Applied mathematics
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'''Applied mathematics''' is a branch of [[mathematics]] that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains.
==Divisions of applied mathematics==
There is no consensus of 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 [[Mathematical analysis|applied analysis]], most notably [[differential equations]], [[approximation theory]] (broadly construed, to include [[Representation (mathematics)|representation]]s, [[asymptotic]] methods, [[Calculus of variations|variational methods]], and [[numerical analysis]]), and [[Probability|applied probability]]. 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]] 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.
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]] that are part of [[pure mathematics]] are now important in applications (such as [[cryptology]]), 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 field of applied mathematics and the many more areas of mathematics that are applicable to real-world problems.
Mathematicians distinguish between applied mathematics, which is concerned with mathematical methods, and applications of mathematics within science and [[Engineering mathematics|engineering]]. A [[biology|biologist]] using a [[Matrix population models|population model]] and applying known mathematics would not be ''doing'' applied mathematics, but rather ''using'' it. However, nonmathematicians do not usually draw this distinction.
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 solution of problems in the [[science]]s and [[engineering]]. These are often considered interdisciplinary programs.
Some mathematicians think that [[statistics]] is a part of applied mathematics. Others think it is a separate discipline. Statisticians in general regard their field as separate from mathematics, and the [[American Statistical Association]] has issued a statement to that effect{{Fact|date=April 2007}}. [[Mathematical statistics]] provides the [[theorem]]s and [[proof (mathematics)|proofs]] that justify statistical procedures and it is based on [[probability theory]], which is in turn based on [[measure theory]].
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]] and [[economics]], [[engineering]], [[physics]], [[psychology]], [[biology]], [[computer science]], and [[mathematical physics]]. Sometimes this is due to these areas having their own specialized mathematical dialects. Often this is the result of efforts of those departments to gain more student credit hours and the funds that go with them.{{POV-statement|date=December 2007}}
==Usefulness of applied mathematics==
Historically, mathematics was most important in the [[natural sciences]] and [[engineering]]. However, in recent years, fields outside of the physical sciences have spawned the creation of new areas of mathematics, such as [[game theory]], which grew out of economic considerations, or [[neural networks]], which arose out of the study of the brain in [[neuroscience]], or [[bioinformatics]], from the importance of analyzing large data sets in biology.
The advent of the computer has created new applications, both in studying and using the new computer technology itself ([[computer science]], which uses [[combinatorics]], [[formal logic]], and [[lattice theory]]), as well as using computers to study problems arising in other areas of science ([[computational science]]), and of course studying the mathematics of computation ([[numerical analysis]]). [[Statistics]] is probably the most widespread application of mathematics in the [[social sciences]], but other areas of math are proving increasingly useful in these disciplines, especially in [[economics]] and [[management science]].
==Status in academic departments==
{{Unreferencedsection|date=February 2008}}
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 [[United Kingdom|UK]] host departments of ''Applied Mathematics and Theoretical Physics'', but it is now much less common to have separate departments of pure and applied mathematics. Schools with separate applied mathematics departments range from [[Brown University]], which has a well-known and large Division of Applied Mathematics that offers degrees through the [[doctorate]], to [[Santa Clara University]], which offers only the [[M.S.]] in applied mathematics{{Fact|date=February 2008}}. Research universities dividing their mathematics department into pure and applied sections include [[Harvard]], [[Caltech]], and [[MIT]].
At some universities there is some tension between applied and pure mathematics departments. One reason is that pure mathematics is often perceived as having a higher intellectual standing. Another reason is a different level of compensation, as applied mathematicians are often paid more. Applied mathematics also enjoys better opportunities to bring external funding from many sources, not limited to the [http://www.nsf.gov/div/index.jsp?org=DMS Division of Mathematical Sciences] at the [[National Science Foundation]] (NSF) like much of pure mathematics. External funding is highly valued at [[research university|research universities]] and is often a condition for faculty advancement. Similar tensions can also exist between statistics and mathematics groups and departments.
== See also ==
*[[Pure mathematics]]
*[[Computational engineering]]
==External links==
{{Wikibookspar|School of Mathematics|Applied Mathematics}}
* 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.
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[[Category:Applied mathematics| ]]
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[[da:Anvendt matematik]]
[[de:Angewandte Mathematik]]
[[hi:व्यावहारिक गणित]]
[[es:Matemática aplicada]]
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[[fa:ریاضیات کاربردی]]
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[[ko:응용수학]]
[[he:מתמטיקה שימושית]]
[[ka:გამოყენებითი მათემატიკა]]
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[[ja:応用数学]]
[[no:Anvendt matematikk]]
[[nov:Aplikat matematike]]
[[pl:Matematyka stosowana]]
[[pt:Matemática aplicada]]
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[[fi:Sovellettu matematiikka]]
[[sv:Tillämpad matematik]]
[[ta:செயல்முறைக் கணிதம்]]
[[th:คณิตศาสตร์ประยุกต์]]
[[vi:Toán học ứng dụng]]
[[uk:Прикладна математика]]
[[vo:Matemat pajäfüköl]]
[[zh:应用数学]]