Everything about Applied Mathematics totally explained
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,
asymptotic methods,
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 wasn't 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 broader sense. It includes the classical areas 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
cryptology), though they're not generally considered to be part of the field of applied mathematics
per se. Sometimes the term
applicable mathematics is used to distinguish between the traditional field of applied mathematics and the many more areas of mathematics that are applicable to real-world problems.
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 wouldn't be
doing applied mathematics, but rather
using it. However, nonmathematicians don't usually draw this distinction.
The success of modern numerical mathematical methods and software has led to the emergence of
computational mathematics,
computational science, and
computational engineering, which use
high performance computing for the simulation of phenomena and solution of problems in the
sciences and
engineering. These are often considered interdisciplinary programs.
Some mathematicians think that
statistics is a part of applied mathematics. Others think it's 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.
Mathematical statistics provides the
theorems and
proofs that justify statistical procedures and it's based on
probability theory, which is in turn based on
measure theory.
The line between applied mathematics and specific areas of application is often blurred. Many universities teach mathematical and statistical courses outside of the respective departments, in departments and areas including
business and
economics,
engineering,
physics,
psychology,
biology,
computer science, and
mathematical physics. Sometimes this is due to these areas having their own specialized mathematical dialects. Often this is the result of efforts of those departments to gain more student credit hours and the funds that go with them.
Usefulness of applied mathematics
Historically, mathematics was most important in the
natural sciences and
engineering. However, in recent years, fields outside of the physical 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, or
bioinformatics, from the importance of analyzing large data sets in biology.
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 (
computational 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's 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's 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
Harvard,
Caltech, and
MIT.
At some universities there's a considerable amount of tension between applied and pure mathematics departments, or between applied and pure groups within a single department. One reason is that pure mathematics is often perceived as having a higher intellectual standing. Another reason is a different level of compensation, as applied mathematicians are often 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.
Further Information
Get more info on 'Applied Mathematics'.
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