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Tensor Calculus-Synge.Pdf DOVERBOOI$ ON MATHEMATICS H,cNDsooxop M.mrmu.mcllRncnons, Milton Abramowitzand lrene A. Stegun.(612724) TulsonAuryss oNM.qMpor..Ds, Richard L. Bishopand Samuell. Goldberg. (6403s6) TlsLEson lNDEmvm lrvrucn{s, G. pettt Bois.(6022F7) VEcronltro TprusonAurysn wmr Appucmons,A. L Borlsenkoand l. E. Tarapov.(6383$2) THs Hrsronyor rHe Crucwus llo lrs CowcsFTUATDevruoruerr, Carl B. Boyer. (6050$4) THe QurumrtvE THEoRyon Onornmy DrrrenEnrnuEeulrrorss: Ax lrrnooucnol, FredBrauer and JohnA. Nohel.(65g4G5) ALcoRlHMsron MmruzqnorvWmrour Deruvlnrru,Richard p. Brent. (4lgg& 3) PRrNcrpLEsoFSrArrsncs, M. G. Bulmer.(6326G3) TupTHeony or Spnvons,6[e Cartan.(640ZGl) AovrNcsDNUMBER THEory Harvey Cohn. (6402&X) Srms-ncsM,cNUAL, Edwin L. Crow,Francis Davis, and MargaretMaxfield. (605e$D FourupnSprues ANo Orrsocont Funcror,rs,Harry F.Davls. (659239) CoMpurr.BrlryAND UNsoLvABrury, Martin Davis.(61471-9) AsyMprotlcMmrops ln Arw-vsn,N. G. de Bruiin. (64221{) PnosLEMsnrGnoup Traony, John D. Dixon.(61574)e THeM.mnu.mcs on Gruuxor SrnArscy,Melvin Dresher.(642lCX) Appuso Pmrw- DmnsNflAL Equmons, paul DuChateauand David Zachmann.(419762> AsyumoncExpnrsror,ts, A. Erd€lyi. (603180) Coupuo<VARIABI rs: HaRuoucltro Aurrnc Fwctons, FrancisJ. Flanigan. (6138&7) On FonuaLly Ur,roEctolslEPnoposmons or pruxctprl M,lrxtu.lttce llrp REI-ArEDSysreus, Kurt G6del.(6698CT) A Hsrony op Gnrer M,mau,mcs,Sir ThomasHeath. (24023€, 2407M) Twovolume set PnoaABrLrry:Euueirrs or rHE MlrrnMAncALTHuony. C. R. Heathcote. (411494) Mrmoosor AppumMarneu.lrrcs, Francis B. Hlldebrand.(67002-3) M.rrneumcsrurro Locrc, Mark Kac and StanislawM. Ulam.(670g54) Paperboundunless otherwise indicated. Available at your book dealer, online at www.doverpubllcadonr.com, or by wriilng tb Dept. 23, Dover Publicatlons,Inc., 3l East2nd Street,Mlneola, Ny lf50l. For current Manufacturcdin the U.S.A. J.L.SYNGE AND A. SCHILD TENSORCALCULUS Mathematicians, theoretical physicists and engineers unacquainted with tensor calculus are at a serious disadvantage in several fields of pure and applied mathematics. They are cut off frorn the study of Reimannian geometry and the general theory of relativity. Even in Euclidean geornetry and Newtonian ntechanics (par- ticularly the mechanics of continua) , they are cornpelled to work in notations which lack the compactness of tensor calculus. This classic text is a fundamental introduction to the subject for the beginning student of absolute differential calculus, and for those interested in the applications of tensor calculus to mathematical physics and engineering. Tensor Calculus contains eight chapters. The first four deal with the basic concepts of tensors, Riernannian spaces, Rienrannian curvature, and spaces of constant curvature. The next three chap- ters are concerned with applications to classical dynamics, hydro- dynamics, elasticity, electromagnetic radiation, and the theorems of Stokes and Green. In the linal chapter, an introduction is given to non-Riemannian spaces including such subjects as affine, Weyl, and projective spaces.There are two appendices which discuss the reduction of a quadratic form and multiple integration. At the conclusion of each chapter a summary of the nrore inportant formulasand a set of exercisesare given. More exercises are scattered throughout the text. The special and general theory of relativity are briefly discussed where applicable.' John Lighton Synge, noted Irish mathematician and physicist, is Senior Professor at the School of Theoretical Physics, Dublin In- stitute for Advanced Studies. The late and distinguished Alfred Schild was Ashbel Smith Professor of Physics at The University of Texas. "This book is an excellent classroorn text, since it is clearly written, contains numerous problenrs and exerciscs, and at the end of each chapter has a sunllllary of the significant results of the chapter." Quarterly <:f .4pplied Mathernatirs. ". a refresh- ing and useful addition to the introducrory literature on tensol' calculus." Mathematical Reuie.us. Unabridged (1978) republicationof the corrected (1969) Preface.Bibliography. Index. ix + 324pp. 5/, x 8V4. Paperbound. ISBN 0-qgh-b3b1a-? Seeevery Dover book in print at www.doverpublications. com +1,q . 15 IN USA +8q.15 IN CANADA llillil86 ilruillililililll I TENSOR CALCULUS TEI\SORCALCULUS by J. L. SYNGE SeniorProfessor, School of TheorcticalPhgsics Dublin lnstitute of AdoancedStudies and A. SCHILD Late Ashbel Smith Professorof Phgsics The Unit>ercitgof Texns DOVER PUBLICATIONS, INC. NEW,YORK Copyright @ Canada, lg4g. All rights reserved under pan American and International Copyright Conventions. This Dover edition, ffrst published in lg7g, is an.unabr_id_gedrepublication of the work originally published by University of Toronto press in lg4d, as No. 5 in its series, Mathematical Expositions. This edition is reprinted from the correited edi- tion published in fg6g. Intematianal Standard B ook N umber : 0486_63612-7 Librarg of CongressCatalog Card Number: 77-94163 Manufactured in the United States of America Dover Publications, Inc. 31 East 2nd Street, Mineola, N.y. 11501 PREFACE MATHEMATICIAN unacquainted with tensor calculus is at a serious disadvantage in several fields of pure and applied mathematics. He is cut off from the study of Rieman- tri.o geometry and the general theory of relativity. Even in Euclidean geometry and Newtonian mechanics (particularly the mechanics of continua) he is compelled to work in notations which lack the compactnessof tensor calculus. This book is intended as a general brief introduction to tensor calculus, without claim to be exhaustive in any particular direction. There is no attempt to be historical or to assign credit to the originators of the various lines of development of the subject. A bibliography at the end gives the leading texts to which tlre reader may turn to trace the development of tensor calculus or to go more deeply into some of the topics. As treat- ments of tensor calculus directd towarrds relativity are compar- atively numerous, we have excluded relativity almost completely, and emphasized the applications to classical mathematical physics. Hourever, by using a metric which may be indefinite, we have given an adequate basis for applications to relativity. Each chapter ends with a summary of the most important formulae and a set of srercises; there are also exercisesscattered through the toct. A number of the orercises appeared on exam- ination paper at tle University of Toronto, and our thanks are due to ttre Univensity of Toronto Press for permission to use them. The book has grovrn out of lechrres delivered over a number of years by one of us (J.L,S,) at the Univereity of Toronto, the Ohio State University, the Carnegie Institute of Technology. Many suggestions from those who attended the lectures have been incorporated into the book. Our special thanks are due to Professors G. E. Albert, B. A. Griffith, E. J. MicHe, M. Wyman, and Mr. C. W. Johnson, each of whom has read the manuscript in whole or in part and made valuable suggestions for its improvement. J. L. SrrgcB A. Scgu.o vlrr COxrfrrs SECTION 3.3. Geodesicdeviation 90 3.4. Riemannian cunrature 93 3.5. Parallel propagation 99 Summary III LO7 ExercisesIII 108 IV. SPECIAL TYPES OF SPACE 4.1.Space of constant currrature 111 4.2. Flat space 118 4.3. Cartesian tensors L27 4.4. A spaceof constant curvattre regarded as a sphere in a flat space 136 Summary IV 138 Exercises IV 139 V. APPLICATIONS TO CI-ASSICAL DYNAMICS 5.1. Physical components of tensors l4Z 5.2. Dynamics of a particle 149 5.3. Dynamics of a rigrd body 1b6 5.4. Moving frames of reference lgz 5.5. Generd dynamical systems t6g Summary V 184 Exercises V 186 VI. APPLICATIONS TO HYDRODYNAMICS, ELASTICITY, AND ELECTROMAGNETIC RADIATION 6.1. Hydrodynamics 190 6.2. Elasticity n2 6.3. Electromagnetic radiation 2t8 Summary VI 2it2 Exercises VI 234 Conmnrrs VII. RELATIVE TENSORS, IDEAS OF VOLUME' GREEN.STOKES THEOREMS 7.L. Relative tensors, generalized Kronecker delta, permutation sYmbol 240 7.2. Change of weight. Differentiation ?48 7.9. Extension 252 7.4. Volume 262 7.5. Stokes'theorem 267 7.6. Green's theorem n5 Summary VII 277 ExercisesVII 278 VIII. NON-RIEMANNIAN SPACES 8.1. Absolute derivative. Spaceswith a linear connection. Paths 282 8.2. Spaces with symmetric connection. Curyature 292 8.3. Weyl spaces. Riemannian spaces. Projective spaces 296 Summary VIII 306 ExercisesVIII 307 Appendix A. Reduction of a Quadratic Form 313 Appendix B. MultiPle Integration 316 BrsLrocRAPEY 319 Ir.roBx 82L TENSOR CALCULUS CHAPTER I SPACES AI{D TENSORS 1.1. The generalizedidea of a space. In dealing with two real variables (the pnessu3eand volume of o 9?s,for srample), it is a common practice to use a geometrical reprlesantation. The variables are represnted by the Cartesian coordinatesof a point in a plane. If we have to deal with three variables, a point in ordinary Euctidean spaceof tbree dimensionsmay be used. The advantagesof such geometrical representation are too well knon'n to require emphasis.The analytic aspectof the problem assistsus with the geometry and vdccscrso,. When the number of rirariablesexceeds t[r€e, the geome- trical representationpresents some difficulty, for we require a spac€of more than thrce dimensions.Although such a space need not be r€garded as having an actual physical existence, it is an ortremely valuable oncept, becausethe language of geometry may be employed with referene to it. With due caution, we may e\rendraw diagrams in this "gpace," or rather we may imagine multidimensional diagrafiIs proiected
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