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Laplacian Operator

Laplacian operator

In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator, with many applications in mathematics and physics. In physics, it is used in modeling of wave propagation and heat flow, forming the Helmholtz equation. It is central in electrostatics, anchoring in Laplace's equation and Poisson's equation. In quantum mechanics, it represents the kinetic energy term of the Schrödinger equation. In mathematics, functions with vanishing Laplacian are called harmonic functions; the Laplacian is at the core of Hodge theory and 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 of the gradient: :

\Delta = \nabla^2 = \nabla \cdot \nabla.

Equivalently, the Laplacian is the sum of all the unmixed second partial derivatives: :

\Delta = \sum_^n \frac .

Here, it is understood that the

x_i

are Cartesian coordinates on the space; the equation takes a different form in spherical coordinates and cylindrical coordinates, as shown below. In the three-dimensional space the Laplacian is commonly written as :

\Delta = 
\frac   +
\frac   +
\frac .

As we shall see later, the Laplacian can be generalized to non-Euclidean spaces, where it may be elliptic or hyperbolic. For example, in the Minkowski space the Laplacian becomes the d'Alembert operator or d'Alembertian :

\square = 
 +
 +
 -
\frac .

The D'Alembert operator is often used to express the Klein-Gordon equation and the four-dimensional wave equation. The sign in front of the fourth term is negative, while it would have been positive in the Euclidean space. The additional factor of c is required because space and time are usually measured in different units; a similar factor would be required if, for example, the x direction was measured in inches, and the y direction was measured in centimeters. Indeed, physicists usually work in units such that c=1 in order to simplify the equation.

Coordinate expressions

In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems. Given a function f, in cylindrical coordinates, one has: :

\Delta f 
=  
  \left( r  \right) 
+  
+ .

In spherical coordinates: :

\Delta f 
=  
  \left( r^2  \right) 
+  
  \left( \sin \theta  \right) 
+  .

The spherical coordinates Laplacian can also be written in this form: :

\Delta f 
=  
  \left( rf \right) 
+  
  \left( \sin \theta  \right) 
+  .

See also the article Nabla in cylindrical and spherical coordinates.

Identities

If f and g are functions, then the Laplacian of the product is given by :

\Delta(fg)=(\Delta f)g+2(\nabla f)\cdot(\nabla g)+f(\Delta g).

Laplace-Beltrami operator

The Laplacian can be extended to functions defined on surfaces, or more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace-Beltrami operator. One defines it, just as the Laplacian, as the divergence of the gradient. To be able to find a formula for this operator, one will need to first write the divergence and the gradient on a manifold. If

g

denotes the (pseudo)-metric tensor on the manifold, one finds that the volume form in local coordinates is given by :

\mathrm_n := \sqrt \;dx^1\wedge \ldots \wedge dx^n

where the

dx^i

are the 1-forms forming the dual basis to the basis vectors :

\partial_i := \frac

for the local coordinate system, and

\wedge

is the wedge product. Here

|g|:=|\det g|

is the absolute value of the determinant of the metric tensor. The divergence of a vector field X on the manifold can then be defined as :

\mathcal_X \mathrm_n = (\mbox X) \; \mathrm_n

where

\mathcal_X

is the Lie derivative along the vector field X. In local coordinates, one obtains :

\mbox X = \frac \partial_i \sqrt  X^i

Here (and below) we use the Einstein notation, so the above is actually a sum in i. The gradient of a scalar function f may be defined through the inner product

\langle\cdot,\cdot\rangle

on the manifold, as :

\langle \mbox f(x) , v_x \rangle = df(x)(v_x)

for all vectors

v_x

anchored at point x in the tangent bundle

T_xM

of the manifold at point x. Here, df is the exterior derivative of the function f; it is a 1-form taking argument

v_x

. In local coordinates, one has :

\left(\mbox f\right)^i = 
\partial^i f = g^ \partial_j f

Combining these, the formula for the Laplace-Beltrami operator applied to a scalar function f is, in local coordinates :

\Delta f = \mbox \; f = 
\frac \partial_i \sqrt \partial^i f

. Here,

g^

are the components of the inverse of the metric tensor

g

, so that

g^g_=\delta^i_k

with

\delta^i_k

the Kronecker delta. Note that the above definition is, by construction, valid only for scalar functions

f:M\rightarrow \mathbb

. One may want to extend the Laplacian even further, to differential forms; for this, one must turn to the Laplace-deRham operator, defined in the next section. One may show that the Laplace-Beltrami operator reduces to the ordinary Laplacian in Euclidean space by noting that it can be re-written using the chain rule as :

\Delta f = \partial_i \partial^i f + (\partial^i f) \partial_i \ln \sqrt.

When

|g| = 1

, such as in the case of Euclidean space, one then easily obtains :

\Delta f = \partial_i \partial^i f

which is the ordinary Laplacian. Using the Minkowski metric with signature (+++-), one regains the D'Alembertian given previously. Note also that by using the metric tensor for spherical and cylindrical coordinates, one can similarly regain the expressions for the Laplacian in spherical and cylindrical coordinates. The Laplace-Beltrami operator is handy not just in curved space, but also in ordinary flat space endowed with a non-linear coordinate system. Note that the exterior derivative d and -div are adjoint: :

\int_M df(X) \;\mathrm_n = - \int_M f \mbox X \;\mathrm_n

(proof) where the last equality is an application of Stokes theorem. Note also, the Laplace-Beltrami operator is symmetric: :

\int_M f\Delta h \;\mathrm_n = 
\int_M \langle \mbox f, \mbox h \rangle \;\mathrm_n = 
\int_M h\Delta f \;\mathrm_n

for functions f and h.

Laplace-de Rham operator

In the general case of differential geometry, one defines the Laplace-de Rham operator as the generalization of the Laplacian. It is a differential operator on the exterior algebra of a differentiable manifold. On a Riemannian manifold it is an elliptic operator, while on a pseudo-Riemannian manifold it is hyperbolic. The Laplace-de Rham operator is defined by :

\Delta= \mathrm\delta+\delta\mathrm = (\mathrm+\delta)^2,\;

where d is the exterior derivative or differential and δ is the codifferential. When acting on scalar functions, the codifferential may be defined as δ = −∗d∗, where ∗ is the Hodge star; more generally, the codifferential may include a sign that depends on the order of the k-form being acted on. One may prove that the Laplace-de Rahm operator is equivalent to the previous definition of the Laplace-Beltrami operator when acting on a scalar function f; see the Laplace operator article proofs for details. Notice that the Laplace-de Rham operator is actually minus the Laplace-Beltrami operator; this minus sign follows from the conventional definition of the properties of the codifferential. Unfortunately, Δ is used to denote both; which can sometimes be a source of confusion.

Properties

Given scalar functions f and h, and a real number a, the Laplace-de Rham operator has the following properties: #

\Delta(af + h) = a\Delta f + \Delta h\!

\Delta(fh) = f \Delta h + 2 \partial_i f \partial^i h + h \Delta f

(proof)

External links

- [http://mathworld.wolfram.com/Laplacian.html MathWorld: Laplacian]

References

- Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. (Provides a basic review of differential geometry in the special case of four-dimensional space-time.)
- Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 . (Provides a general introduction to curved surfaces). Category:Multivariate calculus Category:Riemannian geometry

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Elliptic operator

In mathematics, an elliptic operator is one of the major types of differential operator P. It will be defined on spaces of complex-valued functions, or some more general function-like objects. What is distinctive is that that the coefficients of the highest-order derivatives satisfy a positivity condition. An important example of an elliptic operator is the Laplacian. Equations of the form :

P u = 0 \quad

are called elliptic partial differential equations. Equations involving time, such as the heat equation or the Schrodinger equation also involve elliptic operators (on the LHS, say) as well as a time derivative (as RHS).

Second order operators

For expository purposes, we consider initially a second order linear partial differential operators of the form :

P\phi = \sum_ a_  D_k D_j \phi  + \sum_\ell b_\ell D_\phi  +c \phi

where

D_k = \frac \partial_

. Such an operator is called elliptic iff for every x the matrix of coefficients of the highest order terms :

\begin a_(x) & a_(x) & \cdots & a_(x) \\ a_(x) & a_(x) & \cdots & a_(x) \\
\vdots & \vdots & \vdots & \vdots \\ a_(x) & a_(x) & \cdots & a_(x)  \end

is a positive-definite real symmetric matrix. In particular, for every non-zero vector :

\vec = (\xi_1, \xi_2, \ldots , \xi_n)

the following inequality holds: :

\sum_ a_(x) \xi_k \xi_j > 0. \quad

Example. The negative of the Laplacian in Rⁿ given by :

- \Delta = \sum_^n D_\ell^2

is an elliptic operator.

External links

- [http://eqworld.ipmnet.ru/en/solutions/lpde/lpdetoc3.pdf Linear Elliptic Equations] at EqWorld: The World of Mathematical Equations.
- [http://eqworld.ipmnet.ru/en/solutions/npde/npde-toc3.pdf Nonlinear Elliptic Equations] at EqWorld: The World of Mathematical Equations.

Bibliography

- L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
- D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1983. ISBN 3-540-41160-7
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9 Category:Differential operators Category:Partial differential equations

Mathematics

History

Inspiration, pure and applied mathematics, and aesthetics

Notation, language, and rigor

Is mathematics a science?

Overview of fields of mathematics

Major themes in mathematics

Quantity

Structure

Space

Change

Foundations and methods

Discrete mathematics

Applied mathematics

Important theorems

Important conjectures

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Bibliography

External links

Wave equation

The wave equation is an important partial differential equation which generally describes all kinds of waves, such as sound waves, light waves and water waves. It arises in many different fields, such as acoustics, electromagnetics, and fluid dynamics. Variations of the wave equation are also found in quantum mechanics and general relativity. 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. The general form of the wave equation for a scalar quantity

u

is: :

= c^2 \nabla^2u

Here c is usually a fixed constant, the speed of the wave's propagation (for a sound wave in air this is about 330 m/s, see speed of sound). For the vibration of string this can vary widely: on a spiral spring (a slinky) it can be as slow as a meter per second. If c however is changing in function of the wavelength, it shall be replaced by the phase velocity: :

v_\mathrm = \frac.

Also note that a wave may be superimposed onto another movement (for instance sound propagation in a moving medium like a gas flow). In that case the scalar u will contain a Mach factor (which is positive for the wave moving along the flow and negative for the reflected wave). u = u(x,t), is the amplitude, a measure of the intensity of the wave at a particular location x and time t. For a sound wave in air u is the local air pressure, for a vibrating string it is the physical displacement of the string from its rest position.

\nabla^2

is the Laplace operator with respect to the location variable(s) x. Note that u may be a scalar or vector quantity. The general solution to the one dimensional scalar wave equation was derived by d'Alembert as:

u(x,t) = F(x-ct) + G(x+ct)

where F and G are arbitrary functions, corresponding to a forward-traveling wave, and a back-ward traveling wave, respectively. To determine F and G one must consider the two initial conditions: :

u(x,0)=f(x)

u_t(x,0)=g(x)

Thus the d´Alembert formula becomes:

u(x,t) = \frac + \frac \int_^ g(s) ds

In the classical sense if

f(x) \in C^k

and

g(x) \in C^

then

u(t,x) \in C^k

. The wave equation in the one dimensional case can be derived in the following way: Imagine an array of little weights of mass m interconnected with slinkies of length h . The slinkies have a stiffness of k : :Image:array_of_masses.png Here u (x) measures the distance from the equilibrium of the mass situated at x. The equation of motion for the weight at the location x+h is: :

m= k[u(x+2h,t)-u(x+h,t)-u(x+h,t)+u(x,t)]

where the time-dependence of u(x) has been made explicit. If the array of weights consists of N weights spaced evenly over the length L = N h of total Mass M = N m, and the total stiffness of the array K = k/N we can write the above equation as: :

=

Taking the limit N

\rightarrow \infty

, h

\rightarrow 0

one gets: :

=

L^2

)/M is the square of the propagation speed in this particular case. The basic wave equation is a linear differential equation which means that the amplitude of two waves interacting is simply the sum of the waves. This means also that a behavior of a wave can be analyzed by breaking up the wave into components. The Fourier transform breaks up a wave into sinusoidal components and is useful for analyzing the wave equation. The one-dimensional form can be derived from considering a flexible string, stretched between two points on a x-axis. It is :

= c^2

The general solution to this is a Fourier series: an infinite sum of sine and cosine waves. If the domain of the equation is infinite with no boundary conditions, then D'Alembert's method can be used to solve it. In two dimensions, expanding the Laplacian gives: :

= c^2 \left ( +  \right )

An example of the solution to the 2-D wave equation is the motion of a tightly-stretched drumhead. In this case, rather than sinusoids, the solutions are combinations of Bessel functions. The wave equation is the prototypical example of a hyperbolic partial differential equation. More realistic differential equations for waves allow for the speed of wave propagation to vary with the frequency of the wave, a phenomenon known as dispersion. Another common correction is that, in realistic systems, the speed also can depend on the amplitude of the wave, leading to a nonlinear wave equation: :

= c(u)^2 \left ( +  \right )

In three dimensions, for instance to study the propagation of sound in a space: :

= c^2 \left ( +  + \right )

The elastic wave equation in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion: :

\rho = \bold + ( \lambda + 2\mu )\nabla(\nabla \cdot \bold) - \mu\nabla \times (\nabla \times \bold)

where:
-

\lambda

and

\mu

are the so-called Lamé moduli describing the elastic properties of the medium,
-

\rho

is density,
-

\bold

is the source function (driving force),
- and

\bold

is displacement. Note that in this equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation.

External links

- [http://eqworld.ipmnet.ru/en/solutions/lpde/wave-toc.pdf Linear Wave Equations] at EqWorld: The World of Mathematical Equations.
- [http://eqworld.ipmnet.ru/en/solutions/npde/npde-toc2.pdf Nonlinear Wave Equations] at EqWorld: The World of Mathematical Equations.
- [http://35.9.69.219/home/modules/pdf_modules/m201.pdf MISN-0-201 The Wave Equation and Its Solutions] (PDF file) by William C. Lane for [http://www.physnet.org Project PHYSNET]. Category:Partial differential equations Category:Equations Category:Wave mechanics ko:파동방정식 ja:波動方程式

Heat equation

The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. In the special case of heat propagation in an isotropic and homogeneous medium in the 3-dimensional space, this equation is :

u_t = k ( u_ + u_ + u_ ) \quad

where:
- u(t, x, y, z) is temperature as a function of time and space;
- u_t is the rate of change of temperature at a point over time;
-

u_

u_

, and

u_

are the second spatial derivatives (thermal conductions) of temperature in the x, y, and z directions, respectively
- k is a material-specific constant called thermal diffusivity. The heat equation is a consequence of Fourier's law of cooling (see heat conduction). To solve the heat equation, we also need to specify boundary conditions for u. Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object. The heat equation is the prototypical example of a parabolic partial differential equation. Using the Laplace operator, the heat equation can be generalized to :

u_t = k \Delta u, \quad

where the Laplace operator is taken in the spatial variables. The heat equation governs heat diffusion, as well as other diffusive processes, such as particle diffusion. Although they are not diffusive in nature, some quantum mechanics problems are also governed by a mathematical analog of the heat equation (see below). It also can be used to model some processes in finance.

Solving the heat equation using Fourier series

The following solution technique for the heat equation was proposed by Joseph Fourier in his treatise Théorie analytique de la chaleur, published in 1822. Let us consider the heat equation for one space variable. This could be used to model heat conduction in a rod. The equation is :

(1) \ u_t = k u_ \quad

where u = u(t, x) is a function of two variables t and x. Here
- x is the space variable, so x ∈ [0,L], where L is the length of the rod.
- t is the time variable, so t ≥ 0. We assume the initial condition :

(2) \ u(0,x) = f(x) \quad \forall x \in [0,l] \quad

where the function f is given and the boundary conditions :

(3) \ u(t,0) = 0 = u(t,L) \quad \forall  t > 0 \quad

. Let us attempt to find a solution of (1) which is not identically zero satisfying the boundary conditions (3) but with the following property: u is a product in which the dependence of u on x, t is separated, that is: :

(4) \ u(t,x) = X(x) T(t). \quad

This solution technique is called separation of variables. Substituting u back into equation (1), :

\frac = \frac. \quad

Since the right hand side depends only on x and the left hand side only on t, both sides are equal to some constant value − λ. Thus: : $(5) \ T'(t) = - \lambda kT(t) \quad$ and : $(6) \ X$ (x) = - \lambda X(x). \quad We will now show that solutions for (6) for values of λ ≤ 0 cannot occur: 1. Suppose that λ < 0. Then there exists real numbers B, C such that :

X(x) = B e^ + C e^

. From (3) we get :

X(0) = 0 = X(L) \quad

, and therefore B = 0 = C which implies u is identically 0. 2. Suppose that λ=0. Then there exists real numbers B, C such that :

X(x) = Bx + C \quad

. From equation (3) we conclude in the same manner as in 1 that u is identically 0. 3. Therefore, it must be the case that λ > 0. Then exists there exist real numbers A, B, C such that :

T(t) = A e^ \quad

and :

X(x) = B \sin(\sqrt x) + C \cos(\sqrt x).

From (3) we get C=0 and that for some positive integer n, :

\sqrt = n \frac

. This solves the heat equation in the special case that the dependence of u has the special form (4). In general, the sum of solutions to (1) which satisfy the boundary conditions (3) also satisfies (1) and (3). We can show that the solution to (1), (2) and (3) is given by :

u(x,t) = \sum_^ D_n \left(\sin \frac\right) e^

where :

D_n = \frac \int_0^L f(x) \sin \frac \, dx

Generalizing the solution technique

The solution technique used above can be greatly extended to many other types of equations. The idea is that the operator u_xx with the zero boundary conditions can be represented in terms of its eigenvectors. This leads naturally to one of the basic ideas of the spectral theory of linear self-adjoint operators. Consider the linear operator Δ u = u_{x x}. The infinite sequence of functions :

e_n(x) = \sqrt\sin \frac

for n ≥ 1 are eigenvectors of Δ. Indeed :

\Delta e_n = -\frac e_n.

Moreover, any eigenvector f of Δ with the boundary conditions f(0)=f(L)=0 is of the form e_n for some n ≥ 1. The functions e_n for n ≥ 1 form an orthonormal sequence with respect to a certain inner product on the space of real-valued functions on [0, L]. This means :

\langle e_n, e_m \rangle = \int_0^L e_n(x) e_m(x) dx = \left\

Helmholtz equation

The Helmholtz equation, named for Hermann von Helmholtz, is the following elliptic partial differential equation: :

(\nabla^2 + k^2)\phi = 0

The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. For example, consider the wave equation: :

\left(\nabla^2-\frac\frac\right)u(\mathbf,t)=0

Applying the technique of separation of variables (

u=\phi(\mathbf)T(t)

), we obtain two differential equations: :

\nabla^2\phi + k^2\phi=(\nabla^2 + k^2)\phi=0,\quad \frac + k^2c^2T=0,

where

k

is the separation constant. We see that we now have Helmholtz's equation for the spatial variable

\mathbf

and a second-order ordinary differential equation in time. The solution in time will be a linear combination of sine and cosine functions, while the form of the solution in space will depend on the boundary conditions. Alternatively, integral transforms, such as the Laplace or Fourier transform, are often used to transform a hyperbolic PDE into a form of the Helmholtz equation. Due to its relationship to the wave equation, the Helmholtz equation arises in problems in such areas of physics as the study of electromagnetic radiation, seismology, and acoustics.

Solving the Helmholtz equation using separation of variables

The general solution to the spatial Helmholtz equation :

( \nabla^2 + k^2 ) \phi = 0

can be obtained using separation of variables. In spherical polar coordinates, the solution is: :

\phi (r, \