electrostatic or magnetic scalar potential.
GravitationEdit
Let <math>\mathbf{g}</math> be the gravitational field, <math>\rho</math> the mass density, and <math>G</math> the gravitational constant. Then Gauss's law for gravitation in differential form is<ref name=":0">Template:Cite journal</ref> <math display="block">\nabla\cdot\mathbf g=-4\pi G\rho.</math>
The gravitational field is conservative and can therefore be expressed as the negative gradient of the gravitational potential: <math display="block">\begin{align} \mathbf g &= -\nabla V, \\ \nabla\cdot\mathbf g &= \nabla\cdot(-\nabla V) = -\nabla^2 V, \\ \implies\nabla^2 V &= -\nabla\cdot\mathbf g. \end{align}</math>
Using the differential form of Gauss's law of gravitation, we have <math display="block">\nabla^2 V = 4\pi G\rho,</math> which is Poisson's equation for gravitational fields.<ref name=":0" />
In empty space, <math>\rho=0</math> and we have <math display="block">\nabla^2 V = 0,</math> which is Laplace's equation for gravitational fields.
In the Schwarzschild metricEdit
S. Persides<ref>Template:Cite journal</ref> solved the Laplace equation in Schwarzschild spacetime on hypersurfaces of constant Template:Mvar. Using the canonical variables Template:Mvar, Template:Mvar, Template:Mvar the solution is <math display="block">\Psi(r,\theta,\varphi) = R(r)Y_l(\theta,\varphi),</math> where Template:Math is a spherical harmonic function, and <math display="block"> R(r) = (-1)^l\frac{(l!)^2r_s^l}{(2l)!}P_l\left(1-\frac{2r}{r_s}\right)+(-1)^{l+1}\frac{2(2l+1)!}{(l)!^2r_s^{l+1}}Q_l\left(1-\frac{2r}{r_s}\right). </math>
Here Template:Math and Template:Math are Legendre functions of the first and second kind, respectively, while Template:Math is the Schwarzschild radius. The parameter Template:Mvar is an arbitrary non-negative integer.
See alsoEdit
- 6-sphere coordinates, a coordinate system under which Laplace's equation becomes R-separable
- Helmholtz equation, a generalization of Laplace's equation
- Spherical harmonic
- Quadrature domains
- Potential theory
- Potential flow
- Bateman transform
- Earnshaw's theorem uses the Laplace equation to show that stable static ferromagnetic suspension is impossible
- Vector Laplacian
- Fundamental solution
NotesEdit
<references group="note"/>
ReferencesEdit
SourcesEdit
Further readingEdit
External linksEdit
- Template:Springer
- Laplace Equation (particular solutions and boundary value problems) at EqWorld: The World of Mathematical Equations.
- Example initial-boundary value problems using Laplace's equation from exampleproblems.com.
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
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