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Encyclopedia > 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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...

Contents

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 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. Analysis has its beginnings in the rigorous formulation of calculus. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ... Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Probability is the likelihood that something is the case or will happen. ... Classical mechanics is a model of the physics of forces acting upon bodies. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Fluid mechanics is the subdiscipline of continuum mechanics that studies fluids, that is, liquids and gases. ...


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 are 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. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... Cryptology is an umbrella term for cryptography and cryptanalysis. ...


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. To meet Wikipedias quality standards, this article may require cleanup. ... Biology studies the variety of life (clockwise from top-left) E. coli, tree fern, gazelle, Goliath beetle Biology (from Greek: βίος, bio, life; and λόγος, logos, knowledge), also referred to as the biological sciences, is the study of living organisms utilizing the scientific method. ... Population models are used in population ecology to model the dynamics of wildlife or human populations. ...


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. Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods. ... Computational Science is the use of computers to perform research in other fields. ... It has been suggested that this article or section be merged with Computational Science. ... The field of high performance computing (HPC) comprises computing applications on (parallel) supercomputers and computer clusters. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Engineering is the discipline of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. ...


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[citation needed]. Mathematical statistics provides the theorems and proofs that justify statistical procedures and it is based on probability theory, which is in turn based on measure theory. This article is about the field of statistics. ... The American Statistical Association (ASA) is a scientific and educational society in the United States with the stated mission to promote excellence in the application of statistical science across the wealth of human endeavor. ... Mathematical statistics uses probability theory and other branches of mathematics to study statistics from a purely mathematical standpoint. ... Look up theorem in Wiktionary, the free dictionary. ... In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... In mathematics, a measure is a function that assigns a number, e. ...


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. In economics, a business is a legally-recognized organizational entity existing within an economically free country designed to sell goods and/or services to consumers, usually in an effort to generate profit. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... Engineering is the discipline of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Psychological science redirects here. ... Biology studies the variety of life (clockwise from top-left) E. coli, tree fern, gazelle, Goliath beetle Biology (from Greek: βίος, bio, life; and λόγος, logos, knowledge), also referred to as the biological sciences, is the study of living organisms utilizing the scientific method. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories. ...


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 term natural science as the way in which different fields of study are defined is determined as much by historical convention as by the present day meaning of the words. ... Engineering is the discipline of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. ... Game theory is a branch of applied mathematics that is often used in the context of economics. ... A neural network is an interconnected group of neurons. ... Drawing of the cells in the chicken cerebellum by S. Ramón y Cajal Neuroscience is a field that is devoted to the scientific study of the nervous system. ...


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. Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... Computational Science is the use of computers to perform research in other fields. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... This article is about the field of statistics. ... The social sciences are a group of academic disciplines that study human aspects of the world. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... Management science, or MS, is the discipline of using mathematics, and other analytical methods, to help make better business decisions. ...


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 Harvard and MIT. Brown University is a private university located in Providence, Rhode Island. ... The Santa Clara Mission is a notable on-campus landmark. ... A masters degree is an academic degree usually awarded for completion of a postgraduate course of one or two years in duration. ... Harvard University is a private university in Cambridge, Massachusetts, USA, and a member of the Ivy League. ... Mapúa Institute of Technology (MIT, MapúaTech or simply Mapúa) is a private, non-sectarian, Filipino tertiary institute located in Intramuros, Manila. ...


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 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. The logo of the National Science Foundation The National Science Foundation (NSF) is an independent United States government agency that supports fundamental research and education in all the non-medical fields of science and engineering. ... A university is an institution of higher education and of research, which grants academic degrees. ...


See also

Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... It has been suggested that this article or section be merged with Computational Science. ...

External links

Wikibooks
Wikibooks' School of Mathematics has more about this subject:
Applied Mathematics
  • 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.

  Results from FactBites:
 
NJIT - Graduate Programs: Mathematics (2760 words)
Applied Mathematics is the application of classical and modern mathematical techniques to the solution of practical problems in the physical and biological sciences and engineering.
The applied mathematician develops and analyzes mathematical models of physical and biological phenomena and engineering systems, interprets solutions to mathematical problems and uses the results to identify relationships, patterns, and the effects of altering one or more variables or modeling assumptions.
The Master of Science in Applied Mathematics, with its areas of specialization in analysis, applied mathematics, computational methods, and mathematical biology is designed to serve the needs of students who may be interested in pursuing a doctoral degree in the mathematical, physical, or biological sciences.
  More results at FactBites »

 
 

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