International Series in Pure and Applied Mathematics William Ted Martin, CONSULTING EDITOR VECTOR AND TENSOR ANALYSIS a International Series in Pure and Applied 'Mathematics WILI.LA11 TED MARTIN, Consulting Editor AIILFORS Complex Analysis BELLMAN Stability Theory of Differential Equations Bucx Advanced Calculus CODDnNUTON AND LEVINSON Theory of Ordinary Differential Equations G01.011 13 AND SHANKS Elements of Ordinary Differential Equations GRAVES The Theory of of Real Variables GRIFFIN Elementary Theory of Numbers IIILDEBRAND - rodurl ion to Numerical Analysis Principles of Nunurriral Analysis LAS b;leuu'nts of Pure aunt Applied Mathematics LASS Vector and Tensor Analysis LEIGIITON An Introduction to the Theory of Differential Equations NEHAIU Conformal Mapping NEWELL Vector Analysis ROSSER Logic for 'Mathematicians RUDIN Principles of Mathematical Analysis SNEDDON Elciuents of Partial I)iiTerential Equations Pourier Transforms SNEDDON STOLL Linear Algebra and 'Matrix Theory WEINSTOCK Calculus of Variations VECTOR AND TENSOR ANALYSIS BY HARRY LASS JET PROPULSION LABORATORY Calif. Institute of Tech. 4800 Oak Grove Drive Pasadena, California New York Toronto London McGRAW-HILL BOOK COMPANY, INC. 1950 VECTOR AND TENSOR. ANALYSIS Copyright, 1950, by the '.11e(_;ratt-Bill Book Company, Inc. Printed in the 1 nited States of America. All rights reserved. 't'his hook, or parts thereof, may not be reproduced in any form without, perrnis.sion of the publishers. To MY MOTHER AND FATHER PREFACE This text can be used in a variety of ways. The student totally unfamiliar with vector analysis can peruse Chapters 1, 2, and 4 to gain familiarity with the algebra and calculus of vectors. These chapters cover the ordinary one-semester course in vector analysis. Numerous examples in the fields of differential geometry, electricity, mechanics, hydrodynamics, and elasticity can be found in Chapters 3, 5, 6, and 7, respectively. Those already acquainted with vector analysis who feel that they would like to become better acquainted with the applications of vectors can read the above-mentioned chapters with little difficulty: only a most rudimentary knowledge of these fields is necessary in order that the reader be capable of following their contents, which are fairly complete from an elementary viewpoint. A knowledge of these chapters should enable the reader to further digest the more comprehensive treatises dealing with these sub- jects, some of which are listed in the reference section. It is hoped that these chapters will give the mathematician a brief introduction to elementary theoretical physics. Finally, the author feels that Chapters 8 and 9 deal sufficiently with tensor analysis and Riemannian geometry to enable the reader to study the theory of relativity with a minimum of effort as far as the mathematics involved is concerned. In order to cover such a wide range of topics the treatment has necessarily been brief. It is hoped, however, that nothing has been sacrificed in the way of clearness of ideas. The author has attempted to be as rigorous as is possible in a work of this nature. Numerous examples have been worked out fully in the text. The teacher who plans on using this book as a text can surely arrange the topics to suit his needs for a one-, two-, or even three- semester course. If the book is successful, it is due in no small measure to the composite efforts of those men who have invented and who have vii viii PREFACE applied the vector and tensor analysis. The excellent works listed in the reference section have been of great aid. Finally, I wish to thank Professor Charles de Prima of the California Institute of Technology for his kind interest in the development of this text. HARRY LASS URBANA, ILL. February, 1950 CONTENTS PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTER 1 THE ALGEBRA OF VECTORS . . . . . . . . . . . . . . . . . 1 1. Definition of a vector 2. Equality of vectors 3. Multipli- cation by a scalar 4. Addition of vectors 5. Subtraction of vectors 6. Linear functions 7. Coordinate systems 8. Scalar, or dot, product 9. Applications of the scalar product to space geometry 10. Vector, or cross, product 11. The distributive law for the vector product 12. Examples of the vector product 13. The triple scalar product 14. The triple vector product 15. Applications to spherical trigonometry CHAPTER 2 DIFFERENTIAL VECTOR CALCULUS . . . . . . . . . . . . . 29 16. Differentiation of vectors 17. Differentiation rules 18. The gradient 19. The vector operator del, V 20. The divergence of a vector 21. The curl of a vector 22. Recapitulation 23. Curvi- linear coordinates CHAPTER 3 DIFFERENTIAL GEOMETRY. . . . . . . . . . . . . . . . . 58 24. Frenet-Serret formulas 25. Fundamental planes 26. In- trinsic equations of a curve 27. Involutes 28. Evolutes 29. Spherical indicatrices 30. Envelopes 31. Surfaces and curvi- linear coordinates 32. Length of arc on a surface 33. Surface curves 34. Normal to a surface 35. The second fundamental form 36. Geometrical significance of the second fundamental form 37. Principal directions 38. Conjugate directions 39. Asymptotic lines 40. Geodesics CHAPTER 4 INTEGRATION. . . . . . . . . . . . . . . . . . . . . . . . 89 41. Point-set theory 42. Uniform continuity 43. Some proper- ties of continuous functions 44. Cauchy criterion for sequences 45. Regular area in the plane 46. Jordan curves 47. Functions of bounded variation 48. Arc length 49. The Riemann integral ix