William and Mary

Classical Applied Mathematics

What problems does it solve?

The most traditional role of the applied mathematician in a professional setting has been in the solution of problems arising from physical phenomena and engineering. From its very inception, calculus has been applied to laws of motion and to understanding the consequences of interacting forces. While the early applied mathematicians were necessarily physicists and engineers as well, the modern setting calls for the mathematician to serve as a member of a team of specialists, each bringing a particular talent to bear on problems.

In a broad sense, the applied mathematician is instrumental in designing and analyzing models of systems and in testing and evaluating performance. It is a characteristic of this field that the technical questions readily move across once clearly distinguished boundaries. Whether in research and development or in industrial production, the applied mathematician must interact with engineers, physicists, programmers, and other specialists. The common goal is to find ways to improve quality, reduce cost, and increase productivity. The analytical skills of the mathematician are particularly valuable in consulting for technical services or trouble shooting.

Recent mathematical research in combination with increasing computer sophistication has opened fields that saw little development in the past due to their intractability to classical analytic techniques. These include the solution of problems involving enormous numbers of equations, the numerical simulation of complex systems such as power grids, and the application of control theory and other mathematical tools to the management of traffic or industrial processes.

The tasks of the applied mathematician are as diverse as the constituencies served. The broad category of engineering disciplines is a rich source of mathematical problems. In the aeronautical field a mathematician may help to develop models for atmospheric flight including the analysis of performance in search of optimal trajectories. Biomedical engineers may rely on mathematicians when designing and interpreting theoretical models of chemical and biological processes. A mechanical engineer may require a study of heat transfer by conduction, convection, and radiation resulting from a gas turbine.

Many problems involve scientific or engineering data and the use of computer techniques to answer questions arising in research, plant operations, product distribution systems, inventory controls, and business system analyses. Mathematicians seek efficient and reliable computer programs for the numerical solution of initial value problems or special function routines capable of delivering accurate answers over a wide range of parameters. While the methods most frequently applied are based in ordinary and partial differential equations, there is increasing involvement of probability, statistics, and computing.

What should one study in college?

For a career in classical applied mathematics, a student should obtain a thorough background in calculus, linear algebra, ordinary and partial differential equations, probability, statistics, numerical analysis, and vector calculus. These courses should include some extensive use of computing, or they should be supplemented by appropriate courses in computer sciences. Supporting work should include physics and basic engineering courses.

Additional Resource:

Careers in Applied Mathematics and Computational Mathematics, by the Society for Industrial and Applied Mathematics (SIAM), at the weblink: