MATRIX AND TENSOR CALCULUS With Applications to Mechanics, Elasticity and Aeronautics ARISTOTLE D. MICHAL DOVER PUBLICATIONS, INC. Mineola, New York Bibliographical Note This Dover edition, first published in 2008, is an unabridged republication of the work originally published in 1947 by John Wiley and Sons, Inc., New York, as part of the GALCIT (Graduate Aeronautical Laboratories, California Institute of Technology) Aeronautical Series. Library of Congress Cataloging-in-Publication Data Michal, Aristotle D., 1899- Matrix and tensor calculus: with applications to mechanics, elasticity, and aeronautics I Aristotle D. Michal. - Dover ed. p. em. Originally published: New York: J. Wiley, [1941] Includes index. ISBN-13: 978-0-486-46246-2 ISBN-IO: 0-486-46246-3 I. Calculus of tensors. 2. Matrices. I. Title. QA433.M45 2008 515'.63-dc22 2008000472 Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501 ", To my wiJe Luddye Kennerly Michal EDITOR'S PREFACE The editors believe that the reader who has finished the study of this book will see the full justification for including it in a series of volumes dealing with aeronautical" subjects. " However, the editor's preface usUally is addressed to the reader who starts with the reading of the volume, and therefore a few words on our reasons for including Professor Michal's book on matrices and tensors in the GALCIT series seem to be appropriate. Since the beginnings of the modem age of the aeronautical sciences a close cooperation has existed between applied mathematics and aeronautics. Engineers at large have always appreciated the help of applied mathematics in furnishing them practical methods for numerical and graphical solutions of algebraic and differential equations. How ever, aeronautical and also electrical engineers are faced with problems reaching much further into several domains of modem mathematics. As a matter of fact, these branches of engineering science have often exerted an inspiring influence on the development of novel methods in applied mathematics. One branch of applied mathematics which fits especially the needs of the scientific aeronautical engineer is the matrix and tensor calculus. The matrix operations represent a powerful method for the solution of problems dealing with mechanical systems" with a certain number of degrees of freedom. The tensor calculus gives admirable insight into complex problems of the mechanics of continuous media, the mechanics of fluids, and elastic and plastic media. Professor Michal's course on the subject given in the frame of the war-training program on engineering science and management has found a surprisingly favorable response among engineers of the aero nautical industry in the Southern Californian region. The editors be lieve that the engineers throughout the country will welcome a book which skillfully unites exact and clear presentation of mathematical statements with fitness for immediate practical applications. THEODORE VON KAmIdN CLARK B. MILLIKAN v PREFACE This volume is based on a series of lectures on matrix calculus and tensor calculus, and their applications, given under the sponsorship of the Engineering, Science, and Management War Training (ESMWT) program, from August 1942 to March 1943. The group taking the course included a considerable number of outstanding research en gineers and directors of engineering research and development. I am very grateful to these men who welcomed me and by their interest in my lectures encouraged me. The purpose of this book is to give the reader a working knowledge of the fundamentals of matrix calculus and tensor calculus, which he may apply to his own field. Mathematicians, physicists, meteorologists, and electrical en~eers, as well as mechaiucal and aeronautical e~­ gineers, will discover principles applicable to their respective fields. The last group, for instance, will find material on vibrations, aircraft flutter, elasticity, hydrodynamics, and fluid mechanics. . The book is divided into two independent parts,_ the first dealing with the matrix calculus and its applications, the second with the tensor calculus and its applications. The minimum of mathematical concepts is presented in the introduction to each part, the more ad vanced mathematical ideas being developed as they are needed in connection with the applications in the later chapters. The two-part division of the book is primarily due to the fact that matrix and tensor calculus are essentially two distinct mathematical studies. The matrix calculus is a purely analytic and algebraic sub ject, whereas the tensor calculus is geometric, being connected with transformations of coordinates and other geometric concepts. A care ful reading of the first chapter in each part of the book will, clarify the meaning of the word "tensor," which is occasionally misused in modem scientific and engineering literature. I wish to acknowledge with gratitude the kind cooperation of the Douglas Aircraft Company in making available some of its work in connection with the last part of Chapter 7 on aircraft flutter. It is a pleasure to thank several of my students, especially Dr. J. E. Lipp and Messrs. C. H. Putt and Paul Lieber of the Douglas Aircraft Company, for making available the material worked out by Mr. Lieber and his research group. I am also very glad to thank the members of my seminar on applied mathematics at the California Institute for their helpful suggestions. I wish to make special mention of Dr. C. C. vii viii PREFACE Lin, who not only took an active part in the seminar but who also kindly consented. to have his unpublished researches on some dramatic applications of the tensor calculus to boundary-layer theory in aer.o nautics incorporated. in Chapter 18. This furnishes an application of the Riemannian tensor calculus described in Chapter 17. I should like also to thank Dr. W. Z. Chien for his timely help. I gratefully acknowledge the suggestions of my colleague Prc;Ifessor Clark B. Millikan concerning ways of making the book more useful to aeronautical engineers. . Above all, I am indebted to my distinguished colleague and friend, Professor Theodore von K8.rm8.n, director of the Guggenheim Graduate School of Aeronautics at the California Institute, for honoring me by an invitation to put my lecture notes in book form for publicat,ion in the GALCIT series. I ~ve also the delightful privilege of expressing Karman my indebtedness to Dr. for his inspiring conversations and wise counsel on applied mathematics in general and this volume in particular, and for encouraging me to make contacts with the aircraft industry on an advanced mathematical level. I regret that, in order not to delay unduly the publication of this boQk, I am unable to include some of my more recent unpublished researches on the applications of the tensor calculus of curved infinite dimensional spaces to the vibrations of elastic beams and other elastic media. AmsTOTLE D. MiCHAL CALIFORNIA INsTITUTE OF TECHNOLOGY OcroBI!lB, 1946 CONTENTS PART'I MATRIX CALCULUS AND ITS APPLICATIONS CHA.PTJlB PAGE 1. ALGlilBBAIC PBELlMINARIES Introduction • . . . . • . 1 Definitions and notations . 1 Elementary operations on matrices 2. ALGl!IBBAIC PRELIMINARIES (Continued) Inverse of a matrix and the solution of linear equations • • • • • •• 8 Multiplication of matrices by numbers, and matric polynomials. • •• 11 Characteristic equation of a matrix and the Cayley-Hamilton theorem. 12 3. DIFFERENTIAL AND INTl!lGRAL CALCULUS OF MATBICES Power series in matrices . . • • .--. . • . . . . . . . . • • • 15 Differentiation and integration depending on a numerical variable • 16 4. DIFFERENTIAL AND INTEIlBAL CALCULUS OF MATBICES (Continued) Systems of linear differential equations with constant coefficients 20 Systems of linear differential equations with variable coefficients. 21 5. MATRIX METHODS IN PROBLl!IMS OF SMALL OSCILLATIONS Differential equations pf motion 24 Illustrative example . . • • • . . . . . . . . • • • . 26 6. MATBIX METHODS IN PROBLEMS OF SMALL OsCILLATIONS (Continued) Calculation of frequencies and amplitudes . . . . . . . . . . •. 28 7. MATRIX METHODS IN THE MATHEMATICAL THEORY OF AIBCllAI'T FLUTTER 32 8. MATRIX METHODS IN ELASTIC DEFORMATION THEORY 38 I PART 11 TENSOR CALCULUS AND ITS APPLICATIONS 9. SPACIil LINE ELEMENT IN CURVILINEAB COORDINATES Introductory remarks . • . • • . . 42 Notation and summation coDvention • . • • • • . 42 Euclidean metrio tensor • • . . • • • . . • . • . 44 10. Vl!ICI'OB FIELDS, TENSOR FIELDS, AND EUCLIDEAN GHlWITOFFilL SnmoLS The strain tensor . • • • • • • . . . . • . . . . 48 Scalars, contravariant vectors, and covariant vectors 49 Tensor fields of rank two 50 Euclidean Christoffel symbols 53 ix x CONTENTS CBAPTIlB PAGlIl 11. TENsoR ANALYSIS Covariant difierentiation of vector fields 56 Tensor fields of rank r = p + q, contravariant of rank p and covariant of rank'p. . . • • • • 57 Properties of tensor fields • • . . . . • • . . . . • • • • • • " 59 12. LAPLACE EQUATION, WAVE EQUATION, AND POISSON EQUATION IN QuaY!;' LINlIlAR COORDINATES Some further concepts and remarks on the tensor caloulus 60 Laplace's equation •. . . . . .' 62 Laplace's equation for veotor fields 65 Wave equation .. .' 65 Poisson's equation •..••.• 66 13. SOME ELEMENTARY ApPLICATIONS OF THE TENSOR CALCULUS TO HYDRO- DYNAMICS Navier-Stokes differential equations for the motion of a viscous'fluid • 69 Multiple-point tensor fields. . • . . • . . . • 71 A two-point correlation tensor field in turbulence • . . . • . • 73 14. APPLICATIONS OF THE TENSOR CALCULUS TO ELASTICITY THJiIORY Finite deformation theory of elastic media • 75 Strain tensors in rectangular coordinates • • 77 Change in volume under elastic deformation 79 15. HOMOGENEOUS AND ISOTROPIC 8TaAJNs, STRAIN INV AJUANTS, AND VA RJ- ATION OF STRAIN TENSOR Strain invariants . . . . . . . . . . . • . • 82 Homogeneous and isotropic strains . . • • . . 83 A fundamental theorem on homogeneous strains 84 Variation of the strain tensor. . . . . . . . • 86 16. STRESS TENSOR, ELASTIC POTENTIAL, AND STRESS-8TaAJN RELATIONS Stress tensor . • • . • . • • • . . • • . • 89 Elastic potential. . . . • . . . . ',' . . . • . • • • • . . •. 91 StresHtrain relations for an isotropic medium . . • • • • • . .. 93 17. TENifoR CALCULUS IN RlmMANNJAN SPACliIS AND TBJD FuNDAMENTALS OF CLASSICAL MECHANICS Multidimensional Euclidean spaces . • • • • • • 95 Riemannian geometry. . . . • • . . . . . . • 96 Curved surfaces as examples of RiElmannian spaces 98 The Riemann-Chrlstoffel ourvature ~r • • • • 99 Geodesics. • • . . • • • . • . . . . • . . . . 100 Equations of motion of a dynamical system with n degrees of freedom. 101 CONTENTS xi CBAPTmB PAGE 18. ,ApPLICA!l'IONB OF THE TENSOB CALCULUS TO BOUNDARy-LAYER TBlDOBY "Incompressible and compressible fluids. • • . • . . . . . . • • .. 103 Boundary-layer equations for the steady motion of a homogeneous in- compressible fluid • 104 NOTES ON PART I • • • • III NOTJIlS ON PART IT. . • • 114 RuEBENCES FOB PART I • 124 RmFERENCES FOR PART IT 125 INDEX. • • • .. • • • • • 129