# Applied mathematics

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

Traditionally, applied mathematics consisted of three major areas: Approximation theory (including representation theory and computational methods); differential equations (especially partial differential equations); and applied probability. One could even do so-called "theoretical applied mathematics" in these areas, by performing research on the fundamentals of the subjects.

Many statisticians contend that statistics is a separate discipline from mathematics, but in practice both subjects are often taught in the math department. Engineering mathematics and mathematical physics describe physical processes, and so are almost indistinguishable from theoretical physics. Traditionally, classical mechanics was often taught in applied math departments at American universities rather than in physics departments.

Today the term applied mathematics is often used in a much broader sense. Mathematicians generally consider it incorrect to conflate applied mathematics (a subset of mathematics) with applications of mathematics (the actual act of applying mathematics to real-world problems). But scientists and social scientists who utilize mathematics in their work do not usually make this distinction.

An example may serve to sharpen the somewhat fuzzy lines separating traditional applied mathematics from pure mathematics and distinguishing both of these from applicable mathematics, or the use of mathematics as a tool. A "smooth" function, such as cosx, can be represented by a Taylor series containing a countably infinite number of terms. Pure mathematics is concerned with the problem of proving that the Taylor series exists, and with the closely associated problems of determining its coefficients and its circle of convergence, or the domain in which it is valid. Applied mathematics addresses the more practical problems of how the series may best be calculated; how many terms must be included to achieve a desired level of precision; and how best to tabulate the resulting values, or perhaps encapsulate the Taylor series within a computer algorithm. Finally, a surveyor who consults the tabulated or computerized values of cosx while making trigonometric calculations is not really doing mathematics – he is simply using results the mathematicians have derived to complete a surveying project.

Some branches of mathematics – differential equations (ODEs and PDEs), matrix theory, continuous modelling, probability, and statistics – are widely applicable to many fields of science and technology. Others – such as numerical analysis, scientific computing, information theory, cryptography, graph theory as applied to network analysis, and theoretical computer science – have fueled the rapid proliferation of digital computers. Problems associated with computer technology have, in their turn, provided the motivation for mathematical advances in all these fields. And the increasing power and speed of the computers themselves have opened new possibilities in computational topology and computational geometry.

Both physics and engineering have their own specialized mathematical dialects, including control theory. Advances in the life sciences have stimulated the development of mathematical biology and have recently generated an entirely new field, bioinformatics. Economics, finance, and insurance have spawned several related disciplines that might be characterized as commercial mathematics. Certain special economic problems gave the initial impetus to game theory. Additional problems from business and commerce have driven mathematical research into optimization techniques, including the widely employed methods of operations research and linear programming. As time passes all these optimization methods are finding new applications in a widening circle of disciplines.

## Segregation within Universities

Some universities in the 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. Many research universities divide their mathematics department into pure and applied sections (e.g., MIT).

Fundamental applied mathematics is taught at second-level in some countries, such as Ireland, where it is a minority option at Leaving Certificate.