Beckenbach - Modern mathematics for the engineer. Second series

MODERN MATHEMATICS FOR THE ENGINEER Second Series

MODERN MATHEMATICS FOR THE ENGINEER Second Series

Edited by

EDWIN F. BECKENBACH

With an Introduction by

MAGNUS R. HESTENES

Dover Publications, Inc., Mineola, New York

Bibliographical Note

Modern Mathematics for the Engineer: Second Series, first published by Dover Publications, Inc., in 2013, is an unabridged republication of the work originally published by McGraw-Hill Book Company, New York, in 1961.

International Standard Book Number

ISBN-13: 978-0-486-49747-1

ISBN-10: 0-486-49747-X

Manufactured in the United States by Courier Corporation

49747X01

www.doverpublications.com

The Authors

Arthur Erdlyi, Dr. Ver. Nat., D.Sc., Professor of Mathematics, California Institute of Technology

Bernard Friedman, Ph.D., Professor of Mathematics, University of California, Berkeley

John W. Miles, Ph.D., Professor of Engineering and Geophysics, University of California, Los Angeles

Ralph S. Phillips, Ph.D., Professor of Mathematics, Stanford University

J. Barkley Rosser, Ph.D., Professor of Mathematics, Cornell University

William Feller, Ph.D., Eugene Higgins Professor of Mathematics, Princeton University

David Blackwell, Ph.D., Professor of Statistics, University of California, Berkeley

Richard Bellman, Ph.D., Research Mathematician, The RAND Corporation

George B. Dantzig, Ph.D., Research Mathematician, The RAND Corporation

Samuel Karlin, Ph.D., Professor of Mathematics and Statistics, Stanford University

Stanislaw M. Ulam, Ph.D., Research Advisor, Los Alamos Scientific Laboratory

Raymond Redheffer, Ph.D., Professor of Mathematics, University of California, Los Angeles

Subrahmanyan Chandrasekhar, Ph.D., Sc.D., Morton D. Hull Distinguished Service Professor in the Departments of Physics and Astronomy, University of Chicago

Paul R. Garabedian, Ph.D., Professor of Mathematics, New York University

David Young, Ph.D., Professor of Mathematics and Director of the Computation Center, University of Texas

George Plya, Ph.D., Emeritus Professor of Mathematics, Stanford University

Foreword to the 1961 Edition

The physical and economic world in which the modern engineer operates continues to grow more complex, putting ever greater demands on the mathematical models representing that world. During the five years since University Extension last offered a lecture series in Modern Mathematics for the Engineer, reliance on a variety of these models has grown in an amazing fashion, due in no small measure to the adaptation of advanced mathematical techniques for use in connection with high-speed computing machines. Hilbert-space methods, always meaningful to the mathematicians who developed them, have now become useful also to the engineer and applied scientist, and the acceptance of probabilistic as well as deterministic analyses has become commonplace.

The present series was conceived with the objective of presenting some exciting aspects of modern mathematics. The course was designed for nonspecialists with training in engineering or science, high-school and college teachers of mathematics, and others desiring to remain au courant concerning mathematical developments. The material is intended to be quite understandable in the large, although not necessarily in complete detail, on the basis of the single lectures, and all of it should be applicable either now or in the reasonably near future to science and engineering.

We are pleased to share the stimulating experience of the second Modern Mathematics for the Engineer lecture series with you, the reader, through the pages of this book.

PAUL H. SHEATS

Professor of Education

Dean, University Extension

University of California

L. M. K. BOELTER

Professor of Engineering

Dean, College of Engineering

University of California

Los Angeles

MORROUGH P. OBRIEN

Professor of Engineering

Dean, College of Engineering

University of California

Berkeley

Preface to the 1961 Edition

For several years the University of California, through University Extension, has been conducting a highly successful series of lecture courses devoted to Modern Science for the Engineer. The lectures are available in book form in the University of California Engineering Extension Series, of which six volumes (see page ii) have preceded the present contribution.

The timeliness of the seriesor perhaps the scientific alertness of the Eastern European countriesis attested to by the fact that, during its first three years of existence, the initial volume of Modern Mathematics for the Engineer was translated into Hungarian, Polish, and Russian.

Turning a second time to mathematics, University Extension appointed an Advisory Committee composed of individuals representing various universities and industrial organizations in California to plan the 195859 lecture series.

The Committee adopted as its objective the presentation of some exciting aspects of modern mathematics that either are presently applicable or promise soon to be applicable to science and engineering.

To achieve this objective, the Committee sought to obtain the services of a group of outstanding speakers who were experts on advanced applicable mathematics. The topics chosen were, for the most part, those that have had recent spectacular applications in mathematics, that have been applied or are likely soon to be applied in physical, sociological, and biological sciences, and that involve a degree of mathematical subtlety.

The high success of the course was ensured by the enthusiastic response of the distinguished group of lecturers who participated in the series. It was presented at five locations throughout the state: at Berkeley, Corona, Los Angeles, Palo Alto, and San Diego.

The volume that has resulted is intended for engineers, scientists, mathematicians, students, high-school and college teachers, and others who desire to become or remain informed concerning current applicable mathematical developments.

The first volume of Modern Mathematics for the Engineer was somewhat arbitrarily divided into three parts: Mathematical Models, Probabilistic Problems, and Computational Considerations. The present volume is similarly divided into three parts: Mathematical Methods, Statistical and Scheduling Studies, and Physical Phenomena.

Certainly the foregoing partitioning is not a sharp one; rather, it is one of emphasis. Thus the broad and deep mathematical methods developed in .

is concerned with physical observations, but these observations lead to general conjecturesand the conjectures demand proof, which in turn involves ingenious and penetrating mathematical methods.

There are numerous examples and exercises throughout the book. Some of the exercises are at the ends of chapters, others are interspersed with the text, and still others are incorporated in the text. It is hoped that they will aid the reader in his over-all assimilation of the material and that they will add to the usefulness of the book.

The editor is most grateful to the authors for their excellent and prompt contributions to this volume; to the other Advisory Committee members, John L. Barnes, Clifford Bell, L. M. K. Boelter, George W. Brown, John C. Dillon, Gerald Estrin, James C. Fletcher, Bernard Friedman, Magnus R. Hestenes, John W. Miles, Russell R. ONeill, Louis A. Pipes, C. T. Singleton, Ivan S. Sokolnikoff, Thomas H. Southard, Angus E. Taylor, Charles B. Tompkins, and John D. Williams for their efforts and excellent ideas; to the Course Coordinators, John C. Bowman, Bernard Friedman, Stanley B. Schock, and Victor Twersky for their smooth handling of lecture arrangements; once again to Clifford Bell, the Statewide Coordinator of the course, for his unobtrusive but highly valued leadership; and especially the editor thanks his secretaries, Mildred Webb and Patti Hansen, for their careful and efficient work.