Applied mathematics

Applied mathematics

Overview
Applied mathematics is a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional
Professional
A professional is a person who is paid to undertake a specialised set of tasks and to complete them for a fee. The traditional professions were doctors, lawyers, clergymen, and commissioned military officers. Today, the term is applied to estate agents, surveyors , environmental scientists,...

 specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation and study of mathematical models.
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Encyclopedia
Applied mathematics is a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional
Professional
A professional is a person who is paid to undertake a specialised set of tasks and to complete them for a fee. The traditional professions were doctors, lawyers, clergymen, and commissioned military officers. Today, the term is applied to estate agents, surveyors , environmental scientists,...

 specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation and study of mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

.

Divisions



There is no consensus as to what the various branches of applied mathematics are. Such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees.

Historically, applied mathematics consisted principally of applied analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, most notably differential equations; approximation theory
Approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...

 (broadly construed, to include representation
Representation (mathematics)
In mathematics, representation is a very general relationship that expresses similarities between objects. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the...

s, asymptotic
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...

 methods, variational methods
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

, and numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

); and applied probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

. These areas of mathematics were intimately tied to the development of Newtonian physics, and in fact the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left a legacy as well: until the early 20th century subjects such as classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

 were often taught in applied mathematics departments at American universities rather than in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 departments, and fluid mechanics
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...

 may still be taught in applied mathematics departments. As well as physics, engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 and computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

 have traditionally made use of applied mathematics.


Today, the term applied mathematics is used in a broader sense. It includes the classical areas above, as well as other areas that have become increasingly important in applications. Even fields such as number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

 that are part of pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

 are now important in applications (such as cryptography
Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...

), though they are not generally considered to be part of the field of applied mathematics per se. Sometimes the term applicable mathematics is used to distinguish between the traditional applied mathematics that developed alongside physics and the many areas of mathematics that are applicable to real-world problems today.

Many mathematicians distinguish between applied mathematics, which is concerned with mathematical methods, and the applications of mathematics within science and engineering. A biologist
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

 using a population model
Matrix population models
Population models are used in population ecology to model the dynamics of wildlife or human populations. Matrix population models are a specific type of population model that uses matrix algebra...

 and applying known mathematics would not be doing applied mathematics, but rather using it; however mathematical biologists have posed problems that have stimulated the growth of pure mathematics. However, mathematicians like Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

 and Arnold
Vladimir Arnold
Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...

 deny the existence of "applied mathematics" and claim that there are only "applications of mathematics"; similarly, nonmathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to solve industrial problems is also called industrial mathematics.

The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics
Computational mathematics
Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods. Computation in the research is prominent. Computational mathematics emerged as a distinct part of applied...

, computational science
Computational science
Computational science is the field of study concerned with constructing mathematical models and quantitative analysis techniques and using computers to analyze and solve scientific problems...

, and computational engineering
Computational engineering
Computational science and engineering is a relatively new discipline of engineering. It is typically offered as a masters or doctorate program at several institutions...

, which use high performance computing for the simulation
Simulation
Simulation is the imitation of some real thing available, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system....

 of phenomena and the solution of problems in the science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

s and engineering. These are often considered interdisciplinary disciplines.

Utility



Historically, mathematics was most important in the natural sciences and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

. However, since World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...

, fields outside of the physical sciences have spawned the creation of new areas of mathematics, such as game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

 and social choice theory
Social choice theory
Social choice theory is a theoretical framework for measuring individual interests, values, or welfares as an aggregate towards collective decision. A non-theoretical example of a collective decision is passing a set of laws under a constitution. Social choice theory dates from Condorcet's...

, which grew out of economic considerations, or neural networks
Neural Networks
Neural Networks is the official journal of the three oldest societies dedicated to research in neural networks: International Neural Network Society, European Neural Network Society and Japanese Neural Network Society, published by Elsevier...

, which arose out of the study of the brain in neuroscience
Neuroscience
Neuroscience is the scientific study of the nervous system. Traditionally, neuroscience has been seen as a branch of biology. However, it is currently an interdisciplinary science that collaborates with other fields such as chemistry, computer science, engineering, linguistics, mathematics,...

.

The advent of the computer has created new applications: studying and using the new computer technology itself (computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

), using computers to study problems arising in other areas of science (computational science
Computational science
Computational science is the field of study concerned with constructing mathematical models and quantitative analysis techniques and using computers to analyze and solve scientific problems...

), and studying the mathematics of computation (for example, theoretical computer science
Theoretical computer science
Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....

, computer algebra, numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

). Statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

 is probably the most widespread mathematical science used in the social sciences
Social sciences
Social science is the field of study concerned with society. "Social science" is commonly used as an umbrella term to refer to a plurality of fields outside of the natural sciences usually exclusive of the administrative or managerial sciences...

, but other areas of mathematics are proving increasingly useful in these disciplines, most notably in economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

.

Status in academic departments



Academic institutions are not consistent in the way they group and label courses, programs, and degrees in applied mathematics. At some schools, there is a single mathematics department, whereas others have separate departments for Applied Mathematics and (Pure) Mathematics. It is very common for Statistics departments to be separate at schools with graduate programs, but many undergraduate-only institutions include statistics under the mathematics department.

Many applied mathematics programs (as opposed to departments) consist of primarily cross-listed courses and jointly-appointed faculty in departments representing applications. Some Ph.D. programs in applied mathematics require little or no coursework outside of mathematics, while others require substantial coursework in a specific area of application. In some respects this difference reflects the distinction between "application of mathematics" and "applied mathematics".

Some universities in the UK
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...

 host departments of Applied Mathematics and Theoretical Physics, but it is now much less common to have separate departments of pure and applied mathematics. A notable exception to this is the Department of Applied Mathematics and Theoretical Physics
Faculty of Mathematics, University of Cambridge
The Faculty of Mathematics at the University of Cambridge comprises the Department of Pure Mathematics and Mathematical Statistics and the Department of Applied Mathematics and Theoretical Physics . It is housed in the Centre for Mathematical Sciences site in West Cambridge, alongside the Isaac...

 at the University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...

, housing the Lucasian Professor of Mathematics whose past holders include Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

, Charles Babbage
Charles Babbage
Charles Babbage, FRS was an English mathematician, philosopher, inventor and mechanical engineer who originated the concept of a programmable computer...

, James Lighthill
James Lighthill
Sir Michael James Lighthill, FRS was a British applied mathematician, known for his pioneering work in the field of aeroacoustics.-Biography:...

, Paul Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

 and Stephen Hawking
Stephen Hawking
Stephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...

.

Schools with separate applied mathematics departments range from Brown University
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...

, which has a well-known and large Division of Applied Mathematics that offers degrees through the doctorate
Doctorate
A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder to teach in a specific field, A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder...

, to Santa Clara University
Santa Clara University
Santa Clara University is a private, not-for-profit, Jesuit-affiliated university located in Santa Clara, California, United States. Chartered by the state of California and accredited by the Western Association of Schools and Colleges, it operates in collaboration with the Society of Jesus , whose...

, which offers only the M.S.
Master of Science
A Master of Science is a postgraduate academic master's degree awarded by universities in many countries. The degree is typically studied for in the sciences including the social sciences.-Brazil, Argentina and Uruguay:...

 in applied mathematics . Research universities dividing their mathematics department into pure and applied sections include Harvard and MIT.

In Canada the only school with an Applied Mathematics program colloquially known as "Apple Math" is Queen's University
Queen's University
Queen's University, , is a public research university located in Kingston, Ontario, Canada. Founded on 16 October 1841, the university pre-dates the founding of Canada by 26 years. Queen's holds more more than of land throughout Ontario as well as Herstmonceux Castle in East Sussex, England...

 in the Faculty of Engineering & Applied Science
Queen's Faculty of Engineering & Applied Science
The Faculty of Engineering & Applied Science is the faculty responsible for all students pursuing degrees in the various engineering disciplines at Queen's University. Undergraduate students are represented by the Engineering Society...

.

Other associated mathematical sciences



Applied mathematics is closely related to other mathematical sciences.

Scientific computing


Scientific computing includes applied mathematics (especially numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

), computing science (especially high-performance computing
High-performance computing
High-performance computing uses supercomputers and computer clusters to solve advanced computation problems. Today, computer systems approaching the teraflops-region are counted as HPC-computers.-Overview:...

), and mathematical modelling in a scientific discipline.

Operations research and management science


Operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

 and management science are often taught in faculties of engineering, business, public policy.

Statistics


Applied mathematics has substantial overlap with the discipline of statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions. Statistical theory relies on probability
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 and decision theory
Optimal decision
An optimal decision is a decision such that no other available decision options will lead to a better outcome. It is an important concept in decision theory. In order to compare the different decision outcomes, one commonly assigns a relative utility to each of them...

, and makes extensive use of scientific computing, analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, and optimization; for the design of experiments
Design of experiments
In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...

, statisticians use algebra
Algebraic statistics
Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing....

 and combinatorial design
Combinatorial design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence and construction of systems of finite sets whose intersections have specified numerical properties....

. Applied mathematicians and statistician
Statistician
A statistician is someone who works with theoretical or applied statistics. The profession exists in both the private and public sectors. The core of that work is to measure, interpret, and describe the world and human activity patterns within it...

s often work in a department of mathematical sciences (particularly at colleges and small universities).

Actuarial science


Actuarial science
Actuarial science
Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries. Actuaries are professionals who are qualified in this field through education and experience...

 uses probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, and economic theory.

Economics


A standard classification system in mathematics is the Mathematics Subject Classification
Mathematics Subject Classification
The Mathematics Subject Classification is an alphanumerical classification scheme collaboratively produced by staff of and based on the coverage of the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH...

 (MSC). It is used by many mathematics journals to allow authors to classify their articles. In the MSC Applied mathematics/other classification there is category 91:
Game theory, economics, social and behavioral sciences.


MSC2010 classifications for 'Game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

' are at 91A here. Classifications for 'Mathematical economics
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...

' are at 91B here. In the latter, subjects in economics are listed, as possibly distinguished from mathematical methods that are used to develop them. For example 'Social choice' (91B14) is a subject in economics, but it is developed using methods of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

.

The Handbook of Mathematical Economics series (Elsevier), currently 4 volumes, distinguishes between mathematical methods in economics per 1st-page chapter links in the accompanying footnote and mathematical approaches to different subjects in in economic theory.

Another source with a similar distinction is The New Palgrave: A Dictionary of Economics (1987, 4 vols., 1,300 subject entries). In it, a "Subject Index" includes mathematical entries under 2 headings (vol. IV, pp. 982-3):
Mathematical Economics (24 listed, such as "acyclicity", "aggregation problem
Aggregation problem
An aggregate in economics is a summary measure describing a market or economy. The aggregation problem refers to the difficulty of treating an empirical or theoretical aggregate as if it reacted like a less-aggregated measure, say, about behavior of an individual agent as described in general...

", "comparative statics
Comparative statics
In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter....

", "lexicographic orderings", "linear models", "orderings", and "qualitative economics
Qualitative economics
Qualitative economics refers to representation and analysis of information about the direction of change in some economic variable as related to change of some other economic variable...

")
Mathematical Methods (42 listed, such as "calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

", "catastrophe theory
Catastrophe theory
In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry....

", "combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

," "computation of general equilibrium
Computable general equilibrium
Computable general equilibrium models are a class of economic models that use actual economic data to estimate how an economy might react to changes in policy, technology or other external factors...

", "convexity", "convex programming", and "stochastic optimal control
Optimal control
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...

").


A widely-used classification system in economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

 that includes mathematical methods on the subject is the JEL classification codes
JEL classification codes
Articles in economics journals are usually classified according to the JEL classification codes, a system originated by the Journal of Economic Literature. The JEL is published quarterly by the American Economic Association and contains survey articles and information on recently published books...

. Articles in journals are usually classified according to the system. It was originated by the Journal of Economic Literature
Journal of Economic Literature
The Journal of Economic Literature is a peer-reviewed academic journal on economy published by the American Economic Association. It was established in 1963 as the Journal of Economic Abstracts. As a review journal, it mainly features essays and reviews of recent economic theories...

for classifying new books and articles. The relevant categories are listed below with the same Wikipedia links at JEL classification codes#Mathematical and quantitative methods JEL: C Subcategories. The New Palgrave Dictionary of Economics
The New Palgrave Dictionary of Economics
The New Palgrave Dictionary of Economics , 2nd Edition, is an eight-volume reference work, edited by Steven N. Durlauf and Lawrence E. Blume. It contains 5.8 million words and spans 7,680 pages with 1,872 articles. Included are 1057 new articles and, from earlier, 80 essays that are designated as...

(2008, 2nd ed.) also uses the JEL codes to classify its entries. The corresponding footnotes below have links to abstracts
Abstract (summary)
An abstract is a brief summary of a research article, thesis, review, conference proceeding or any in-depth analysis of a particular subject or discipline, and is often used to help the reader quickly ascertain the paper's purpose. When used, an abstract always appears at the beginning of a...

 of The New Palgrave Online for each JEL category (10 or fewer per page, similar to Google
Google
Google Inc. is an American multinational public corporation invested in Internet search, cloud computing, and advertising technologies. Google hosts and develops a number of Internet-based services and products, and generates profit primarily from advertising through its AdWords program...

 searches).
JEL: C02 - Mathematical Methods
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...

 (following JEL: C00 - General and JEL: C01 - Econometrics
Econometrics
Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...

)

JEL: C6 - Mathematical Methods
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...

 and Programming
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....

JEL: C60 - General
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...

JEL: C61 - Optimization techniques
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....

; Programming models
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

; Dynamic analysis
JEL: C62 - Existence
Existence theorem
In mathematics, an existence theorem is a theorem with a statement beginning 'there exist ..', or more generally 'for all x, y, ... there exist ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not...

 and stability
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...

 conditions of equilibrium
Economic equilibrium
In economics, economic equilibrium is a state of the world where economic forces are balanced and in the absence of external influences the values of economic variables will not change. It is the point at which quantity demanded and quantity supplied are equal...

JEL: C63 - Computational techniques
Computational economics
Computational economics is a research discipline at the interface between computer science and economic and management science. Areas encompassed include agent-based computational modeling, computational modeling of dynamic macroeconomic systems and transaction costs, other applications in...

JEL: C67 - Input–output models
Input-output model
In economics, an input-output model is a quantitative economic technique that represents the interdependencies between different branches of national economy or between branches of different, even competing economies. Wassily Leontief developed this type of analysis and took the Nobel Memorial...

JEL: C68 - Computable General Equilibrium models
Computable general equilibrium
Computable general equilibrium models are a class of economic models that use actual economic data to estimate how an economy might react to changes in policy, technology or other external factors...


JEL: C7 - Game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

 and Bargaining theory
Bargaining
Bargaining or haggling is a type of negotiation in which the buyer and seller of a good or service dispute the price which will be paid and the exact nature of the transaction that will take place, and eventually come to an agreement. Bargaining is an alternative pricing strategy to fixed prices...

JEL: C70 - General
JEL: C71 - Cooperative game
Cooperative game
In game theory, a cooperative game is a game where groups of players may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players...

s
JEL: C72 - Noncooperative games
JEL: C73 - Stochastic
Stochastic game
In game theory, a stochastic game, introduced by Lloyd Shapley in the early 1950s, is a dynamic game with probabilistic transitions played by one or more players. The game is played in a sequence of stages. At the beginning of each stage the game is in some state...

 and Dynamic games; Evolutionary games
Evolutionary game theory
Evolutionary game theory is the application of Game Theory to evolving populations of lifeforms in biology. EGT is useful in this context by defining a framework of contests, strategies and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith...

; Repeated Games
JEL: C78 - Bargaining theory
Bargaining
Bargaining or haggling is a type of negotiation in which the buyer and seller of a good or service dispute the price which will be paid and the exact nature of the transaction that will take place, and eventually come to an agreement. Bargaining is an alternative pricing strategy to fixed prices...

; Matching theory


Mathematical methods listed in economics texts include those of:
  • Alpha C. Chiang and Kevin Wainwright, 2005. Fundamental Methods of Mathematical Economics, McGraw-Hill Irwin.. Contents.
  • Akira Takayama, 1985. Mathematical Economics, 2nd ed. Cambridge. Contents.
  • Stephen Glaister
    Stephen Glaister
    Stephen Glaister CBE is Professor of Transport and Infrastructure at the Department of Civil & Environmental Engineering, Imperial College London where he is also director of the Railway Technology Strategy Centre; a former board member of Transport for London and director of the RAC Foundation...

    , 1984. Mathematical Methods for Economists, 3rd ed., Blackwell. Contents.
  • Michael Carter, 2001. Foundations of Mathematical Economics, MIT Press. Contents.
  • D. Wade Hands, 2004. Introductory Mathematical Economics, 2nd ed. Oxford. Contents.

Other disciplines


The line between applied mathematics and specific areas of application is often blurred. Many universities teach mathematical and statistical courses outside of the respective departments, in departments and areas including business
Business
A business is an organization engaged in the trade of goods, services, or both to consumers. Businesses are predominant in capitalist economies, where most of them are privately owned and administered to earn profit to increase the wealth of their owners. Businesses may also be not-for-profit...

, engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

, psychology
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...

, biology
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

, computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, and mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

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See also


  • Society for Industrial and Applied Mathematics
    Society for Industrial and Applied Mathematics
    The Society for Industrial and Applied Mathematics was founded by a small group of mathematicians from academia and industry who met in Philadelphia in 1951 to start an organization whose members would meet periodically to exchange ideas about the uses of mathematics in industry. This meeting led...



External links


  • The Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to promoting the interaction between mathematics and other scientific and technical communities. Aside from organizing and sponsoring numerous conferences, SIAM
    Society for Industrial and Applied Mathematics
    The Society for Industrial and Applied Mathematics was founded by a small group of mathematicians from academia and industry who met in Philadelphia in 1951 to start an organization whose members would meet periodically to exchange ideas about the uses of mathematics in industry. This meeting led...

    is a major publisher of research journals and books in applied mathematics.