Applied mathematics

applied mathematicianappliedapplications of mathematicsApplied MathIndustrial Mathematicsapplied mathematiciansApplied and Computational Mathematicsapplicable mathematicsapplied mathematical researchmathematics
Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry.wikipedia
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Engineering

engineerengineersengineered
Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Many universities teach mathematical and statistical courses outside the respective departments, in departments and areas including business, engineering, physics, chemistry, psychology, biology, computer science, scientific computation, and mathematical physics.
The discipline of engineering encompasses a broad range of more specialized fields of engineering, each with a more specific emphasis on particular areas of applied mathematics, applied science, and types of application.

Mathematics

mathematicalmathmathematician
Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry.
Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory.

Pure mathematics

pureabstract mathematicspure mathematician
In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake.
It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference than a rigid subdivision of mathematics.

Henri Poincaré

PoincaréJules Henri PoincaréH. Poincaré
Many mathematicians distinguish between "applied mathematics”, which is concerned with mathematical methods, and the "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, mathematical biologists have posed problems that have stimulated the growth of pure mathematics. Mathematicians such as Poincaré and Arnold deny the existence of "applied mathematics" and claim that there are only "applications of mathematics." Similarly, non-mathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to solve industrial problems is also called "industrial mathematics".
As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics.

Computational mathematics

computationalcomputational mathematical programmingcomputational mathematician
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 the solution of problems in the sciences and engineering.
Computational applied mathematics consists roughly of using mathematics for allowing and improving computer computation in applied mathematics.

Mathematical analysis

analysisclassical analysisanalytic
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.
It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, mechanical engineering, electrical engineering, and particularly, quantum field theory.

James Lighthill

Sir James LighthillMichael James LighthillLighthill
A notable exception to this is the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, housing the Lucasian Professor of Mathematics whose past holders include Isaac Newton, Charles Babbage, James Lighthill, Paul Dirac and Stephen Hawking.
Sir Michael James Lighthill, (23 January 1924 – 17 July 1998) was a British applied mathematician, known for his pioneering work in the field of aeroacoustics.

Theoretical computer science

theoretical computer scientisttheoreticalcomputer science
The advent of the computer has enabled new applications: studying and using the new computer technology itself (computer science) to study problems arising in other areas of science (computational science) as well as the mathematics of computation (for example, theoretical computer science, computer algebra, numerical analysis).
The field is broadly defined and includes foundations in computer science, applied mathematics, animation, statistics, biochemistry, chemistry, biophysics, molecular biology, genetics, genomics, ecology, evolution, anatomy, neuroscience, and visualization.

University of Cambridge

Cambridge UniversityCambridgeUniversity
A notable exception to this is the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, housing the Lucasian Professor of Mathematics whose past holders include Isaac Newton, Charles Babbage, James Lighthill, Paul Dirac and Stephen Hawking.
From the time of Isaac Newton in the later 17th century until the mid-19th century, the university maintained an especially strong emphasis on applied mathematics, particularly mathematical physics.

Physics

physicistphysicalphysicists
This history left a pedagogical legacy in the United States: 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. Many universities teach mathematical and statistical courses outside the respective departments, in departments and areas including business, engineering, physics, chemistry, psychology, biology, computer science, scientific computation, and mathematical physics.
The approach is similar to that of applied mathematics.

Asymptotic analysis

asymptoticasymptoticallyasymptotics
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.

Differential equation

differential equationsdifferentialsecond-order differential equation
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.
The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering.

Operations research

operational researchoperation researchoperational analysis
Operations research and management science are often taught in faculties of engineering, business, and public policy.
It is often considered to be a sub-field of applied mathematics.

Game theory

gamegame theoristgame theoretic
However, since World War II, fields outside the physical sciences have spawned the creation of new areas of mathematics, such as game theory and social choice theory, which grew out of economic considerations.
As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors.

Mathematical optimization

optimizationmathematical programmingoptimal
Statistical theory relies on probability and decision theory, and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorial design.
The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics.

Economics

economiceconomisteconomic theory
Statistics is probably the most widespread mathematical science used in the social sciences, but other areas of mathematics, most notably economics, are proving increasingly useful in these disciplines.
Game theory is a branch of applied mathematics that considers strategic interactions between agents, one kind of uncertainty.

Mathematical economics

mathematical economisteconomicsmathematical
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics.
By convention, these applied methods are beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods.

Mathematical finance

financial mathematicsquantitative financequantitative trading
Quantitative finance is now taught in mathematics departments across universities and mathematical finance is considered a full branch of applied mathematics.
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.

Management science

Management Sciencesmanagement theorymanagement scientist
Operations research and management science are often taught in faculties of engineering, business, and public policy.
The field was initially an outgrowth of applied mathematics, where early challenges were problems relating to the optimization of systems which could be modeled linearly, i.e., determining the optima (maximum value of profit, assembly line performance, crop yield, bandwidth, etc. or minimum of loss, risk, costs, etc.) of some objective function.

Brown University

BrownCollege in the English Colony of Rhode Island and Providence PlantationsBrown Association for Cooperative Housing
Schools with separate applied mathematics departments range from Brown University, which has a large Division of Applied Mathematics that offers degrees through the doctorate, to Santa Clara University, which offers only the M.S. in applied mathematics.
20 different master's degrees are offered as well as Ph.D. degrees in over 40 subjects ranging from applied mathematics to public policy.

Mathematical physics

mathematical physicistmathematicalmathematical physicists
Many universities teach mathematical and statistical courses outside the respective departments, in departments and areas including business, engineering, physics, chemistry, psychology, biology, computer science, scientific computation, and mathematical physics.
It is a branch of applied mathematics, but deals with physical problems.

Engineering mathematics

Mathematical EngineeringApplied Mathematics Engineering
Engineering mathematics is a branch of applied mathematics concerning mathematical methods and techniques that are typically used in engineering and industry.

Computational engineering

Computational Science and EngineeringEngineeringComputational Engineering Science
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 the solution of problems in the sciences and engineering.

Society for Industrial and Applied Mathematics

SIAMSIAM ReviewSociety for Industrial and Applied Mathematics (SIAM)
The society supports educational institutions promoting applied mathematics.

Statistician

statisticiansStatistical theoristsstatistical analyst
Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions.
Additionally, there is a substantial number of people who use statistics and data analysis in their work but have job titles other than statistician, such as actuaries, applied mathematicians, economists, data scientists, data analysts (predictive analytics), financial analysts, psychometricians, sociologists, epidemiologists, and quantitative psychologists.