Differential equation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quadratic_irrational_number

Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are soluble by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.

Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

History

Differential equations came into existence with the invention of calculus by Isaac Newton and Gottfried Leibniz. In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum, Newton listed three kinds of differential equations:

${\displaystyle \begin{array}{rl}\frac{dy}{dx}& =f(x)\\ \frac{dy}{dx}& =f(x,y)\\ x_1\frac{\mathrm{\partial }y}{\mathrm{\partial }{x}_1}& +x_2\frac{\mathrm{\partial }y}{\mathrm{\partial }{x}_2}=y\end{array}}$

In all these cases, y is an unknown function of x (or of x₁ and x₂), and f is a given function.

He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.

Jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is an ordinary differential equation of the form

${\displaystyle y^{\prime }+P(x)y=Q(x)y^n}$

for which the following year Leibniz obtained solutions by simplifying it.

Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.^[5][6][7][8] In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.

The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics.

In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat), in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his heat equation for conductive diffusion of heat. This partial differential equation is now a common part of mathematical physics curriculum.

Example

In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time.

In some cases, this differential equation (called an equation of motion) may be solved explicitly.

An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.

Types

Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

Ordinary differential equations

Main articles: Ordinary differential equation and Linear differential equation

An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.

Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of integrals.

Most ODEs that are encountered in physics are linear. Therefore, most special functions may be defined as solutions of linear differential equations (see Holonomic function).

As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical methods are commonly used for solving differential equations on a computer.

Partial differential equations

Main article: Partial differential equation

A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant computer model.

PDEs can be used to describe a wide variety of phenomena in nature such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. Stochastic partial differential equations generalize partial differential equations for modeling randomness.

Non-linear differential equations

Main article: Non-linear differential equations

A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations.

Equation order and degree

The order of the differential equation is the highest order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only first-order derivatives is a first-order differential equation, an equation containing the second-order derivative is a second-order differential equation, and so on.

When it is written as a polynomial equation in the unknown function and its derivatives, its degree of the differential equation is, depending on the context, the polynomial degree in the highest derivative of the unknown function, or its total degree in the unknown function and its derivatives. In particular, a linear differential equation has degree one for both meanings, but the non-linear differential equation ${\displaystyle y^{\prime }+y^2=0}$ is of degree one for the first meaning but not for the second one.

Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin-film equation, which is a fourth order partial differential equation.

Examples

In the first group of examples u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones.

• Heterogeneous first-order linear constant coefficient ordinary differential equation:

  ${\displaystyle \frac{du}{dx}=cu+x^2.}$

• Homogeneous second-order linear ordinary differential equation:

  ${\displaystyle \frac{d^{2}u}{d{x}^2}-x\frac{du}{dx}+u=0.}$

• Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:

  ${\displaystyle \frac{d^{2}u}{d{x}^2}+{\omega }^{2}u=0.}$

• Heterogeneous first-order nonlinear ordinary differential equation:

  ${\displaystyle \frac{du}{dx}=u^2+4.}$

• Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length L:

  ${\displaystyle L\frac{d^{2}u}{d{x}^2}+g\sin u=0.}$

In the next group of examples, the unknown function u depends on two variables x and t or x and y.

• Homogeneous first-order linear partial differential equation:

  ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+t\frac{\mathrm{\partial }u}{\mathrm{\partial }x}=0.}$

• Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:

  ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}=0.}$

• Homogeneous third-order non-linear partial differential equation, the KdV equation:

  ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}=6u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}-\frac{{\mathrm{\partial }}^{3}u}{\mathrm{\partial }{x}^3}.}$

Existence of solutions

Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.

For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point ${\displaystyle (a,b)}$ in the xy-plane, define some rectangular region ${\displaystyle Z}$, such that ${\displaystyle Z=[l,m]\times [n,p]}$ and ${\displaystyle (a,b)}$ is in the interior of ${\displaystyle Z}$. If we are given a differential equation $\frac{dy}{dx}=g(x,y)$ and the condition that ${\displaystyle y=b}$ when ${\displaystyle x=a}$, then there is locally a solution to this problem if ${\displaystyle g(x,y)}$ and $\frac{\mathrm{\partial }g}{\mathrm{\partial }x}$ are both continuous on ${\displaystyle Z}$. This solution exists on some interval with its center at ${\displaystyle a}$. The solution may not be unique. (See Ordinary differential equation for other results.)

However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:

${\displaystyle f_n(x)\frac{d^{n}y}{d{x}^n}+\cdots +f_1(x)\frac{dy}{dx}+f_0(x)y=g(x)}$

such that

${\displaystyle \begin{array}{rlrlrlr}y(x_0)& =y_0,& y^{\prime }(x_0)& =y_0^{\prime },& y^{″}(x_0)& =y_0^{″},& \dots \end{array}}$

For any nonzero ${\displaystyle f_n(x)}$, if ${\displaystyle \{f_0,f_1,\dots \}}$ and ${\displaystyle g}$ are continuous on some interval containing ${\displaystyle x_0}$, ${\displaystyle y}$ is unique and exists.

Related concepts

• A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.
• Integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals.
• An integro-differential equation (IDE) is an equation that combines aspects of a differential equation and an integral equation.
• A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion equations.
• A stochastic partial differential equation (SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications in quantum field theory and statistical mechanics.
• An ultrametric pseudo-differential equation is an equation which contains p-adic numbers in an ultrametric space. Mathematical models that involve ultrametric pseudo-differential equations use pseudo-differential operators instead of differential operators.
• A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.

Connection to difference equations

See also: Time scale calculus

The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

Applications

The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.

The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See List of named differential equations.

Software

Some CAS software can solve differential equations. These are the commands used in the leading programs:

• Maple: dsolve
• Mathematica: DSolve[]
• Maxima: ode2(equation, y, x)
• SageMath: desolve()
• SymPy: sympy.solvers.ode.dsolve(equation)
• Xcas: desolve(y'=k*y,y)

Laplace operator

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Laplace_operator

 

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols ${\displaystyle \mathrm{\nabla }\cdot \mathrm{\nabla }}$, ${\displaystyle {\mathrm{\nabla }}^2}$ (where ${\displaystyle \mathrm{\nabla }}$ is the nabla operator), or ${\displaystyle \mathrm{\Delta }}$. 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 Δf (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).

The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution. Solutions of Laplace's equation Δf = 0 are called harmonic functions and represent the possible gravitational potentials in regions of vacuum.

The Laplacian occurs in many differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials; the diffusion equation describes heat and fluid flow; the wave equation describes wave propagation; and the Schrödinger equation describes the wave function in quantum mechanics. In image processing and computer vision, the Laplacian operator has been used for various tasks, such as blob and edge detection. The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology.

Definition

The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence (${\displaystyle \mathrm{\nabla }\cdot }$) of the gradient (${\displaystyle \mathrm{\nabla }f}$). Thus if ${\displaystyle f}$ is a twice-differentiable real-valued function, then the Laplacian of ${\displaystyle f}$ is the real-valued function defined by:

$${\displaystyle \mathrm{\Delta }f={\mathrm{\nabla }}^{2}f=\mathrm{\nabla }\cdot \mathrm{\nabla }f}$$

(1)

where the latter notations derive from formally writing:

$${\displaystyle \mathrm{\nabla }=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}\right).}$$

Explicitly, the Laplacian of f is thus the sum of all the unmixed second partial derivatives in the Cartesian coordinates x_i:

$${\displaystyle \mathrm{\Delta }f=\sum _{i=1}^n\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}_i^2}}$$

(2)

As a second-order differential operator, the Laplace operator maps C^k functions to C^k−2 functions for k ≥ 2. It is a linear operator Δ : C^k(Rⁿ) → C^k−2(Rⁿ), or more generally, an operator Δ : C^k(Ω) → C^k−2(Ω) for any open set Ω ⊆ Rⁿ.

Motivation

Diffusion

In the physical theory of diffusion, the Laplace operator arises naturally in the mathematical description of equilibrium. Specifically, if u is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of u through the boundary ∂V of any smooth region V is zero, provided there is no source or sink within V:

$${\displaystyle {\int }_{\mathrm{\partial }V}\mathrm{\nabla }u\cdot \mathbf{n}dS=0,}$$

where n is the outward unit normal to the boundary of V. By the divergence theorem,

$${\displaystyle {\int }_V\mathrm{div}\mathrm{\nabla }udV={\int }_{\mathrm{\partial }V}\mathrm{\nabla }u\cdot \mathbf{n}dS=0.}$$

Since this holds for all smooth regions V, one can show that it implies:

$${\displaystyle \mathrm{div}\mathrm{\nabla }u=\mathrm{\Delta }u=0.}$$

The left-hand side of this equation is the Laplace operator, and the entire equation Δu = 0 is known as Laplace's equation. Solutions of the Laplace equation, i.e. functions whose Laplacian is identically zero, thus represent possible equilibrium densities under diffusion.

The Laplace operator itself has a physical interpretation for non-equilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, in a sense made precise by the diffusion equation. This interpretation of the Laplacian is also explained by the following fact about averages.

Averages

Given a twice continuously differentiable function ${\displaystyle f:{\mathbb{R}}^n\to \mathbb{R}}$ and a point ${\displaystyle p\in {\mathbb{R}}^n}$. Then, the average value of ${\displaystyle f}$ over the ball with radius ${\displaystyle h}$ centered at ${\displaystyle p}$ is:

$${\displaystyle {\overline{f}}_B(p,h)=f(p)+\frac{\mathrm{\Delta }f(p)}{2(n+2)}{h}^2+o(h^2)\text{for}h\to 0}$$

Similarly, the average value of ${\displaystyle f}$ over the sphere (the boundary of a ball) with radius ${\displaystyle h}$ centered at ${\displaystyle p}$ is:

$${\displaystyle {\overline{f}}_S(p,h)=f(p)+\frac{\mathrm{\Delta }f(p)}{2n}{h}^2+o(h^2)\text{for}h\to 0.}$$

Density associated with a potential

If φ denotes the electrostatic potential associated to a charge distribution q, then the charge distribution itself is given by the negative of the Laplacian of φ:

$${\displaystyle q=-{\epsilon }_0\mathrm{\Delta }\phi ,}$$

where ε₀ is the electric constant.

This is a consequence of Gauss's law. Indeed, if V is any smooth region with boundary ∂V, then by Gauss's law the flux of the electrostatic field E across the boundary is proportional to the charge enclosed:

$${\displaystyle {\int }_{\mathrm{\partial }V}\mathbf{E}\cdot \mathbf{n}dS={\int }_V\mathrm{div}\mathbf{E}dV=\frac{1}{{\epsilon }_0}{\int }_{V}qdV.}$$

where the first equality is due to the divergence theorem. Since the electrostatic field is the (negative) gradient of the potential, this gives:

$${\displaystyle -{\int }_V\mathrm{div}(\mathrm{grad}\phi )dV=\frac{1}{{\epsilon }_0}{\int }_{V}qdV.}$$

Since this holds for all regions V, we must have

$${\displaystyle \mathrm{div}(\mathrm{grad}\phi )=-\frac{1}{{\epsilon }_0}q}$$

The same approach implies that the negative of the Laplacian of the gravitational potential is the mass distribution. Often the charge (or mass) distribution are given, and the associated potential is unknown. Finding the potential function subject to suitable boundary conditions is equivalent to solving Poisson's equation.

Energy minimization

Another motivation for the Laplacian appearing in physics is that solutions to Δf = 0 in a region U are functions that make the Dirichlet energy functional stationary:

$${\displaystyle E(f)=\frac{1}{2}{\int }_U\Vert \mathrm{\nabla }f{\Vert }^{2}dx.}$$

To see this, suppose f : U → R is a function, and u : U → R is a function that vanishes on the boundary of U. Then:

$${\displaystyle {\frac{d}{d\epsilon }|}_{\epsilon =0}E(f+\epsilon u)={\int }_U\mathrm{\nabla }f\cdot \mathrm{\nabla }udx=-{\int }_{U}u\mathrm{\Delta }fdx}$$

where the last equality follows using Green's first identity. This calculation shows that if Δf = 0, then E is stationary around f. Conversely, if E is stationary around f, then Δf = 0 by the fundamental lemma of calculus of variations.

Coordinate expressions

Two dimensions

The Laplace operator in two dimensions is given by:

In Cartesian coordinates,

$${\displaystyle \mathrm{\Delta }f=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{y}^2}}$$

where x and y are the standard Cartesian coordinates of the xy-plane.

In polar coordinates,

$${\displaystyle \begin{array}{rl}\mathrm{\Delta }f& =\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r\frac{\mathrm{\partial }f}{\mathrm{\partial }r}\right)+\frac{1}{r^2}\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{\theta }^2}\\ & =\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{r}^2}+\frac{1}{r}\frac{\mathrm{\partial }f}{\mathrm{\partial }r}+\frac{1}{r^2}\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{\theta }^2},\end{array}}$$

where r represents the radial distance and θ the angle.

Three dimensions

See also: Del in cylindrical and spherical coordinates

In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.

In Cartesian coordinates,

$${\displaystyle \mathrm{\Delta }f=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{z}^2}.}$$

In cylindrical coordinates,

$${\displaystyle \mathrm{\Delta }f=\frac{1}{\rho }\frac{\mathrm{\partial }}{\mathrm{\partial }\rho }\left(\rho \frac{\mathrm{\partial }f}{\mathrm{\partial }\rho }\right)+\frac{1}{{\rho }^2}\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{\phi }^2}+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{z}^2},}$$

where ${\displaystyle \rho }$ represents the radial distance, φ the azimuth angle and z the height.

In spherical coordinates:

$${\displaystyle \mathrm{\Delta }f=\frac{1}{r^2}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r^2\frac{\mathrm{\partial }f}{\mathrm{\partial }r}\right)+\frac{1}{r^2\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta \frac{\mathrm{\partial }f}{\mathrm{\partial }\theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{\phi }^2},}$$

or

$${\displaystyle \mathrm{\Delta }f=\frac{1}{r}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{r}^2}(rf)+\frac{1}{r^2\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta \frac{\mathrm{\partial }f}{\mathrm{\partial }\theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{\phi }^2},}$$

by expanding the first term, these expressions read

$${\displaystyle \mathrm{\Delta }f=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{r}^2}+\frac{2}{r}\frac{\mathrm{\partial }f}{\mathrm{\partial }r}+\frac{1}{r^2\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta \frac{\mathrm{\partial }f}{\mathrm{\partial }\theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{\phi }^2},}$$

where φ represents the azimuthal angle and θ the zenith angle or co-latitude.

In general curvilinear coordinates (ξ¹, ξ², ξ³):

$${\displaystyle \mathrm{\Delta }=\mathrm{\nabla }{\xi }^m\cdot \mathrm{\nabla }{\xi }^n\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\xi }^m\mathrm{\partial }{\xi }^n}+{\mathrm{\nabla }}^2{\xi }^m\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }^m}=g^{mn}\left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\xi }^m\mathrm{\partial }{\xi }^n}-{\mathrm{\Gamma }}_{mn}^l\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }^l}\right),}$$

where summation over the repeated indices is implied, g^mn is the inverse metric tensor and Γ^l _mn are the Christoffel symbols for the selected coordinates.

N dimensions

In arbitrary curvilinear coordinates in N dimensions (ξ¹, ..., ξ^N), we can write the Laplacian in terms of the inverse metric tensor, ${\displaystyle g^{ij}}$:

$${\displaystyle \mathrm{\Delta }=\frac{1}{\sqrt{detg}}\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }^i}\left(\sqrt{detg}{g}^{ij}\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }^j}\right),}$$

from the Voss-Weyl formula^[3] for the divergence.

In spherical coordinates in N dimensions, with the parametrization x = rθ ∈ R^N with r representing a positive real radius and θ an element of the unit sphere S^N−1,

$${\displaystyle \mathrm{\Delta }f=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{r}^2}+\frac{N-1}{r}\frac{\mathrm{\partial }f}{\mathrm{\partial }r}+\frac{1}{r^2}{\mathrm{\Delta }}_{S^{N-1}}f}$$

where Δ_S^N−1 is the Laplace–Beltrami operator on the (N − 1)-sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as:

$${\displaystyle \frac{1}{r^{N-1}}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r^{N-1}\frac{\mathrm{\partial }f}{\mathrm{\partial }r}\right).}$$

As a consequence, the spherical Laplacian of a function defined on S^N−1 ⊂ R^N can be computed as the ordinary Laplacian of the function extended to R^N∖{0} so that it is constant along rays, i.e., homogeneous of degree zero.

Euclidean invariance

The Laplacian is invariant under all Euclidean transformations: rotations and translations. In two dimensions, for example, this means that:

$${\displaystyle \mathrm{\Delta }(f(x\cos \theta -y\sin \theta +a,x\sin \theta +y\cos \theta +b))=(\mathrm{\Delta }f)(x\cos \theta -y\sin \theta +a,x\sin \theta +y\cos \theta +b)}$$

for all θ, a, and b. In arbitrary dimensions,

$${\displaystyle \mathrm{\Delta }(f\circ \rho )=(\mathrm{\Delta }f)\circ \rho }$$

whenever ρ is a rotation, and likewise:

$${\displaystyle \mathrm{\Delta }(f\circ \tau )=(\mathrm{\Delta }f)\circ \tau }$$

whenever τ is a translation. (More generally, this remains true when ρ is an orthogonal transformation such as a reflection.)

In fact, the algebra of all scalar linear differential operators, with constant coefficients, that commute with all Euclidean transformations, is the polynomial algebra generated by the Laplace operator.

Spectral theory

See also: Hearing the shape of a drum and Dirichlet eigenvalue

The spectrum of the Laplace operator consists of all eigenvalues λ for which there is a corresponding eigenfunction f with:

$${\displaystyle -\mathrm{\Delta }f=\lambda f.}$$

This is known as the Helmholtz equation.

If Ω is a bounded domain in Rⁿ, then the eigenfunctions of the Laplacian are an orthonormal basis for the Hilbert space L²(Ω). This result essentially follows from the spectral theorem on compact self-adjoint operators, applied to the inverse of the Laplacian (which is compact, by the Poincaré inequality and the Rellich–Kondrachov theorem). It can also be shown that the eigenfunctions are infinitely differentiable functions. More generally, these results hold for the Laplace–Beltrami operator on any compact Riemannian manifold with boundary, or indeed for the Dirichlet eigenvalue problem of any elliptic operator with smooth coefficients on a bounded domain. When Ω is the n-sphere, the eigenfunctions of the Laplacian are the spherical harmonics.

Vector Laplacian

The vector Laplace operator, also denoted by ${\displaystyle {\mathrm{\nabla }}^2}$, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field, returning a vector quantity. When computed in orthonormal Cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.

The vector Laplacian of a vector field ${\displaystyle \mathbf{A}}$ is defined as

$${\displaystyle {\mathrm{\nabla }}^2\mathbf{A}=\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{A})-\mathrm{\nabla }\times (\mathrm{\nabla }\times \mathbf{A}).}$$

This definition can be seen as the Helmholtz decomposition of the vector Laplacian.

In Cartesian coordinates, this reduces to the much simpler form as

$${\displaystyle {\mathrm{\nabla }}^2\mathbf{A}=({\mathrm{\nabla }}^2{A}_x,{\mathrm{\nabla }}^2{A}_y,{\mathrm{\nabla }}^2{A}_z),}$$

where ${\displaystyle A_x}$, ${\displaystyle A_y}$, and ${\displaystyle A_z}$ are the components of the vector field ${\displaystyle \mathbf{A}}$, and ${\displaystyle {\mathrm{\nabla }}^2}$ just on the left of each vector field component is the (scalar) Laplace operator. This can be seen to be a special case of Lagrange's formula; see Vector triple product.

For expressions of the vector Laplacian in other coordinate systems see Del in cylindrical and spherical coordinates.

Generalization

The Laplacian of any tensor field ${\displaystyle \mathbf{T}}$ ("tensor" includes scalar and vector) is defined as the divergence of the gradient of the tensor:

$${\displaystyle {\mathrm{\nabla }}^2\mathbf{T}=(\mathrm{\nabla }\cdot \mathrm{\nabla })\mathbf{T}.}$$

For the special case where ${\displaystyle \mathbf{T}}$ is a scalar (a tensor of degree zero), the Laplacian takes on the familiar form.

If ${\displaystyle \mathbf{T}}$ is a vector (a tensor of first degree), the gradient is a covariant derivative which results in a tensor of second degree, and the divergence of this is again a vector. The formula for the vector Laplacian above may be used to avoid tensor math and may be shown to be equivalent to the divergence of the Jacobian matrix shown below for the gradient of a vector:

$${\displaystyle \mathrm{\nabla }\mathbf{T}=(\mathrm{\nabla }{T}_x,\mathrm{\nabla }{T}_y,\mathrm{\nabla }{T}_z)=\left[\begin{array}{ccc}{T}_{xx}& T_{xy}& T_{xz}\\ T_{yx}& T_{yy}& T_{yz}\\ T_{zx}& T_{zy}& T_{zz}\end{array}\right],\text{ where }{T}_{uv}\equiv \frac{\mathrm{\partial }{T}_u}{\mathrm{\partial }v}.}$$

And, in the same manner, a dot product, which evaluates to a vector, of a vector by the gradient of another vector (a tensor of 2nd degree) can be seen as a product of matrices:

$${\displaystyle \mathbf{A}\cdot \mathrm{\nabla }\mathbf{B}=\left[\begin{array}{ccc}{A}_x& A_y& A_z\end{array}\right]\mathrm{\nabla }\mathbf{B}=\left[\begin{array}{ccc}\mathbf{A}\cdot \mathrm{\nabla }{B}_x& \mathbf{A}\cdot \mathrm{\nabla }{B}_y& \mathbf{A}\cdot \mathrm{\nabla }{B}_z\end{array}\right].}$$

This identity is a coordinate dependent result, and is not general.

Use in physics

An example of the usage of the vector Laplacian is the Navier-Stokes equations for a Newtonian incompressible flow:

$${\displaystyle \rho \left(\frac{\mathrm{\partial }\mathbf{v}}{\mathrm{\partial }t}+(\mathbf{v}\cdot \mathrm{\nabla })\mathbf{v}\right)=\rho \mathbf{f}-\mathrm{\nabla }p+\mu \left({\mathrm{\nabla }}^2\mathbf{v}\right),}$$

where the term with the vector Laplacian of the velocity field ${\displaystyle \mu \left({\mathrm{\nabla }}^2\mathbf{v}\right)}$ represents the viscous stresses in the fluid.

Another example is the wave equation for the electric field that can be derived from Maxwell's equations in the absence of charges and currents:

$${\displaystyle {\mathrm{\nabla }}^2\mathbf{E}-{\mu }_0{ϵ}_0\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}=0.}$$

This equation can also be written as:

$${\displaystyle ◻\mathbf{E}=0,}$$

where

$${\displaystyle ◻\equiv \frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2,}$$

is the D'Alembertian, used in the Klein–Gordon equation.

Generalizations

A version of the Laplacian can be defined wherever the Dirichlet energy functional makes sense, which is the theory of Dirichlet forms. For spaces with additional structure, one can give more explicit descriptions of the Laplacian, as follows.

Laplace–Beltrami operator

Main article: Laplace–Beltrami operator

The Laplacian also can be generalized to an elliptic operator called the Laplace–Beltrami operator defined on a Riemannian manifold. The Laplace–Beltrami operator, when applied to a function, is the trace (tr) of the function's Hessian:

$${\displaystyle \mathrm{\Delta }f=\mathrm{tr}(H(f))}$$

where the trace is taken with respect to the inverse of the metric tensor. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields, by a similar formula.

Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exterior derivative, in terms of which the "geometer's Laplacian" is expressed as

$${\displaystyle \mathrm{\Delta }f=\delta df.}$$

Here δ is the codifferential, which can also be expressed in terms of the Hodge star and the exterior derivative. This operator differs in sign from the "analyst's Laplacian" defined above. More generally, the "Hodge" Laplacian is defined on differential forms α by

$${\displaystyle \mathrm{\Delta }\alpha =\delta d\alpha +d\delta \alpha .}$$

This is known as the Laplace–de Rham operator, which is related to the Laplace–Beltrami operator by the Weitzenböck identity.

D'Alembertian

The Laplacian can be generalized in certain ways to non-Euclidean spaces, where it may be elliptic, hyperbolic, or ultrahyperbolic.

In Minkowski space the Laplace–Beltrami operator becomes the D'Alembert operator ${\displaystyle ◻}$ or D'Alembertian:

$${\displaystyle ◻=\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}.}$$

It is the generalization of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time-independent functions. The overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the usual convention in high-energy particle physics. The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in the wave equations, and it is also part of the Klein–Gordon equation, which reduces to the wave equation in the massless case.

The additional factor of c in the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, the x direction were measured in meters while the y direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that c = 1 in order to simplify the equation.

The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds.