In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study inpure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.
Historically, applied mathematics consisted principally ofapplied analysis, most notablydifferential equations;approximation theory (broadly construed, to includerepresentations,asymptotic methods,variational methods, andnumerical analysis); and appliedprobability. These areas of mathematics related directly to the development ofNewtonian 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 asclassical mechanics were often taught in applied mathematics departments at American universities rather than inphysics departments, andfluid mechanics may still be taught in applied mathematics departments.[1]Engineering andcomputer science departments have traditionally made use of applied mathematics.
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 asnumber theory that are part ofpure mathematics are now important in applications (such ascryptography), though they are not generally considered to be part of the field of applied mathematicsper 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. Abiologist using apopulation model and applying known mathematics would not bedoing applied mathematics, but ratherusing it; however, mathematical biologists have posed problems that have stimulated the growth of pure mathematics. Mathematicians such asPoincaré andArnold 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]
Sometimes, the termapplicable 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.[3]
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.[3] 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.[4] 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 ascryptography), though they are not generally considered to be part of the field of applied mathematicsper se. Such descriptions can lead toapplicable mathematics being seen as a collection of mathematical methods such asreal analysis,linear algebra,mathematical modelling,optimisation,combinatorics,probability andstatistics, which are useful in areas outside traditional mathematics and not specific tomathematical physics.
Other authors prefer describingapplicable mathematics as a union of "new" mathematical applications with the traditional fields of applied mathematics.[4][5][6] With this outlook, the terms applied mathematics and applicable mathematics are thus interchangeable.
Historically, mathematics was most important in thenatural sciences andengineering. However, sinceWorld War II, fields outside the physical sciences have spawned the creation of new areas of mathematics, such asgame theory andsocial 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 asmathematical finance anddata science.
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".
TheBrown University Division of Applied Mathematics is the oldest applied math program in the U.S.[18][19]
Schools with separate applied mathematics departments range fromBrown University, which has a large Division of Applied Mathematics that offers degrees through thedoctorate, toSanta Clara University, which offers only theM.S. in applied mathematics.[20] Research universities dividing their mathematics department into pure and applied sections includeMIT. Students in this program also learn another skill (computer science, engineering, physics, pure math, etc.) to supplement their applied math skills.
An essential discipline for many fields in engineering is that ofcontrol engineering. The associated mathematical theory of this specialism iscontrol theory, a branch of applied mathematics that builds off the mathematics ofdynamical systems. Control theory has played a significant enabling role in modern technology, serving a foundational role inelectrical, mechanical and aerospace engineering. Like continuum mechanics, control theory has also become a field of mathematical research in its own right, with mathematicians such asAleksandr Lyapunov,Norbert Wiener,Lev Pontryagin andfields medallistPierre-Louis Lions contributing to its foundations.
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 onprobability anddecision theory, and makes extensive use of scientific computing, analysis, andoptimization; for thedesign of experiments, statisticians usealgebra andcombinatorial design. Applied mathematicians andstatisticians often work in a department of mathematical sciences (particularly at colleges and small universities).
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics.[28][29][30] The applied methods usually refer to nontrivial mathematical techniques or approaches. Mathematical economics is based on statistics, probability, mathematical programming (as well as othercomputational 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.[31]
TheSociety 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 is a major publisher of research journals and books in applied mathematics.
