Applied mathematics

== 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.