Applied mathematics is a branch of mathematics that focuses on mathematical methods that are usually used in science, engineering, business, and industry. Therefore, “applied mathematics” is a mathematical science with specialized knowledge. The term “applied mathematics” also explains the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation and study of mathematical models. In the past, practical applications have encouraged the development of mathematical theories, which then became the focus of study in pure mathematics, where mathematics is developed mainly for its own sake. Therefore, the activity of applied mathematics is crucially connected with research in pure mathematics.
Historically, applied mathematics consisted mainly of applied analysis, most notably differential equations; approximation theory; and applied probability. These areas of mathematics were intimately tied to the development of Newtonian physics, and in fact the distinction between the mathematicians and physicists was not distinctly drawn before the mid-19th century. This history left a legacy as well: until the early 20th century subjects such as classical mechanics were frequently taught in applied mathematics departments at American universities rather than in physics departments. Fluid mechanics might still be taught in applied mathematics departments. Engineering and computer science departments have traditionally made use of the applied mathematics.
Image Caption: Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming. Credit: Paddington_station_location_map.svg/Wikipedia (CC BY-SA 2.0)