Applied mathematics — Safekipedia Discoverer

Applied mathematics is the use of mathematical methods in many different areas, like physics, engineering, medicine, biology, finance, business, computer science, social science, and industry. It mixes the ideas of math with knowledge from these fields to solve real-world problems. People who work in applied mathematics use math to create models that help us understand and predict things in the world around us.

Historically, real-life problems have helped create new math theories. These theories then became important areas of study in pure mathematics, where math is explored for its own beauty and logic. This means that applied mathematics and pure mathematics are closely linked, with each helping to grow the other.

History

A numerical solution to the heat equation on a pump casing model using the finite element method.

Applied mathematics has a long history, starting with areas like solving equations, making approximations, and studying chance. These math topics helped shape physics, especially the work of Newton. For a long time, the lines between math and physics were blurry. In fact, in the past, subjects like classical mechanics and fluid mechanics were often taught in math classes, especially in the United States, before they moved to physics or engineering departments. Today, many fields, including engineering and computer science, still rely on applied math.

Main articles: Applied analysis, Differential equations, Approximation theory, Probability, Newtonian physics, Classical mechanics, Fluid mechanics, Engineering, Computer science

Divisions

Applied mathematics uses math to solve problems in many fields like physics, engineering, and biology. Even some areas of pure mathematics, like number theory, have become useful for real-world applications such as cryptography.

There isn't a clear agreement on what counts as applied mathematics. Some mathematicians say that applied mathematics is just about using math in science and engineering, while others think every use of math counts. Today, we also have fields like computational mathematics and computational science, which use powerful computers to simulate and solve problems.

Utility

Mathematical finance is concerned with the modelling of financial markets.

Mathematics has been very important in science and engineering for a long time. After World War II, new areas of math began to grow from other fields like economics, such as game theory and social choice theory. Today, math is used in many areas, including finance and data science.

Computers have also opened up new ways to use math. We now study how computers work and solve problems in science using math. Statistics, a type of math, is widely used in social sciences to help understand data.

Status in academic departments

The Brown University Division of Applied Mathematics is the oldest applied math program in the U.S.

Academic institutions organize applied mathematics differently. Some schools have a single mathematics department, while others have separate departments for Applied Mathematics and Pure Mathematics. Statistics is often a separate department in schools with graduate programs, but may be included in the mathematics department at undergraduate-only institutions.

Applied mathematics programs often include courses and faculty shared with other departments that use mathematics, such as physics or engineering. Some Ph.D. programs in applied mathematics focus mainly on mathematics, while others require additional coursework in a specific field of application. For example, the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge is a famous department that has housed notable professors like Isaac Newton and Stephen Hawking. Schools with separate applied mathematics departments range from Brown University, which offers degrees up to the doctorate, to smaller schools like Santa Clara University, which offers a master’s degree in applied mathematics. Some research universities, like MIT, divide their mathematics department into pure and applied sections.

Associated mathematical sciences

Applied mathematics connects with many other areas of math. It is used in engineering, where math helps design buildings, machines, and airplanes. Engineers study how materials move and flow, which uses special math areas like partial differential equations.

Applied mathematics has substantial overlap with statistics.

Applied math also works with computer science. Computer science uses logic, algebra, and discrete math to create programs and solve problems. It also overlaps with statistics, where math helps understand data and make predictions. For example, probability helps guess outcomes, and optimization finds the best solutions.

Other fields like business, biology, and physics also use applied math. Whether it's calculating risks in finance or understanding patterns in nature, applied math provides tools to solve real-world problems across many areas.

Applied Mathematics Societies

The Society for Industrial and Applied Mathematics is an international group for people who love using math to solve real-world problems. As of 2024, it has 14,000 members around the world. The American Mathematics Society also has a special group just for applied mathematics.