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 applied analysis, most notably differential equations, approximation theory (broadly construed, to include representations, variational methods, and numerical analysis), and 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 much broader sense. It includes the classical areas of analysis such as differential equations, as well as linear algebra, numerical analysis, probability, operations research, and other areas. Recently, fields such as number theory and topology, often thought to be pure mathematics, have become increasingly important as applications, though they are not generally considered to be part of the field of applied mathematics per se.

Mathematicians distinguish between applied mathematics, which is concerned with mathematical methods, and applications of mathematics within science and engineering. A biologist using a population model and applying known mathematics would not be doing applied mathematics, but rather using it. However, nonmathematicians do not usually draw this distinction.

Computational Mathematics

The success of modern numerical software has led to the emergence of omputational science as a distinct part of applied mathematics. Computational science (and computational engineering) utilizes high performance computing for the simulation of phenomena and solution of problems in the sciences and engineering, and is often considered an interdisciplinary program.

Statistics

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. As a practical matter, classes in basic statistics are often taught by applied mathematicians.

Related areas of Application

The line between applied mathematics and specific areas of application is often blurred. Many universities teach mathematical courses outside of mathematics departments, in departments including business and economics, engineering, physics, psychology, or statistics. Sometimes this is due to these areas having their own specialized mathematical dialects. This is especially true in computer science, economics, engineering, and physics.

Usefulness of Applied Mathematics

Historically, mathematics was most important in the natural sciences and engineering. However, in recent years, fields outside of the hard 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.

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, 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

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 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. Research universities dividing their mathematics department into pure and applied sections include MIT.

At some universities there is a considerable amount of tension between applied and pure mathematics departments, or between applied and pure groups within a single department. One reason is a different level of compensation, faculty specializing in applied mathematics are typically paid more. Applied mathematics also enjoys better opportunities to bring external funding from many sources, not limited to the Division of Mathematical Sciences at the National Science Foundation (NSF) like much of pure mathematics. External funding is highly valued at 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

• The 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, SIAM is a major publisher of research journals and books in applied mathematics.