Applied mathematics
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| Applied mathematics | |
|---|---|
| Name | Applied mathematics |
| Key people | Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, John von Neumann |
| Related fields | Physics, Engineering, Computer science, Statistics, Operations research |
Applied mathematics. It is the application of mathematical methods by diverse fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.
Overview and scope
The primary goal is to develop and apply mathematical tools to solve real-world problems arising in science, commerce, and society. Historically, it grew from fields like classical mechanics and astronomy, with foundational figures including Isaac Newton and Gottfried Wilhelm Leibniz developing calculus to describe motion. Modern scope extends far beyond, interfacing with computational science and data science. Key institutions promoting its development include the Society for Industrial and Applied Mathematics and the Institute for Mathematics and its Applications.
Core disciplines and branches
Several well-established areas form the backbone. Numerical analysis focuses on algorithms for solving mathematical problems numerically, crucial for simulations. Applied probability and stochastic processes model randomness in systems, foundational for queuing theory. Mathematical physics uses advanced mathematics for theories like quantum mechanics and general relativity. Optimization seeks the best solution from a set of alternatives, central to operations research. Other vital branches include dynamical systems, control theory, and fluid dynamics.
Methods and techniques
Practitioners employ a vast toolkit. Mathematical modeling translates a real-world system into equations. Scientific computing and computational mathematics implement numerical methods on computers, using languages like Fortran and MATLAB. Asymptotic analysis approximates complex equations. Statistical inference draws conclusions from data. Techniques from partial differential equations describe continuous systems, while discrete mathematics and graph theory model networks. The finite element method and Monte Carlo method are quintessential computational techniques.
Applications in science and engineering
It is indispensable in the physical sciences. In physics, it underpins electromagnetism, thermodynamics, and particle physics. Engineering applications span civil engineering (structural analysis), aerospace engineering (computational fluid dynamics), and electrical engineering (signal processing). It drives innovation in materials science and nanotechnology. Major projects like the Manhattan Project and the Apollo program relied heavily on applied mathematical work, with contributions from figures like John von Neumann and Richard Feynman.
Applications in other fields
Its reach extends broadly. In biology and medicine, it fuels mathematical biology, epidemiology, and biomechanics, modeling phenomena from the Human Genome Project to the spread of diseases. Economics and finance use mathematical finance, econometrics, and game theory, with foundational models like the Black–Scholes equation. It is critical in computer science for cryptography, algorithm design, and artificial intelligence. Other applications include psychometrics, linguistics, and even digital humanities.
Relationship with pure mathematics
The boundary is fluid and synergistic. While pure mathematics is driven by internal logic and abstraction, applied mathematics is motivated by external problems. However, discoveries in number theory become essential for cryptography, and abstract group theory finds use in quantum physics. Many mathematicians, such as Henri Poincaré and David Hilbert, made profound contributions to both. Institutions like the Clay Mathematics Institute recognize deep interconnections, and fields like topology now inform data analysis.
Education and careers
Typical education involves advanced degrees focusing on both theory and implementation. Many universities, including the Massachusetts Institute of Technology, Stanford University, and the University of Cambridge, offer specialized programs. Careers are diverse, with applied mathematicians working as quants on Wall Street, researchers at Los Alamos National Laboratory or NASA, data scientists at Google or Meta Platforms, and consultants in industries ranging from pharmaceuticals to logistics. Professional societies like the American Mathematical Society provide critical forums and publications.