User:Michael P. Barnett/Applied mathematics

Applied mathematics consists of formulas and other mathematical information that has practical application. The "other" consists of theorems, methods of reasoning and algorithms.

Mathematics faculty are often grouped in administrative units that are called "Applied" and "Pure", respectively, by reference to the branches of mathematics that they address. For example, numerical analysis is considered "applied", and topology is considered "pure".

All (or most) of the work that is done in an administrative unit with the name "applied" has practical application. But so does some of the work in almost every branch of mathematics that is classified as "pure". Also, practical applications are found, quite often, for work that was considered pure when it was published, sometimes decades later.

Universities that have units with the precise name "Department of applied mathematics" (in the U.S., in early 2011) include Colorado and Washington. Brown University has a "Division of Applied Mathematics".

Applied mathematics is surveyed in compendia that include Pearson's[5] and Rektorys'.[6] The key term "applied mathematics" brings up hundreds of titles on the web sites of major publishers that include Oxford University Press,[7] Cambridge University Press[8] and Elsevier.[9] Many of these are oriented to particular areas of application. They vary widely in sophistication and choice of material. Hundreds of journals publish applied mathematical methods and applications. Most of these focus on particular methods or particular areas of application.

The names of many university departments now join "Applied mathematics" with other topics -- for example "Applied mathematics and theoretical physics"[10] at the University of Cambridge, and "Department of Applied Mathematics & Statistics"[11] at SUNY Stony Brook. Often, it would be artificial to try discussing separately an application and the mathematical methods that solved it. For example, the equation of lunar motion, the Brusselator equation and the magnetron equation are known by the applications that give rise to them.

Several names besides "applied" connote bodies of mathematics that have practical applications. Some of these come from books that were widely used. They include Mathematics of physics and chemistry,[12] Engineering mathematics, and Industrial mathematics. The adjectives Theoretical and Mathematical are applied interchangeably to chemistry, physics and biology, and these terms cover work that qualifies as applied mathematics. So do Finite mathematics[13] and Applicable mathematics (derived from the Angewandte Matematik of continental Europe -- see page 21 of the Rektorys compendium[6]) Parts of Statistics, Mathematical logic and Computational mathematics qualify as applied mathematics, too. Mathematical modeling uses methods of applied mathematics.

Topics that are covered in the compendia and other works cited above include basic arithmetic, algebra and trigonometry, plane curves, solid geometry, differential geometry, ordinary and partial differential equations (further categorized as linear and non-linear), special functions of mathematical physics, theory of the potential, vector and tensor calculus, calculus of variations, eigenvalue and eigenfunction theory, classical mechanics, linear algebra, methods of quantum mechanics and statistical mechanics, integral equations, group theory, series expansion, linear programming, combinatorics, applied probability and statistics, infinite series and products, Fourier analysis, conformal mapping, polynomial systems, differential and integral calculus of one a single variable and of several variables, orthogonal systems and vector spaces. Some of these overlap, and the list is just illustrative and by no means exhaustive.

further references will be put in asap