Applied mathematics

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Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming. Vehicle Routing Problem Example.svg
Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.

Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science 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.


A numerical solution to the heat equation on a pump casing model using the finite element method. Elmer-pump-heatequation.png
A numerical solution to the heat equation on a pump casing model using the finite element method.

Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and applied probability. These areas of mathematics related directly 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 pedagogical legacy in the United States: until the early 20th century, subjects such as classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and fluid mechanics may still be taught in applied mathematics departments. [1] Engineering and computer science departments have traditionally made use of applied mathematics.


Fluid mechanics is often considered a branch of applied mathematics and mechanical engineering. HD-Rayleigh-Taylor.gif
Fluid mechanics is often considered a branch of applied mathematics and mechanical engineering.

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 that are part of pure mathematics are now important in applications (such as cryptography), 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.

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 using a population model 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". [2]

The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics, computational science, and computational engineering, which use high-performance computing for the simulation of phenomena and the solution of problems in the sciences and engineering. These are often considered interdisciplinary.


Mathematical finance is concerned with the modelling of financial markets. Market Data Index NYA on 20050726 202628 UTC.png
Mathematical finance is concerned with the modelling of financial markets.

Historically, mathematics was most important in the natural sciences and engineering. However, since World War II, fields outside the physical sciences have spawned the creation of new areas of mathematics, such as game theory and social choice theory, 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 (computer science) to study problems arising in other areas of science (computational science) as well as the mathematics of computation (for example, theoretical computer science, computer algebra, [3] [4] [5] [6] numerical analysis [7] [8] [9] [10] ). Statistics is probably the most widespread mathematical science used in the social sciences, but other areas of mathematics, most notably economics, are proving increasingly useful in these disciplines.

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 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 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 host departments of Applied Mathematics and Theoretical Physics, [11] [12] [13] 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 at the University of Cambridge, housing the Lucasian Professor of Mathematics whose past holders include Isaac Newton, Charles Babbage, James Lighthill, Paul Dirac and Stephen Hawking.

Schools with separate applied mathematics departments range from Brown University, which has a large Division of Applied Mathematics that offers degrees through the doctorate, to Santa Clara University, which offers only the M.S. in applied mathematics. [14] Research universities dividing their mathematics department into pure and applied sections include MIT. Brigham Young University also has an Applied and Computational Emphasis (ACME), a program that allows students to graduate with a Mathematics degree, with an emphasis in Applied Math. 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 has substantial overlap with statistics. Oldfaithful3.png
Applied mathematics has substantial overlap with statistics.

Applied mathematics is associated with the following mathematical sciences:

Scientific computing

Scientific computing includes applied mathematics (especially numerical analysis [7] [8] [9] [10] [15] ), computing science (especially high-performance computing [16] [17] ), and mathematical modelling in a scientific discipline.

Computer science

Computer science relies on logic, algebra, discrete mathematics such as graph theory, [18] [19] and combinatorics.

Operations research and management science

Operations research [20] and management science are often taught in faculties of engineering, business, and public policy.


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 probability and decision theory, and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorial design. 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. [21]

Mathematical economics

Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. [22] [23] [24] 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, another part of applied mathematics. [25]

According to the Mathematics Subject Classification (MSC), mathematical economics falls into the Applied mathematics/other classification of category 91:

Game theory, economics, social and behavioral sciences

with MSC2010 classifications for 'Game theory' at codes 91Axx and for 'Mathematical economics' at codes 91Bxx.

Applicable mathematics

Applicable mathematics is a subdiscipline of applied mathematics, although there is no consensus as to a precise definition. [26] 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.

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. [26] 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. [27] 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 cryptography), 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, linear algebra, mathematical modelling, optimisation, combinatorics, probability and statistics, which are useful in areas outside traditional mathematics and not specific to mathematical physics.

Other authors prefer describing applicable mathematics as a union of "new" mathematical applications with the traditional fields of applied mathematics. [27] [28] [29] With this outlook, the terms applied mathematics and applicable mathematics are thus interchangeable.

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, engineering, physics, chemistry, psychology, biology, computer science, scientific computation, and mathematical physics.

See also

Related Research Articles

Computer science Study of the foundations and applications of computation

Computer science is the study of algorithmic processes, computational machines and computation itself. As a discipline, computer science spans a range of topics from theoretical studies of algorithms, computation and information to the practical issues of implementing computational systems in hardware and software.

Discrete mathematics Study of discrete mathematical structures

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus or Euclidean geometry. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.

Mathematics Field of study

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition.

Computer science is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. One well known subject classification system for computer science is the ACM Computing Classification System devised by the Association for Computing Machinery.

Outline of academic disciplines Overviews of and topical guides to academic disciplines

An academic discipline or field of study is a branch of knowledge, taught and researched as part of higher education. A scholar's discipline is commonly defined by the university faculties and learned societies to which he/she belongs and the academic journals in which he/she publishes research.

Lists of mathematics topics cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. Lists cover aspects of basic and advanced mathematics, methodology, mathematical statements, integrals, general concepts, mathematical objects and reference tables. They also cover equations named after people, societies, mathematicians, journals and meta-lists.

Feng Kang was a Chinese mathematician. He was elected an academician of the Chinese Academy of Sciences in 1980. After his death, the Chinese Academy of Sciences established the Feng Kang Prize in 1994 to reward young Chinese researchers who made outstanding contributions to computational mathematics.

Computational science, also known as scientific computing or scientific computation (SC), is a rapidly growing field that uses advanced computing capabilities to understand and solve complex problems. It is an area of science which spans many disciplines, but at its core, it involves the development of models and simulations to understand natural systems.

Mathematics encompasses a growing variety and depth of subjects over its history, and comprehension of it requires a system to categorize and organize these various subjects into a more general areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different purposes they serve.

Stanley Osher American mathematician

Stanley Osher is an American mathematician, known for his many contributions in shock capturing, level-set methods, and PDE-based methods in computer vision and image processing. Osher is a professor at the University of California, Los Angeles (UCLA), Director of Special Projects in the Institute for Pure and Applied Mathematics (IPAM) and member of the California NanoSystems Institute (CNSI) at UCLA. He has a daughter, Kathryn, and a son, Joel.

Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science as a "third way" besides theoretical and experimental sciences, computational mechanics was widely considered to be a sub-discipline of applied mechanics. It is now considered to be a sub-discipline within computational science.

Engineering mathematics is a branch of applied mathematics concerning mathematical methods and techniques that are typically used in engineering and industry. Along with fields like engineering physics and engineering geology, both of which may belong in the wider category engineering science, engineering mathematics is an interdisciplinary subject motivated by engineers' needs both for practical, theoretical and other considerations outwith their specialization, and to deal with constraints to be effective in their work.

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 arising in engineering analysis and design as well as natural phenomena. CSE has been described as the "third mode of discovery".

Department of Mathematics, University of Manchester

The Department of Mathematics at the University of Manchester is one of the largest unified mathematics departments in the United Kingdom, with over 90 academic staff and an undergraduate intake of roughly 400 students per year and approximately 200 postgraduate students in total. The School of Mathematics was formed in 2004 by the merger of the mathematics departments of University of Manchester Institute of Science and Technology (UMIST) and the Victoria University of Manchester (VUM). In July 2007 the department moved from the Mathematics Tower into a purpose-designed building─the first three floors of the Alan Turing Building─on Upper Brook Street. In a Faculty restructure in 2019 the School of Mathematics reverted to the Department of Mathematics. It is one of five Departments that make up the School of Natural Sciences, which together with the School of Engineering now constitutes the Faculty of Science and Engineering at Manchester.

Computational mathematics

Computational mathematics involves mathematical research in mathematics as well as in areas of science where computing plays a central and essential role, and emphasizes algorithms, numerical methods, and symbolic computations.

Institute of Mathematics and Applications, Bhubaneswar

The Institute of Mathematics and Applications (IMA), located in Bhubaneswar, Odisha, in India, is an academic institution that was established by the Government of Odisha in 1999. Its dual purposes are to conduct advanced research in pure and applied mathematics, and to provide postgraduate education leading to Masters and PhD degrees in mathematics, computation, computational finance and data science. The institute also runs training programs in schools aimed at increasing mathematics awareness and leading to competitions such as the Mathematics Olympiads. It is affiliated to Utkal University which is one of the largest affiliating universities.

The following outline is provided as an overview of and topical guide to formal science:


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  10. 1 2 Linz, P. (2019). Theoretical numerical analysis. Courier Dover Publications.
  11. For example see, The Tait Institute: History (2nd par.). Accessed Nov 2012.
  12. Dept of Applied Mathematics & Theoretical Physics. Queen's University, Belfast.
  13. DAMTP Belfast ResearchGate page.
  14. Santa Clara University Dept of Applied Mathematics, archived from the original on 2011-05-04, retrieved 2011-03-05
  15. Today, numerical analysis includes numerical linear algebra, numerical integration, and validated numerics as subfields.
  16. Hager, G., & Wellein, G. (2010). Introduction to high performance computing for scientists and engineers. CRC Press.
  17. Geshi, M. (2019). The Art of High Performance Computing for Computational Science, Springer.
  18. West, D. B. (2001). Introduction to graph theory (Vol. 2). Upper Saddle River: Prentice Hall.
  19. Bondy, J. A., & Murty, U. S. R. (1976). Graph theory with applications (Vol. 290). London: Macmillan.
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  21. Boland, P. J. (2007). Statistical and probabilistic methods in actuarial science. CRC Press.
  22. Wainwright, K. (2005). Fundamental methods of mathematical economics/Alpha C. Chiang, Kevin Wainwright. Boston, Mass.: McGraw-Hill/Irwin,.
  23. Na, N. (2016). Mathematical economics. Springer.
  24. Lancaster, K. (2012). Mathematical economics. Courier Corporation.
  25. Roberts, A. J. (2009). Elementary calculus of financial mathematics (Vol. 15). SIAM.
  26. 1 2 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. ISBN   9400945043, 9789400945043.
  27. 1 2 Survey of Applicable Mathematics, pg xvii (Foreword). K. Rektorys; 2nd edition, illustrated. Springer, 2013. ISBN   9401583080, 9789401583084.
  29. INTERNATIONAL CONFERENCE ON APPLICABLE MATHEMATICS (ICAM-2016). Archived 2017-03-23 at the Wayback Machine The Department of Mathematics, Stella Maris College.

Further reading

Applicable mathematics