Applied mathematics is the application of mathematical methods by different fields such as

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

medicine Medicine is the science and practice of caring for a patient, managing the diagnosis, prognosis, prevention, treatment, palliation of their injury or disease, and promoting their health. Medicine encompasses a variety of health care pr ...

biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...

, finance, business,

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...

, and

industry Industry may refer to: Economics * Industry (economics), a generally categorized branch of economic activity * Industry (manufacturing), a specific branch of economic activity, typically in factories with machinery * The wider industrial sector ...

. Thus, applied mathematics is a combination of

mathematical science The mathematical sciences are a group of areas of study that includes, in addition to mathematics, those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper. Statisti ...

and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

History

Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods,

variational methods The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

, and

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...

); and applied

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...

. These areas of mathematics related directly to the development of

Newtonian physics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mec ...

, and in fact, the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left a pedagogical legacy in the United States: until the early 20th century, subjects such as

classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...

were often taught in applied mathematics departments at American universities rather than in

departments, and

fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...

may still be taught in applied mathematics departments.

Engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

and

departments have traditionally made use of applied mathematics.

Divisions

Today, the term "applied mathematics" is used in a broader sense. It includes the classical areas noted above as well as other areas that have become increasingly important in applications. Even fields such as

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

that are part of pure mathematics are now important in applications (such as

cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...

), though they are not generally considered to be part of the field of applied mathematics ''per se''. 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. Many mathematicians distinguish between "applied mathematics", which is concerned with mathematical methods, and the "applications of mathematics" within science and engineering. A

biologist A biologist is a scientist who conducts research in biology. Biologists are interested in studying life on Earth, whether it is an individual Cell (biology), cell, a multicellular organism, or a Community (ecology), community of Biological inter ...

using a

population model A population model is a type of mathematical model that is applied to the study of population dynamics. Rationale Models allow a better understanding of how complex interactions and processes work. Modeling of dynamic interactions in nature can ...

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. Mathematicians such as Poincaré and Arnold deny the existence of "applied mathematics" and claim that there are only "applications of mathematics." Similarly, non-mathematicians 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 science, and

computational engineering Computational science and engineering (CSE) is a relatively new discipline that deals with the development and application of computational models and simulations, often coupled with high-performance computing, to solve complex physical problems ...

, which use

high-performance computing High-performance computing (HPC) uses supercomputers and computer clusters to solve advanced computation problems. Overview HPC integrates systems administration (including network and security knowledge) and parallel programming into a mult ...

for the

simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the s ...

of phenomena and the solution of problems in the sciences and engineering. These are often considered interdisciplinary.

Applicable mathematics

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, although there is no consensus as to a precise definition. Mathematicians often distinguish between "applied mathematics" on the one hand, and the "applications of mathematics" or "applicable mathematics" both within and outside of science and engineering, on the other.Perspectives on Mathematics Education: Papers Submitted by Members of the Bacomet Group, pgs 82-3.
Editors: H. Christiansen, A.G. Howson, M. Otte. Volume 2 of Mathematics Education Library; Springer Science & Business Media, 2012. , 9789400945043. Some mathematicians emphasize the term applicable mathematics to separate or delineate the traditional applied areas from new applications arising from fields that were previously seen as pure mathematics. For example, from this viewpoint, an ecologist or geographer using population models and applying known mathematics would not be doing applied, but rather applicable, mathematics. Even fields such as number theory that are part of pure mathematics are now important in applications (such as

), though they are not generally considered to be part of the field of applied mathematics ''per se''. Such descriptions can lead to ''applicable mathematics'' being seen as a collection of mathematical methods such as

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

mathematical modelling A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...

optimisation Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

, combinatorics,

and statistics, which are useful in areas outside traditional mathematics and not specific to

mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...

. Other authors prefer describing ''applicable mathematics'' as a union of "new" mathematical applications with the traditional fields of applied mathematics.Survey of Applicable Mathematics, pg xvii (Foreword).
K. Rektorys; 2nd edition, illustrated. Springer, 2013. , 9789401583084. With this outlook, the terms applied mathematics and applicable mathematics are thus interchangeable.

Utility

Market Data Index NYA on 20050726 202628 UTC

Historically, mathematics was most important in the

natural sciences Natural science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatab ...

and

. However, since

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposing ...

, fields outside the physical sciences have spawned the creation of new areas of mathematics, such as game theory and

social choice theory Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.Amartya Sen (2008). "Soci ...

, which grew out of economic considerations. Further, the utilization and development of mathematical methods expanded into other areas leading to the creation of new fields such as mathematical finance and data science. The advent of the computer has enabled new applications: studying and using the new computer technology itself (

) to study problems arising in other areas of science (computational science) as well as the mathematics of computation (for example,

theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the ...

, computer algebra,

Stoer, J., & Bulirsch, R. (2013). Introduction to numerical analysis. Springer Science & Business Media.Conte, S. D., & De Boor, C. (2017). Elementary numerical analysis: an algorithmic approach. Society for Industrial and Applied Mathematics.Greenspan, D. (2018). Numerical Analysis. CRC Press.Linz, P. (2019). Theoretical numerical analysis. Courier Dover Publications.). Statistics is probably the most widespread

used in the

social sciences Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of so ...

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 separated 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 primarily of cross-listed courses and jointly appointed faculty in departments representing applications. Some Ph.D. programs in applied mathematics require little or no coursework outside 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 U.K. 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 Department may refer to: * Departmentalization, division of a larger organization into parts with specific responsibility Government and military *Department (administrative division), a geographical and administrative division within a country, ...

at the

University of Cambridge The University of Cambridge is a public collegiate research university in Cambridge, England. Founded in 1209 and granted a royal charter by Henry III in 1231, Cambridge is the world's third oldest surviving university and one of its most pr ...

, housing the

Lucasian Professor of Mathematics The Lucasian Chair of Mathematics () is a mathematics professorship in the University of Cambridge, England; its holder is known as the Lucasian Professor. The post was founded in 1663 by Henry Lucas, who was Cambridge University's Member of Pa ...

whose past holders include

Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...

, Charles Babbage,

James Lighthill Sir Michael James Lighthill (23 January 1924 – 17 July 1998) was a British applied mathematician, known for his pioneering work in the field of aeroacoustics and for writing the Lighthill report on artificial intelligence. Biography J ...

Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...

, and Stephen Hawking. Brown_University_Applied_Mathematics_building

Schools with separate applied mathematics departments range from Brown University, which has a large Division of Applied Mathematics that offers degrees through the

doctorate A doctorate (from Latin ''docere'', "to teach"), doctor's degree (from Latin ''doctor'', "teacher"), or doctoral degree is an academic degree awarded by universities and some other educational institutions, derived from the ancient formalism ''li ...

, to Santa Clara University, which offers only the M.S. in applied mathematics. Research universities dividing their mathematics department into pure and applied sections include

MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...

. Students in this program also learn another skill (computer science, engineering, physics, pure math, etc.) to supplement their applied math skills.

Associated mathematical sciences

Applied mathematics is associated with the following mathematical sciences:

Engineering and technological engineering

With applications of ''applied geometry'' together with applied chemistry.

Scientific computing

Scientific computing Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disc ...

includes applied mathematics (especially

computing science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

(especially

), and mathematical modelling in a scientific discipline.

Computer science

Computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...

relies on

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...

, discrete mathematics such as

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, and combinatorics.

Operations research and management science

Operations research Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve decis ...

and

management science Management science (or managerial science) is a wide and interdisciplinary study of solving complex problems and making strategic decisions as it pertains to institutions, corporations, governments and other types of organizational entities. It is ...

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

Statistics

Applied mathematics has substantial overlap with the discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions. Statistical theory relies on

and

decision theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...

, and makes extensive use of scientific computing, analysis, and

optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

; for the

design of experiments The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...

, statisticians use

and

combinatorial design Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of ''balance'' and/or ''symmetry''. These co ...

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

Actuarial science

Actuarial science applies probability, statistics, and economic theory to assess risk in insurance, finance and other industries and professions.

Mathematical economics

Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. The applied methods usually refer to nontrivial mathematical techniques or approaches. Mathematical economics is based on statistics, probability, mathematical programming (as well as other computational methods), operations research, game theory, and some methods from mathematical analysis. In this regard, it resembles (but is distinct from)

financial mathematics Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...

, another part of applied mathematics.Roberts, A. J. (2009). Elementary calculus of financial mathematics (Vol. 15). SIAM. According to the

Mathematics Subject Classification The Mathematics Subject Classification (MSC) 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), mathematical economics falls into the Applied mathematics/other classification of category 91: :Game theory, economics, social and behavioral sciences wit
MSC2010
classifications for ' Game theory' at code
91Axx
and for 'Mathematical economics' at code
91Bxx

Other disciplines

The line between applied mathematics and specific areas of application is often blurred. Many universities teach mathematical and statistical courses outside the respective departments, in departments and areas including business,

, chemistry,

psychology Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries between ...

scientific computation Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disc ...

, and

References

External links

* * Th
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 Thailand ( ), historically known as Siam () and officially the Kingdom of Thailand, is a country in Southeast Asia, located at the centre of the Indochinese Peninsula, spanning , with a population of almost 70 million. The country is bo ...

is a major publisher of research journals and books in applied mathematics.
The Applicable Mathematics Research Group
at

Notre Dame University The University of Notre Dame du Lac, known simply as Notre Dame ( ) or ND, is a private Catholic research university in Notre Dame, Indiana, outside the city of South Bend. French priest Edward Sorin founded the school in 1842. The main campu ...

Centre for Applicable Mathematics
at

Liverpool Hope University , mottoeng=Hope to all who need it , established=1844 – Saint Katharine's College (as Warrington Training College)1856 – Notre Dame College (as Our Lady's Training College)1964 – Christ's College1979 – Liverpool Institute of Higher Edu ...

Applicable Mathematics research group
at

Glasgow Caledonian University Glasgow Caledonian University ( gd, Oilthigh Chailleannach Ghlaschu, ), informally GCU, Caledonian or Caley, is a public university in Glasgow, Scotland. It was formed in 1993 by the merger of The Queen's College, Glasgow (founded in 1875) and G ...

{{DEFAULTSORT:Applied Mathematics

History

Divisions

Applicable mathematics

Utility

Status in academic departments

Associated mathematical sciences

Engineering and technological engineering

Scientific computing

Computer science

Operations research and management science

Statistics

Actuarial science

Mathematical economics

Other disciplines

See also

References

Further reading

Applicable mathematics

External links