Applied mathematics

{{short description|Application of mathematical methods to other fields}}
{{for|the company|Applied Maths}}
[[File:Vehicle Routing Problem Example.svg|thumb|right|Efficient solutions to the [[vehicle routing problem]] require tools from [[combinatorial optimization]] and [[integer programming]].]]

'''Applied mathematics''' is the application of [[mathematics|mathematical method]]s by different fields such as [[physics]], [[engineering]], [[medicine]], [[biology]], [[finance]], [[business]], [[computer science]], and [[Industrial sector|industry]]. Thus, applied mathematics is a combination of [[mathematical science]] and specialized knowledge. The term "applied mathematics" also describes the [[profession|professional specialty]] in which mathematicians work on practical problems by formulating and studying [[mathematical model]]s.

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 ==
[[File:Elmer-pump-heatequation.png|thumb|right|A numerical solution to the [[heat equation]] on a pump casing model using the [[finite element method]].]]

Historically, applied mathematics consisted principally of [[Mathematical analysis|applied analysis]], most notably [[differential equations]]; [[approximation theory]] (broadly construed, to include [[Representation (mathematics)|representation]]s, [[Asymptotic analysis|asymptotic]] methods, [[Calculus of variations|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.<ref name="Stolz2002">
{{citation|author=Stolz, M.|title=The History Of Applied Mathematics And The History Of Society|journal=Synthese|volume=133|issue=1|pages=43–57|year=2002|doi=10.1023/A:1020823608217|s2cid=34271623}}</ref> [[Engineering]] and [[computer science]] departments have traditionally made use of applied mathematics.

As time passed, Applied Mathematics grew alongside the advancement of science and technology. With the advent of modern times, the application of mathematics in fields such as science, economics, technology, and more became deeper and more timely. The development of computers and other technologies enabled a more detailed study and application of mathematical concepts in various fields.

Today, Applied Mathematics continues to be crucial for societal and technological advancement. It guides the development of new technologies, economic progress, and addresses challenges in various scientific fields and industries. The history of Applied Mathematics continually demonstrates the importance of mathematics in human progress.

== Divisions ==
[[File:HD-Rayleigh-Taylor.gif|left|thumb|[[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''.

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 [[Matrix population models|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 [[Henri Poincaré|Poincaré]] and [[Vladimir Arnold|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".<ref>{{citation | author=University of Strathclyde | title=Industrial Mathematics | url=http://www.maths.strath.ac.uk/applying/postgraduate/research_topics/industrial_mathematics | date=17 January 2008 | access-date=8 January 2009 | archive-url=https://archive.today/20120804104748/http://www.maths.strath.ac.uk/applying/postgraduate/research_topics/industrial_mathematics | archive-date=2012-08-04 | url-status=dead }}</ref>

The success of modern numerical mathematical methods and software has led to the emergence of [[computational mathematics]], [[computational science]], and [[computational engineering]], which use [[supercomputer|high-performance computing]] for the [[simulation]] 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.<ref name=OtteEtAl/>

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.<ref name=OtteEtAl>
[https://books.google.com/books?id=VgLZBAAAQBAJ&q=applicable+mathematics&pg=PA83 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.</ref> 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.<ref name=rektorys/> 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.<ref name=rektorys>[https://books.google.com/books?id=-sztCAAAQBAJ&q=applicable+mathematics&pg=PR17 Survey of Applicable Mathematics, pg xvii (Foreword). ] K. Rektorys; 2nd edition, illustrated. Springer, 2013. {{ISBN|9401583080}}, 9789401583084.</ref><ref>[https://www.math.ust.hk/~mahsieh/APMATH.htm  THOUGHTS ON APPLIED MATHEMATICS.]</ref><ref>[http://stellamariscollege.org/documents/icaml.pdf  INTERNATIONAL CONFERENCE ON APPLICABLE MATHEMATICS (ICAM-2016).] {{Webarchive|url=https://web.archive.org/web/20170323142900/http://stellamariscollege.org/documents/icaml.pdf |date=2017-03-23 }} The  Department  of  Mathematics,  Stella  Maris College.</ref> With this outlook, the terms applied mathematics and applicable mathematics are thus interchangeable.

== Utility ==
[[File:Market Data Index NYA on 20050726 202628 UTC.png|right|thumb|[[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]],<ref>Von Zur Gathen, J., & Gerhard, J. (2013). Modern computer algebra. Cambridge University Press.</ref><ref>Geddes, K. O., Czapor, S. R., & Labahn, G. (1992). Algorithms for computer algebra. Springer Science & Business Media.</ref><ref>Albrecht, R. (2012). Computer algebra: symbolic and algebraic computation (Vol. 4). Springer Science & Business Media.</ref><ref>Mignotte, M. (2012). Mathematics for computer algebra. Springer Science & Business Media.</ref> [[numerical analysis]]<ref name="stoer">Stoer, J., & Bulirsch, R. (2013). Introduction to numerical analysis. Springer Science & Business Media.</ref><ref name="conte">Conte, S. D., & De Boor, C. (2017). Elementary numerical analysis: an algorithmic approach. [[Society for Industrial and Applied Mathematics]].</ref><ref name="green">Greenspan, D. (2018). Numerical Analysis. CRC Press.</ref><ref name="linz">Linz, P. (2019). Theoretical numerical analysis. Courier Dover Publications.</ref>). [[Statistics]] is probably the most widespread [[mathematical science]] used in the [[social sciences]].

== 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 [[United Kingdom|U.K]]. host departments of ''Applied Mathematics and Theoretical Physics'',<ref>For example see, [http://www.tait.ac.uk/History.html The Tait Institute: History (2nd par.)]. Accessed Nov 2012.</ref><ref>[http://www.am.qub.ac.uk Dept of Applied Mathematics & Theoretical Physics.] [[Queen's University, Belfast]].</ref><ref>[https://www.researchgate.net/institution/Queens_University_Belfast/department/Department_of_Applied_Mathematics_Theoretical_Physics DAMTP Belfast ResearchGate page].</ref> but it is now much less common to have separate departments of pure and applied mathematics. A notable exception to this is the [[Faculty of Mathematics, University of Cambridge|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]].
[[File:Brown_University_Applied_Mathematics_building.jpg|thumb|The [[Brown University]] Division of Applied Mathematics is the oldest applied math program in the U.S.<ref name=":9">{{Cite book|last=Suzuki|first=Jeff|url=https://books.google.com/books?id=lew5IC5piCwC|title=Mathematics in Historical Context|date=2009-08-27|publisher=MAA|isbn=978-0-88385-570-6|pages=374|language=en}}</ref><ref>{{Cite journal|last1=Greenberg|first1=John L.|last2=Goodstein|first2=Judith R.|date=1983-12-23|title=Theodore von Kármán and Applied Mathematics in America|url=https://www.ams.org/publicoutreach/math-history/hmath2-greenberg.pdf|journal=Science|volume=222|issue=4630|pages=1300–1304|doi=10.1126/science.222.4630.1300|pmid=17773321|bibcode=1983Sci...222.1300G |s2cid=19738034}}</ref>]]
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 [[Master of Science|M.S.]] in applied mathematics.<ref>{{citation|title=Santa Clara University Dept of Applied Mathematics |url=http://www.scu.edu/academics/bulletins/undergraduate/Department-of-Applied-Mathematics.cfm |access-date=2011-03-05 |url-status=dead |archive-url=https://web.archive.org/web/20110504005925/http://www.scu.edu/academics/bulletins/undergraduate/Department-of-Applied-Mathematics.cfm |archive-date=2011-05-04 }}</ref> Research universities dividing their mathematics department into pure and applied sections include [[MIT]]. Students in this program also learn another skill (computer science, engineering, physics, pure math, etc.) to supplement their applied math skills.

== Associated mathematical sciences ==
[[File:Oldfaithful3.png|thumb|right|Applied mathematics has substantial overlap with statistics.]]

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]] includes applied mathematics (especially [[numerical analysis]]<ref name="stoer"/><ref name="conte"/><ref name="green"/><ref name="linz"/><ref>Today, numerical analysis includes [[numerical linear algebra]], [[numerical integration]], and [[validated numerics]] as subfields.</ref>), [[computing science]] (especially [[high-performance computing]]<ref>Hager, G., & Wellein, G. (2010). Introduction to high performance computing for scientists and engineers. CRC Press.</ref><ref>Geshi, M. (2019). The Art of High Performance Computing for Computational Science, Springer.</ref>), and mathematical modelling in a scientific discipline.

===Computer science===
[[Computer science]] relies on [[logic]], [[algebra]], [[discrete mathematics]] such as [[graph theory]],<ref>West, D. B. (2001). Introduction to graph theory (Vol. 2). Upper Saddle River: Prentice Hall.</ref><ref>Bondy, J. A., & Murty, U. S. R. (1976). Graph theory with applications (Vol. 290). London: Macmillan.</ref> and [[combinatorics]].

===Operations research and management science===
[[Operations research]]<ref>Winston, W. L., & Goldberg, J. B. (2004). Operations research: applications and algorithms (Vol. 3). Belmont: Thomson Brooks/Cole.</ref> and [[management science]] are often taught in faculties of engineering, business, and public policy.

===Statistics===
Applied mathematics has substantial overlap with the discipline of statistics. [[Statisticians|Statistical theorists]] study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions. Statistical theory relies on [[Probability theory|probability]] and [[optimal decision|decision theory]], and makes extensive use of scientific computing, analysis, and [[Mathematical optimization|optimization]]; for the [[design of experiments]], statisticians use [[algebraic statistics|algebra]] and [[combinatorial design]]. Applied mathematicians and [[statistician]]s 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.<ref>Boland, P. J. (2007). Statistical and probabilistic methods in actuarial science. CRC Press.</ref>

===Mathematical economics===
[[Mathematical economics]] is the application of mathematical methods to represent theories and analyze problems in economics.<ref>Wainwright, K. (2005). Fundamental methods of mathematical economics/Alpha C. Chiang, Kevin Wainwright. Boston, Mass.: McGraw-Hill/Irwin,.</ref><ref>Na, N. (2016). Mathematical economics. Springer.</ref><ref>Lancaster, K. (2012). Mathematical economics. Courier Corporation.</ref> 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 economics|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.<ref>Roberts, A. J. (2009). Elementary calculus of financial mathematics (Vol. 15). SIAM.</ref>

According to the [[Mathematics Subject Classification]] (MSC), mathematical economics falls into the [[Mathematics Subject Classification#Applied mathematics / other|Applied mathematics/other]] classification of category 91:
:Game theory, economics, social and behavioral sciences

with [http://msc2010.org/mscwiki/index.php?title=MSC2010 MSC2010] classifications for '[[Game theory]]' at codes [http://msc2010.org/mscwiki/index.php?title=91Axx 91Axx] {{Webarchive|url=https://web.archive.org/web/20150402092147/http://msc2010.org/mscwiki/index.php?title=91Axx |date=2015-04-02 }} and for 'Mathematical economics' at codes [http://msc2010.org/mscwiki/index.php?title=91Bxx 91Bxx] {{Webarchive|url=https://web.archive.org/web/20150402155902/http://msc2010.org/mscwiki/index.php?title=91Bxx |date=2015-04-02 }}.

===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]], [[information theory]], and [[mathematical physics]].

== See also ==
{{Portal|Mathematics}}
* [[Analytics]]
* [[Applied science]]
* [[Engineering mathematics]]
* [[Society for Industrial and Applied Mathematics]]
{{clear}}

== References ==
{{reflist}}

==Further reading==
===Applicable mathematics===
*[https://web.archive.org/web/20140407103033/http://www2.moreheadstate.edu/mejam/index.aspx?id=5096 The Morehead Journal of Applicable Mathematics] hosted by [[Morehead State University]]
*[http://www.worldscientific.com/series/scam Series on Concrete and Applicable Mathematics] by [[World Scientific]]
*[https://web.archive.org/web/20140513162457/http://www.barnesandnoble.com/s/?series_id=194353 Handbook of Applicable Mathematics Series] by [[Walter Ledermann]]

==External links==
{{Wikiversity|School:Mathematics|Applicable Mathematics}}
{{Wikibooks|Applicable Mathematics}}
*{{Commons category-inline}}
* The [http://www.siam.org/ 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, [[Society for Industrial and Applied Mathematics|SIAM]] is a major publisher of research journals and books in applied mathematics.
*[https://web.archive.org/web/20130329132423/http://math.nd.edu/research/research-groups-in-mathematics/applicable-mathematics/ The Applicable Mathematics Research Group] at [[Notre Dame University]] (archived 29 March 2013)
*[https://web.archive.org/web/20180401213544/http://www.hope.ac.uk/research/researchcentres/researchcentredetails/centreforapplicablemathematics/ Centre for Applicable Mathematics] at [[Liverpool Hope University]] (archived 1 April 2018)
*[https://web.archive.org/web/20160304194828/http://www.gcu.ac.uk/ebe/aboutus/subjectgroups/applicablemathematics/ Applicable Mathematics research group] at [[Glasgow Caledonian University]] (archived 4 March 2016)

{{Industrial and applied mathematics}}
{{Areas of mathematics}}
{{Authority control}}

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[[Category:Applied mathematics| ]]