Introduction to the 1961 Edition

MAGNUS R. HESTENES

PROFESSOR OF MATHEMATICS
UNIVERSITY OF CALIFORNIA, LOS ANGELES

During the last decade, there has been a remarkable expansion in the demand for advanced mathematics by the engineer. This demand has arisen in part because of the increasing complexities created by technological progress.

In advanced design it is frequently necessary to study a carefully constructed mathematical model before creating the physical model. The modern rockets and satellites, for example, could not have been built and successfully launched without careful mathematical analysis of the physical problem at hand.

Not only does mathematics enter into initial planning and design; it enters into testing programs as well. Data are collected and interpreted in accordance with a statistical theory. Once a product has been designed and tested, a mathematical theory of quality control is frequently used in the manufacture of the product.

More recently, mathematics has been found to be a useful tool in the field of production planning.

Thus mathematics enters into all phases of engineering and production.

The modern high-speed computing machine is playing an ever-increasing role in physical, biological, and social sciences, in engineering, and in business. The effective use of computers requires the aid of persons with a high degree of mathematical training and proficiency. Almost every branch of mathematics has been used on problems that have been successfully attacked with the help of computing machines.

These machines can be used for experimentation as well as for solving intricate mathematical problems. For example, a traffic problem has been simulated on a computing machine, and experiments have suggested means of traffic control that have significantly increased the flow of traffic in a congested area.

It is clear that persons responsible for the operations of computing machines must be proficient in mathematics and must have on hand source material necessary for the solution of their problems.

The role of mathematics in engineering has been most aptly presented by Dr. Royal Weller in the Introduction to “Modern Mathematics for the Engineer,” First Series.

The topics selected for the present volume have been chosen to complement those found in the first volume. As is customary in mathematics, the mathematical theory is presented largely without regard to applications, except as illustrations of the theory. This is done in order to bring out the mathematical structure and in order to facilitate applications to problems that actually are of similar basic mathematical structure but that bear little superficial similarity to the applications being described. An attempt is made also to call attention to various mathematical fields that undoubtedly will play an important role in the science and engineering of the future.

Though it is not possible to cover all the phases of mathematics that are important for applications, it is hoped that this volume and its predecessor have exposed many of the most important and promising ones, have laid a firm foundation in modern mathematical thinking, and will stimulate the reader to further study of this most interesting and useful field of human endeavor.