DIFFERENTIAL ANDVARIATIONAL PRINCIPLES byDavidLovelock Professor of Mathematics Universityof Arizona andHannaRund Professor of Mathematics Universityof Arizona and Adjunct Professor of Applied Mathematics Universityof Waterloo(Ontario) DOVERPUBLICATIONS,INC.,NewYork Copyright1975,1989byDavidLovelock andHannoRund. AllrightsreservedunderPanAmericanandInternationalCopyright Conventions. ThisDoveredition,firstpublishedin1989,isanunabridged,corrected republication of the work firstpublishedby John Wiley&Sons ("A Wiley-IntersciencePublication'1,NewYork,1975.For thiseditiontheAppendix has been completely revisedby theauthors. Manufactured in theUnited States of America Dover Publications, Inc.,31East 2ndStreet,Mineola,N.Y.1150 I Libraryof Congress Cataloging-in-Publication Data Lovelock,David,1938-Tensors, differential forms,and variational principles. Reprint.Originally published:NewYork: Wiley,1975. Includes index. Bibliography:p. I. Calculusof tensors.2.Differential forms.3.Calculusof variations.I. Rund, Hanno. II. Title. QA433.L671989515'.6388-31014 ISBN 0-486-65840-6 PREFACE The objective of thisbook istwofold.Firstly, it isour aim to present a self-contained,reasonablymodern account of tensor analysisandthecalculus of exterior differentialforms,adapted to the needsof physicists,engineers, and applied mathematicians in general.Secondly,however,it isanticipated thatasubstantialpartof thematerialincludedinthelaterchaptersisof interestalsotothosewhohavesomepreviousknowledgeof tensorsand differentialforms:wereferinparticular totheremarkableinteractionbe-tweentheconceptof invarianceandthecalculusof variations,whichhas profound implications in almost allphysical fieldtheories. The generalapproach and organization of the opening chapters isdeter-mined almost exclusivelybythe requirementsof our firstobjective.In fact, thesechaptersarebasedoncoursespresentedduringthepastdecadeat several universitiesto audiences with widely varying interests and academic backgrounds.Accordinglytheprerequisitesconsistmerelyof basiclinear algebra,advanced calculus of severalrealvariables,and some veryelemen-taryaspectsof thetheoryof differentialequations;thefirstfivechapters therefore constitute a one-semester course which should be readily accessible to senior undergraduates majoring in mathematics, physics, or some branch of engineering.Initiallythepaceisverygradual,if notleisurely,withem-phasison motivation with the aid of simple physical examples,whileat the sametime asystematic effort ismade to proceedtothecoreof thesubject matterbywayof successiveabstractionfromconcretesituations.Inthis respect,therefore, our treatment does not follow the historical development of thecalculusof tensorsandforms,whoseoriginsaredeeplyrootedin differential geometry, which does, in fact,provide their most natural setting. However,sincetheapplicationsoftensorsandformshavemeanwhile spreadtoentirelydifferentareas,thebasicapproachpresentedbelow hasintentionallybeendivestedofmanyofthecustomarytrappingsof metric differential geometry. Thelaterpartsof thebook,beginning withChapter 6,aredevotedpri-marily to the second of the aforementioned objectives. It had been observed repeatedlybyDavidHilbertthattheeffectsof invariancepostulateson v vi PREFACE variationalprinciplesareofasurprisinglyprofoundandfar-reaching nature,particularly insofar as relativistic fieldtheories are concerned. Since nopreviousknowledgeof thecalculusof variationsispresupposed,this subjectisdevelopedabinitioforsingleaswellasmultipleintegrals,with specialemphasisonthevariousinvariancerequirementswhichmaybe imposed on the fundamental(or action) integral.Much of this treatment is based on therelativelyrecent direct methodsof Constantin Caratheodory, insteadof ontheclassicaltheoryof thefirstandsecondvariationsof the fundamentalintegral;thustheapproachof Caratheodory,whichisun-doubtedlymoreilluminating and powerfulthan thetraditionalprocedure, should berendered accessible to awide classof readers. t Averyimportantinstanceoftheconsequencesoftheimpositionof specialinvariancerequirementson variationalprinciplesisrepresentedby thefamoustheoremsof Noether,which,whenappliedtophysicalfield theories,predicttheexistenceandprecisenatureofconservationlaws. Insteadof followingtheoriginalprocedureofEmmyNoether,whichis basedonsomedeepandconceptuallydifficultresultsinthecalculusof variations,itwillbeseenthatthesetheoremscanbederiveddirectlyand almost effortlessly bymeans of elementary tensorial techniques. Inviewof thefactthatmanyapplicationsof theseideasareconcerned with thegeneraltheory of relativity,a chapter on Riemannian spaceswith indefinitemetricsisincluded;itshouldbeemphasized,however,thatitis onlyat thisstagethattheconcept of ametricisintroducedand usedina systematicmanner.Thus,ontheonehand,theaestheticappealof metric differentialgeometry,absent fromtheearlier chapters, isrecaptured, while ontheotherhand,theprerequisitesforthefinalchapterareestablished. Thischapterisconsiderablymorespecializedthantheremainderof the book:itcontains,interalia,fairlyrecentresultswhichareof interestpri-marily to relativists. Toagreat extenttheemphasisinthisbookison analyticaltechniques. Thus a large number of problems is included, ranging from routine manipu-lativeexercisestotechnicallydifficultproblemsofthekindfrequently encounteredbythose whousetensortechniquesinthecourse of theirre-searchactivities.Indeed,someconsiderabletroublehasbeentakento collect many useful results of apurely technical nature, whichgenerally are not discussedinthestandard literature,but whichformpart of the almost indispensable"folklore"knowntomostexpertsinthefield.Despitethis emphasis on technique,however,every effort hasbeen made to maintain an acceptablelevelof rigorcommensuratewiththeclassicalbackgroundon t Incidentally,it should bepointed out that this approach alsoplays an increasingly significant role in the theory of optimal control (which isnot touched upon in this book). PREFACEvii which the analysis isbased.Moreover,the modem, more sophisticated and abstract approachtothetheoryof tensorsand formson manifolds,which islessdependentontheuseof coordinatesystems,isdiscussedbrieflyin the Appendix, which,it ishoped,willassistthe reader inbridging the deep chasmbetweenclassicaltensoranalysisandthefundamentalsofmore recent global theories. The first drafts of the manuscript of this book were scrutinized at various stagesbyProfessorsW.C.SalmonandD.Trifanof theUniversityof Arizona, and byProfessor G. W.Homdeski of theUniversity of Waterloo. It isa pleasure to acknowledge the many valuable suggestions received from thesecolleagues.For assistancewiththearduoustaskof proof-readingat various stages,weare indebted to the following:S.Aldersley,M. J.Boyle, andR.J.McKellarattheUniversityof Waterloo,andP.L.Nashand W.E.Smith at the University of Arizona. Last,butnotleast,wearedeeplygratefultoBeatriceShubeof Wiley-Interscience forher constant encouragement and invaluable advice. Tucson,Arizona January1975 DAviDLoVELOCK HANNORUND SUGGESTIONSFOR THE GENERALUSE OF THE BOOK Chapters1-5constituteaone-semesterseniorundergraduateorjunior graduatecourse.Sections4.3,4.4,5.4,and5.7maycon-ceivablybeomittedsincetheremainderof thispart of the book isindependent of their contents. Chapters 6-8constitute a one-semester graduate course, in which Sections 6.4,6.6,7.4,7.5,and 7.6 maybeomitted. An introductory course,specifically designed fortheneedsof relativists, could be based on Chapters 1-3, together with Sections 4.1, 4.2, 5.1, 5.2,5.3, 5.4,5.6,6.1,6.2,7.1,7.2,and 7.3. References to equations are of the form (N.M.P), where Nand Mindicate the corresponding chapter and section, respectively.If N coincides with the chapter at hand it isomitted. Difficultproblemsaremarkedwithanasterisk;insomecasesexplicit bibliographicalreferencesaregiven.Noattemptwasmadetocompilean exhaustivebibliography;thelatterconsistsentirelyof generalreferences and individual papers cited in the text. CONTENTS Chapter1.Preliminary observations 1.1.Simple examples of tensors in physics and geometry,I 1.2.Vector components in curvilinear coordinate systems, 6 1.3.Some elementary properties of determinants, 13 Problems, 16 Chapter 2.Affine tensor algebra in Euclideangeometry 2.1.Orthogonal transformations in E3,19 2.2.Transformation properties of affine vector components and related concepts,22 2.3.General affinetensor algebra,31 2.4.Transition to nonlinear coordinate transformations, 36 2.5.Digression: Parallel vector fieldsin Enreferred to curvilinear coordinates, 45 Problems, 50 Chapter 3.Tensor analysis onmanifolds 3.1.Coordinate transformations on differentiable manifolds, 55 3.2.Tensor algebra on manifolds, 58 3.3.Tensor fieldsand their derivatives, 65 3.4.Absolute differentials of tensor fields, 72 3.5.Partial covariant differentiation, 75 3.6.Repeated covariant differentiation, 81 3. 7.Parallel vector fields, 84 3.8.Properties of the curvature tensor,91 Problems, 95 1 18 54 ix X CONTENTS Chapter 4.Additionaltopics from thetensor calculus 4.1.Relative tensors,I 02 4.2.The numerical relative tensors,109 4.3.Normal coordinates, 117 4.4.The Lie derivative, 121 Problems, 126 Chapter 5.Thecalculus of differential forms 5.1.The exterior (or wedge)product of differential forms,131 5.2.Exterior derivatives of p-forms,136 5.3.The lemma of Poincare and its converse, 141 5.4.Systems of total differential equations,147 5.5.The theorem of Stokes, 156 5.6.Curvature formson differentiable manifolds, 164 5. 7.Subs paces of a differentiable manifold, 170 Problems, 175 Chapter 6.Invariantproblems in the calculus of variations 6.1.The simplest problem in the calculus of variations; invariance requirements, 182 6.2.Fields of extremals, 187 6.3.In variance properties of the fundamentalintegral: the theorem of Noether for single integrals,20 I 6.4.Integral invariants and the independent Hilbert integral,207 6.5.Multiple integral problems in the calculus of variations,214 6.6.The theorem of Noether for multiple integrals, 226 6.7.Higher-order problems in the calculus of variations, 231 Problems, 236 Chapter 7.Riemanniangeometry 7.1.Introduction of a metric,240 7.2.Geodesics, 250 7.3.Curvature theory of Riemannian spaces, 257 7.4.Subspaces of a Riemannian manifold, 267 7.5.Hypersurfaces of a Riemannian manifold,273 7.6.The divergence theorem forhypersurfaces of a Riemannian manifold, 281 Problems,287 101 130 181 239 CONTENTS Chapter 8.Invariant variational principles and physical field theories 8.1.Invariant fieldtheories,299 8.2.Vector fieldtheory, 300 8.3.Metric fieldtheory, 305 8.4.The fieldequations of Einstein in vacuo,314 8.5.Combined vector-metric fieldtheory,323 Problems, 326 Appendix.Tensors and forms on differentiable manifolds A.l.Differentiable manifolds and their tangent spaces, 332 A.2.Tensor algebra,339 A.3.Tensor fields:Exterior derivatives,Liebrackets,and Lie derivatives, 344 A.4.Covariant differentiation: torsion and curvature, 349 Bibliography Index xi 298 331 353 359 TENSORS, DIFFERENTIAL FORMS, AND VARIATIONAL PRINCIPLES 1 PRELIMINARY OBSERVATIONS One of the principal advantages of classicalvector analysis derives fromthe factthat it enables one to expressgeometricalor physical relationships ina concise manner whichdoes not depend on the introduction of a coordinate system.However,formanypurposesofpureandappliedmathematics theconceptofavectoristoolimitedinscope,andtoaverysignificant extent,thetensorcalculusprovidestheappropriate generalization.It,too, possessestheadvantageofaconcisenotation,andtheformulationof its basicdefinitionsissuchastoallowforeffortlesstransitionsfromagiven coordinate system to another, while in general the inspection of any relation involving tensors permits an inference to be drawn immediately as to whether or not that relation isvalid in allallowable coordinate systems. Theobjectiveofthischapterisessentiallymotivational.Afewsimple physicaland geometricalsituations arebrieflydescribedinorder toreveal theinadequacyofthevectorconceptundercertaincircumstances.In particular, whereas ina three-dimensionalspace a vector isuniquely deter-minedbyitsthreecomponentsrelativetosomecoordinatesystem,there existimportant physicalandgeometricalentitieswhichrequiremorethan three components fortheir complete specification.Tensorsare examplesof suchquantities;however,thedefinitionofthetensorconceptisdeferred untilChapters2and3,andaccordinglyanyreaderwhoisreasonably familiar with the elementary ideastouched upon below may confine himself toa verycasual survey of the contents of thischapter. 1.1SIMPLEEXAMPLESOFTENSORSINPHYSICSANDGEOMETRY Although it is not feasibleto define the concept of a tensor at this stage, we brieflydescribeafewelementaryexampleswhichservetoindicatequite 2PRELIMINARYOBSERVATIONS clearlythat, inthe case of many basic geometricalor physicalapplications, it isnecessarytointroduce quantities whichare more generalthan vectors. Whereas a vector possesses three components in a three-dimensionalspace, weshallbeconfrontedwithentitieswhichpossessmorethanthreecom-ponents in such spaces. Thefollowingnotationisadoptedinthissection.Supposethat,ina Euclideanspace3,wearegivenarectangularcoordinatesystemwith origin at a fixedpoint 0of E3.The coordinates of an arbitrary point P of E3 relativetothiscoordinatesystemaredenotedby(x1,x2,x3),andtheunit vectors in the directions of the positive Ox1-,Ox2-,and Ox3-axes arerepre-sented bye1,e2,and e3.The latter formabasisof E3 inthe sense that any vector A can be expressed in the form (1.1) in which Ap A2,A3 denote the three components of A, these being the lengths of theprojectionsof A onto e1,e2,e3,respectively.The lengthIAIof A is givenby IAI2 =(Atf + (Azf+ (A3)2. In particular, for the position vectorr of P we have r=x1e1 + x2e2 + x3e3, and, writing r=Ir I forthe sake of brevity, r2 =(x1f+ (x2f+ (x3f. EXAMPLE1THE STRESSTENSOROF ELASTICITY (1.2) (1.3) (1.4) Letusconsideranarbitraryelasticmediumwhichisat restsubjecttothe action of body and/or external loading forces, sothat in its interior there will existinternalforces.ThestressvectoracrossasurfaceelementASinthe interiorof thebodyatapointPis definedbythe limitAF/ASasAS-+ 0, where AF denotes the resultant of the internal forces distributed over the sur-faceelementAS.More precisely,letus considerasmallcubeof theelastic material,whoseedgesareparalleltothecoordinateaxes.The facesof the cubewhoseoutwardnormalsaree1,e2,e3,arerespectivelydenotedby S ~ >S 2,S 3Let Fdenote the stress over S(k=1,2,3) (see Figure 1).Each of these vectors can be decomposed into components in accordance with ( 1.1): (1.5) where rki represents the jth component U =1,2,3) of the stress over the face Sk.Thequantitiesrkiarecalledthecomponentsof thestresstensor:they completelyspecifytheinternalforcesoftheelasticmediumateachpoint. 1.1SIMPLEEXAMPLESOFTENSORSINPHYSICSANDGEOMETRY3 s, Fig.I It shouldbeemphasizedthat thisdescriptiondependsonasystemof nine componentsrki(Thesearenotgenerallyindependentbutsatisfythesym-metry condition rk;= 'ik provided that certain conditions are satisfied,but this phenomenon isnot relevant to this discussion.) Asa result of the forcesacting on the body, a deformation of the medium occurs, which is described as follows.A point P of the medium, whose initial coordinates are (xi, x2,x3),will be transferred to a position with coordinates V, y2,y3),and weshall write ui=yi- xi U =1,2,3).If it isassumed that uiisacontinuouslydifferentiablefunctionof position,onemaydefinethe quantities (1.6) whichrepresentthecomponentsof thestraintensor.(Wedonot motivate thisdefinitionhere,the readerbeingreferredtotextsonthetheory of elas-ticity,e.g.,GreenandZerna[1].)Againweremarkthatthespecification of sikdependson ninecomponents(apartfromsymmetry).Moreover,one of the basic physical assumptions of the theory iscontained in the so-called generalizedHooke'slaw,whichstatesthat thereexistsalinearrelationship betweenthecomponentsofstressandthecomponentsofstrain.Thisis expressed inthe form 33 'ik=IIcjklmslm l=lm=l u, k=1,2,3),(1.7) 4PRELIMINARYOBSERVATIONS inwhichthecoefficientsciklmareof necessityfour-indexsymbols(ascon-trastedwithordinaryvectors,whosecomponentsaremerelyone-index symbols). It shouldbeabundantlyclearfromthisverycursorydescriptionthat ordinary vectortheory istotally inadequate forthe purposes of an analysis of this kind. EXAMPLE2THE INERTIATENSOROFARIGIDBODY Letusconsiderarigidbody rotatingabout afixedpoint0initsinterior. The angular momentum about 0of a particle of mass m of the body located at apoint P withposition vector rrelativeto0isgivenby rxp,wherep= m drjdtisthelinearmomentumoftheparticle.If rodenotestheangular velocityvectorwehavethatdrjdt= roxr,andconsequentlytheangular momentumoftheparticleabout0ism[rx(roxr)].Thusifthemass density of the body isp, its total angular momentum about 0isgiven by the integral H=fp[rX(roxr)] dV,(1.8) in which dV denotes the volume element, and where the integration isto be performed over the entire rigidbody, (e.g.,see Goldstein [1]). InordertoobtainamoreusefulexpressionforH, weintroducearec-tangular coordinate system with basis vectors e1,e2,e3 and origin at 0, this systembeingfixedrelativetothe rigidbody.Intermsof (1.3)wehave,by definition of vector product, roxr= e1(w2x3 - w3x2)+ e2(w3x1 - w1x3)+ e3(w1x2 - w2x1), and, after alittle simplification, rx(roxr)=e1{w1[(x2f+ (x3f]- w2x1x2- w3x1x3} + e2{-w1x1x2 + w2[(x3f+ (x1f]- w3x2x3} + e3{-w1X1X3 - W2X2X3 + w3[(x1)2 + (x2f]}.(1.9) Thisexpressionistobesubstitutedin(1.8).Sincetheangularvelocity components w1,w2,w3 are independent of position, the followingintegrals willoccur, forwhich aspecial notation isintroduced: /22=Jp[(x3)2 + (x1f] dV,} I 33=Jp[(x1f+ (x2f] dV, (1.10) 1.1SIMPLEEXAMPLESOFTENSORSINPHYSICSANDGEOMETRY5 together with - Jpx'x'dV ~ I , . ,':" ~_- Jpx'x'dV ~1,.,1 I23- - fpxXdV- I32.J Il2= (1.11) Intermsof thisnotation,then,thesubstitutionof (1.9)in(1.8)yieldsthe followingexpressionforthetotalangularmomentumof ourrigidbody about the point 0: H= ei(Jllwl+ I12w2+ I13w3)+ e2(I21W1+ I22W2+ I23w3) + e3(J31w1+ I32w2+ I33w3) 33333 =e1 L I 1hwh+ e2 L I 2hwh+ e3 L I 3hwh=LL Iihwhei.(1.12) h= Ih=Ih= Ij=Ih=I This isthe formulathat wehave been seeking; it clearly permits the com-putation of theangular momentum Hforany angular velocity rowhenever thequantities(1.10),theso-calledmomentsofinertia,togetherwiththe quantities(1.11),theso-calledcoefficientsof inertia,areknown.However, these quantitiesare defined uniquelybythestructure of therigidbody,the position of the point 0, and the directionsel' e2,e3 of theaxesthrough0. Thisphenomenon clearly exhibitsthe fundamentallyimportant role played by the coefficients Iihwhich are known collectively asthe components of the inertia tensor.Again it should be stressed that these are two-index quantities. The inertia tensor may also be used to calculate the total kinetic energyT of therigidbodyduetoitsrotationalmotion.Thekineticenergyof the particle of mass mlocated at Pis tm ldrjdt 12,and bya process of integration similartothat sketchedabove it iseasilyshownthat 133 T= 2 LL Iihwiwh. j= Ih= I (1.13) Let ussuppose, forthe moment,that theinertia tensorof arigidbody is knownforagivensetof axesthroughapoint0.Foradifferentphysical problem involving the same rigid body it may be necessary to determine the inertiatensorforanother setof axesthrougha different centerof rotation. Thusit wouldbe desirabletohavesomeknowledgeof thebehaviorof the componentsof theinertiatensorunderachangeof rectangularaxes.This state of affairsexemplifiesthe central theme of the tensor calculus, which is, infact,deeplyconcernedwiththetransformationpropertiesof entitiesof this kind. 6PRELIMINARYOBSERVATIONS EXAMPLE3THEVECTORPRODUCT LetA,B betwo vectorswithacommonpoint of applicationPinE3,the angle between these vectors (defined by a rotation from A to B) being denoted by 0. If 0 #0 and 0#n, these vectors span a plane II through P.The vector product C=Ax Bat Pis defined geometricallyasfollows:(1)the magni-tude IC I of CisgivenbyIA I IB I sin 0;(2)thevectorCisnormaltoIIin such a mannerthat A,B,Cformaright-handedsystem.Thisdefinitionis equivalent to the analytical expression C= e1(A2B3- A3B2)+ e2(A3B1- A1B3)+ e3(A1B2- A2B1).(1.14) It shouldberemarkedthattheabovegeometricaldefinitionof AxB ismeaningfulsolelyforthe caseof athree-dimensionalspace,because ina higher-dimensionalspacetheplaneIIdoesnotpossessauniquenormal. However,the formula(1.14)suggestsquite clearlythat in ann-dimensional space with n~3 weshould define the quantities U, h=1,... , n).(1.15) These two-index symbols represent the components of an entity which isan obviousgeneralizationtondimensionsofthethree-dimensionalvector product.However,this entity isnot avector unlessn=3.The geometrical reason forthis isevident from the remark made above; the analytical reason emergesfromthefactthatthecomponentsC 1,C 2,C 3 of Casgivenby (1.14) are related to the quantities (1.15) forn=3by (1.16) or (1.17) wherej, h, k is an even permutation of the integers 1, 2, 3, and this construction is not feasible if the three indices j, h,k are allowed to assume integer values greater than 3.Thusthe generalization (1.15)of the vector product isrepre-sented by quantities which are not vector components; in fact, it will be seen that (1.15)exemplifiesa certain type of tensor. 1.2VECTORCOMPONENTSINCURVILINEARCOORDINATESYSTEMS Let usregard a given vector A,with fixedpoint of application Pin 3,as a directedsegmentintheusualsense.Relativetoarectangularcoordinate systemwithoriginat0andbasisvectorse1,e2,e3,wecanrepresentA according to (1.1),thelengthlA I of A being givenby(1.2).Again,it should beemphasizedthat the componentsA1,A2,A3 of A whichappearin(1.1) 1.2VECTORCOMPONENTSINCURVILINEARCOORDINATESYSTEMS7 areobtainedbyaprocessof projectionontotheaxes,whichimpliesthat these components, and hence the expression forthe length IA I,are indepen-dent of thelocation of the point of application P. However,whenweattempt tocarryoutasimilarprogram relativetoa curvilinearsystemof coordinates,the position isnot quite sosimple.Since in this case there does not exist a universal triple of coordinate axes, we cannot definethecomponentsofAbymeansofadirectprocessof projection; moreover,itisnot immediatelyevidentwhatformthecounterpart of the formula (1.1)should assume.In this connection it should beremarked that ( 1.1) isnot a vector equation of the type A=B, which isindependent of the choice of the coordinate system: on the contrary, it is a nonvectorial relation which depends crucially on the factthat it refersto arectangular system. In order to illustrate these remarks we shall consider in the first instance a special type of curvilinear coordinates, namely, a spherical polar coordinate system whose pole islocated at the origin 0of the givenrectangular system andwhosepolaraxiscoincideswithe3.Thesphericalpolarcoordinates (p,0,)are related tothe rectangular coordinates (xi,x2, x3) according to the transformation x1 =p sin 0 cos,} x2 =p sin 0 sin , x3 =p cos 0, (2.1) asisimmediatelyevidentfromFigure2.It shouldbenotedthatp0, 0::;;0::;;n,0::;;0,0h2,... ,h,=1,... , n).(3.11) From the transformation law (3.6)itthen followsthat (3.12) in any other rectangular coordinate system obtained from the firstby means of the orthogonal transformation (3.4). Thus if an affine tensor vanishes in a given coordinate system, it vanishes in all other systems related to the firstby anorthogonal transformation. 34AFFINETENSORALGEBRAINEUCLIDEANGEOMETRY Accordingly,whenever a geometrical or physicalsituation isdescribed in agivenrectangularcoordinatesystembymeansof atensorequation such as(3.11 ),thesame state of affairsisdescribed in anyother admissible coor-dinate system by the corresponding tensor equation (3.12); it follows that the validityof theassertioncontained in (3.11)isindependentof the choiceof coordinates (provided that theyare obtained fromeachotherbyan ortho-gonaltransformation).Thisispreciselywhatonewouldexpectof ageo-metrical theorem or of a physical law. Therearecertainoperationsof an algebraicnaturewhichonecanper-formwithtensors.Wedealhereverybrieflywithoperationsof thiskind; an alternative discussion ispresented later under more general conditions. Firstly, when the components of an affinetensor of given rankr are each multipliedbythesamenonzeronumberorscalar,theresultingquantities are once more the components of an affine tensor of the same rankr.This is immediately evidentfromthe transformation law (3.6)when the right-hand side of the latter is multiplied by a nonvanishing scalar, which, by virtue of (3.7),istantamounttomultiplication of itsleft-handsideby(/). Secondly,whentherespectivecomponentsof twoaffinetensorsof the samerankareadded,theresultingsumsareagainthecomponentsof an affinetensorof thesamerank.Thisfollowsdirectlyfromthefactthatthe left- andright-handsidesof thetransformationlaw(3.6)arelinearinthe various components. For instance, let~ k hand Sikh denote the components of twotensorsof rank3,sothattheirtransformsunder(3.4)arerespectively given by nnnnnn IJkh=LLL ailakpahq Tzpq, sjkh=IIIajlakpahqslpq(3.13) 1=1p=1q=11=1p=1q=1 Thus if wewrite (3.14) in our respective coordinate systems,wefindbymeans of (3.13)that 111111111111 jkh=LLL ailakpahq('I;pq+ Slpq)=LLL ailakpahq Wzpq,(3.15) 1=1p=1q=11=1p=1q=1 which shows that the quantitiesWzpqare in factthe components of an affine tensor of rank 3. From these two statements we infer that the set of all affine tensors ofrank r constitutesavectorspaceovertherealnumbers.Thedimensionalityof this vector space isequal to the number of distinct components of these tensors, namely,n'.In particular, the set of allaffinevectors formsan n-dimensional vector space. 2.3GENERALAFFINETENSORALGEBRA35 From this itisimmediately evident that theaddition of tensors of different rank cannot possibly giveriseto a tensorial quantity. For example, if one were toaddoneof theequations(3.13)tothetransformationlawof arank2 tensor, say, nn f:jk=LL ailakp VzP, (3.16) 1=1p=l the result cannot be put into tensorial form.Indeed, while a combination such asljk+Uikmakes sense in that the indicesj, k on Vand U respectively are tacitly assumed to have the same values, this is not the case for a combination such asljk+1jkhbecause the index h on Thasno counterpart on V,which therefore precludes an identification of this kind. Itis,however,permissibletomultiplythecomponentsoftensorsof arbitrary ranks, forthis process givesrise to new tensors.For instance, if we were to multiply the left-hand side of(3.16) by the component Ah of an affine vector,this component being given byAh=I;= 1 ahqAq,wewould obtain nnn f:jkAh=LLL ailakpahq VzpAq, 1=1p=lq=l whichshowsthatthequantitiesVzpAqformthecomponentsofanaffine tensor of rank 3. It is easily verified that, in general, the quantities consisting of the products of thecomponentsof twotensorsof rankrandsrespectivelyagaincon-stitute the components of a tensor of rank r + s. Thereisanotheroperation,peculiartotensoralgebra,whichisof con-siderableimportanceincertainmanipulations.Foranyaffinetensorof rank r~2,say,1j,h-.. ip"'io"ir'one may select a pair of indices, say,jP andjq, assign to these pairs of indices in turn the values (1,1),(2,2),... , (n,n),and form the sums of then components thus obtained. These quantities constitute thecomponentsof anaffinetensorofrankr- 2.Thisiseasilyverified directly by means of (3.2) when the operation indicated above is carried out in respect of the transformation law (3.6).This process is generally referred to as the contraction of tensors, and weshall illustrate it forthe case of the trans-formation law (3.13) of a tensor of rank 3. Suppose, then, that we put k=h on the left-hand side of(3.13), after which wesum over h: 1111111111 ~ I I+~ 2 2+ +~ n n=L ~ h h=LLLL ajlahpahq Tzpq h=Ih= II= Ip= Iq= I 36 AFFINETENSORALGEBRAINEUCLIDEANGEOMETRY We then apply the orthogonality condition (3.2) to the sum I:= 1 ahpahqthus obtaining ht1 ~ h h= ,t1 Pt1 qt1 ai,bpq 1~ p q= ,t1 ai(t1 r;PP} whichindicatesthatthequantitiesI;= 1 r;PPformthecomponentsof an affinetensor of rank1 =3 - 2. It canbestatedasageneralrulethattheprocessof contractionapplied onceto a tensorof rankr givesriseto a newtensorof rankr- 2. In particular, if the process of contraction isapplied to an affine tensor of rank2,theresultisan affinescalar.Thisestablishesdirectly,forinstance, theinvariantnatureof thetraceC11 +C22 + ... +cnnof amatrix(Ch), whose entries are the components of an affinetensor. As a final example we note that, if Iikdenotes the components of a tensor of rank2,whilewh,w1 arethecomponents of a vector,thequantitiesIikwhw1 constitute the components of a tensor of rank 4.Contraction over the indices U,h)yieldsa rank 2 tensor,and subsequent contraction over (I,k)givesrise to ascalar.Thus theinvariant nature of the expression (1.1.13),namely, isestablished effortlessly without calculation. 2.4TRANSITIONTONONLINEARCOORDINATETRANSFORMATIONS It wasstressed repeatedly that the tensor concept withwhichwehavebeen concerned thus far is restricted in the sense that we are dealing with affine ten-sorswhicharedefinedrelativetoorthogonaltransformations,thelatter being linearbydefinition. However, it is imperative that we should divest ourselves of this restriction. First,it isoften necessaryfromapurely practicalpointof viewto usecur-vilinearcoordinates,thetransitiontowhichinvolvesnonlinearcoordinate transformations.Second,more often than not, one isconcerned with tensor analysisonmanifolds(suchascurvedsurfaces)onwhichorthonormal coordinate systems simply cannot bedefined; accordingly the theory devel-opedthusfarisentirelyinadequate forsituationsof thiskind.Thisobser-vation,farmoresothanthefirst,providesthemotivationforthetypeof generalization which wenowattempt to achieve. Letusendeavortopinpointthosefeaturesof our earlieranalysiswhich proved tobethemostessentialfortheobjectivesthusfarattained.One of these isthe factthat thetransformation law of tensors isalways linearinthe 2.4TRANSITIONTONONLINEARCOORDINATETRANSFORMATIONS37 components of thetensors:this allows us to definethe processes of multiplic-ationbyscalarsandadditionof tensorsof thesamerank,whichinturn permits usto regard all affinetensors of a given rank as elements of avector space.Also, the success of the operation of contraction depends vitally on the relation(3.2)satisfiedbythecoefficientsaihwhichappearinthetransfor-mation law. It is evident, therefore, that if we can extend our theory such as to preservethesetwofeatures,agreatdeal of our basicanalysis can begener-alized as required. WeshallconsidertwocoordinatesystemsinEn,relativetowhichthe coordinatesof apointParedenotedby(x1, ,xn)and(x1,,.xn),re-spectively.Thesetwon-tuplesof realnumbersarerelated to eachother by transformation equations of the type { ~ . ~ .~ .~ . ~ ~ ~ . ! : : : : :. ~ n ~ : .xn=fn(x'' ... ' xn), in whichthefunctions f I' ... 'rare ndistinctfunctionsof nvariables.In futureweshalldenotethen-tuple(x1,,xn)by asingleletterxh,itbeing understood that allindices j, h,k,... assume thevalues1 to n.Similarly we shall also write the above system of n transformation equations in the form of asingle relation, namely, as". " _xi=Ji(xh)(j=1,... , n). Furthermore, since thenumerical value of each _xiisgiven by Ji,there isno need for two distinct symbols x andf, and accordingly, in order to economize on symbols, our transformation formulaehenceforth are written in the form _xi=_xi(xh).(4.1) It isnotassumedthatthefunctionsxi(xh)arenecessarilylinearinthe coordinates xh[as isthe case in (3.4),which isaspecial case of (4.1 )].Atthis stage, only two assumptions are made: 1.Thefunctionsxi(xh)areof classC2 in aregionG of En.[A function g(xh) is said to be of class cP in G if it possesses continuous partial derivatives up to and including the pth order with respectto allits arguments xhin G.] 2.The functionaldeterminant o(x1, ,.xn) o(x1, 'xn) ox'ox' o.xnoxn ox1oxn (4.2) 38 AFFINETENSORALGEBRAINEUCLIDEANGEOMETRY isnonvanishing on a subregion of G;this implies (see,e.g.,Apostol [1]) that thereexistsaregionR~Ginwhichthetransformation (4.1)possessesan inverse, expressed in the form (4.3) itagainbeingunderstoodthatthisrepresentsasystemof nequations,of whichtheright-handsidesarefunctionsof thenvariablesxi.Indeed,by virtue of the inverse functiontheorem itfollowsfromour assumptionsthat these functionsare also of classC2. Henceforth our considerations will be restricted to theregionRinwhich the inverse(4.3)exists:thisistobeunderstoodthroughoutwithoutspecific mention. Forfuturereferencewenotethefollowingelementaryfact.Letussub-stitute thefunctionsxi(xh) in (4.3),whichyieldsasystem of identitiesin the variables xk,namely, which,written out more fully,isof the form ........................................ ' Bymeans of the chain rule wemay differentiate each of these identities with respectto xk.For the hth identity wethus obtain oxhoxhox1 oxhoxnnoxhoxj oxk=ox1 oxk+ ... + oxnoxk= j ~ loxi oxk' Butthevariablesx1, ,xnareindependent,sothatoxhjoxk= bhkand accordingly one has (4.4) Similarly (4.5) Afterthesepreparations wenow recallthat,in thecase of alineartrans-formation, the transformation law (3.6) of an affine tensor can be expressed in the form (3.9), which is possible because of the identifications (2.18) and (2.19) 2.4TRANSITIONTONONLINEARCOORDINATETRANSFORMATIONS39 of thecoefficientsaikwiththepartialderivativesof therelevantcoordinate transformation,namely,aik=oxijoxkandaih=oxh/oxi.Obviouslythis preservesthelinear character of thetensortransformationlaw asfarasthe tensor components are concerned; thussincethepartial derivativesoxijoxk areatour disposalinanycase,thissuggeststhatweshouldformulatethe transformationlawfortensorsinrespectof thegeneralcoordinatetrans-formation (4.1) in terms of these derivatives exactly asin (3.9).Furthermore, the counterpart of the orthogonality condition (3.2) assumes the form (4.4) or (4.5) after the replacement of the coefficients ajhby the oxi1oxh has been made, and,aswehaveseenabove,inthis formthecrucialcondition(3.2)isalways satisfied. Fromapurelyanalyticalpointofviewthissuggestionseemstobea satisfactory one, atleast at firstsight.It implies that weshould, forinstance, define a vector bythe requirement that its components Ah should transform according to the relation (4.6) relativetocoordinatetransformation(4.1).However,beforeproceeding further,weshouldverifythatthisprogramisinaccordancewithourgeo-metricalapproachofSection1.2totheconceptof thecomponentsof a vector in curvilinear coordinates. Weshalltherefore consider aspecialcase in 3,namely,our purelygeo-metricalconstructionof thecomponentsof avectorAinsphericalpolar coordinates.In this contextweshall denote the latter as follows: x'=p, (4.7) andthetransformation(1.2.1)fromthesecurvilinearcoordinatestothe appropriate rectangular coordinates isgiven by X1 = X1 sin x2 cos x3,} x2 =x1 sin x2 sin x3, x3 =x'cos x2, (4.8) this being a special case of (4.3).The matrix of the derivativesoxh/oxi of this transformation is - x'sin x2 sin x3) x'sin x;cos x3 .(4.9) 40 AFFINETENSORALGEBRAINEUCLIDEANGEOMETRY A simple calculation yieldsthe inverse: sin x2 cos x3 sin x2 sin x3 _xl (4.10) = cos x2 cos x3 cos x2 sin x3 xlxl sin x2 sin x3 cosx3 x1 sin x2 x1 sin x2 0 Thus,substitutingfrom(4.10)in (4.6)weinferthat,accordingtothispre-scription,thecomponentsAiof thevectorAaregiveninsphericalpolar coordinates by A 1 =sin x2 cos x3 A 1 + sin x2 sin x3 A 2 + cos x2 A 3, - cos x2 cos x3 Acos x2 sin x3 Asin x2 A2=II+I2- ---I-A3, XXX(4.11) Thisisseentocoincidewithour previousexpressions(1.2.7)forthecom-ponents of A insphericalpolar coordinates whenwereverttotheoriginal notationbymeansof (4.7).Accordinglythisconclusionseemstoconfirm, at least from a geometrical point of view,that the suggested definition based on (4.6)isasatisfactory one. Nevertheless,alittlereflectionshowsthatourapproachrequiressome additional refinements.For instance, if weconsider a setof components Bk, satisfying thetransformation law (4.6),namely, - na.x' B,=I-akBk, k= IX (4.12) the products of (4.6)and (4.12)yield (4.13) which exemplifieswhatwould seemtobean acceptable transformation law for a rank 2 tensor.However, if we contract over the indicesj, I (in an attempt to obtain an inner product), wefindthat n- - nnna.xja.xj AiBi =axhaxkAhBk, (4.14) in which the expression on the right-hand side cannot, in general, be simplified 2.4TRANSITIONTONONLINEARCOORDINATETRANSFORMATIONS41 byan application of (4.4) or (4.5).Thus the process of contraction, ascarried outinthiscontext,doesnotgiverisetoacorrespondingscalar.[Thefact that this difficulty does not arise in the case of linear orthogonal coordinate transformations is due to the factthat, according to (2.18) and (2.19), wehave 8xi8xh 8xh=aih=8xi (4.15) forsuchtransformations;ifitwerepossibletosubstitute(4.15)in(4.14) theidentity (4.4) could bereadily applied.] Clearlyourprogramstilllackssomevitalingredient,foritisessential thatsomehow oneshould beableto constructscalarsbymeans of suitable combinationsof tensors.Thusweapproachthisproblemfromaslightly differentanglebyfirstconsideringscalarsandcertainvectorsassociated with scalars. A function(xh)of thecoordinatesxhissaidtobeascalarorinvariant underthetransformation(4.1)ifitstransform(/)(xi)possessesthesame numerical value; that is,if (4.16) Hereitistobeclearlyunderstood thattheargumentsxi are relatedto the argumentsxhaccordingto(4.1):both setsof coordinatesrefertothesame pointP of En. Withanydifferentiablescalarfunctiononeusuallyassociatestheso-calledgradientvector:then components of thelatteraredefinedtobethe n partial derivatives8!8xh.Thequestion thatimmediately arises concerns the transformation properties of these derivatives, for, if they are to constitute the components of a vector in the senseof the proposed transformation law (4.6),thelattershould besatisfiedby8!8xh.In order totestthiscriterion, we differentiate (4.16) partially with respect to xi, the chain rule being applied tothe right-hand side, which gives 8(/)n8xh8 8xi=h ~ l8xi8xh' (4.17) Thiswouldbe inaccordance with(4.6)onlyif one could identify8xh/8xi with8xi/8xh,whichisnotgenerallypossible.Thus(4.17)exemplifiesa transformation law of thetype (4.18) whichis distinct from (4.6).However, the example (4.17) is obviously a signifi-cantonewhichcannotbeignored.Furthermore,if wecombinetwosets of 42 AFFINETENSORALGEBRAINEUCLIDEANGEOMETRY components,thefirstof whichsatisfy(4.6),thesecond(4.18),weobtain quantities of the type (4.19) and if the process of contraction over jand I is carried out in this instance we obtain, using (4.4)on the right-hand side, n- - nnnoxi oxknnn oxhoxiAhCk= bhkAhCk=(4.20) In contrastto(4.14)thisresultisindeedverysatisfactory,foritindicates thatwehavesucceededinconstructinganinvariantresemblingtheinner product of twovectors. This state of affairs indicates quite clearly that wemust consider two types of transformationlaws,namely,(4.6)and(4.18),andaccordinglywemust definetwodistinctkindsof vectors.[Thisdistinctiondoesnotariseinthe theory of linear orthogonal transformations since in the case of the latter the condition (4.15)issatisfiedwhichdeletesthedistinctionbetween(4.6)and (4.18).] Asregardsnomenclature,weshalldistinguishbetweenthesetwokinds ofvectorsbycallingtheformercontravariantandthelattercovariant. Thisdistinctionmustalsobereflectedinournotation:henceforthcontra-variantvectorsareendowedwithsuperscripts,andcovariantvectorsare distinguishedbysubscripts.Thisnotationisusedconsistently;itformsan essential aspect of tensor calculus. The results of these considerations may now be crystallized in the following twodefinitions. DEFINITION Asetof nquantities(A 1, ,A")issaidtoconstitutethecomponentsof a contravariantvectoratapointPwithcoordinates(x 1, ,x")if,underthe transformation (4.1),these quantities transform accordingto therelations n::>-j Ai=Lh= IOXh' (4.21) inwhichthecoefficients oxijoxh areto beevaluated at P. DEFINmON Asetof nquantities( C 1, ,Cn)issaidtoconstitutethecomponentsof a covariantvectorat apointPwithcoordinates (x1, ,x")if,underthetrans-2.4TRANSITIONTONONLINEARCOORDINATETRANSFORMATIONS43 formation(4.1 ),these quantities transform according to therelations noxh 'cj =I::>-jch, h=!uX (4.22) inwhichthe coefficients oxh/oxJ are to beevaluated atP. Remark 1.For future reference wenote that co- and contravariant vectors mayalsobe called type (0,1)and type (1,0)tensors,respectively. Remark2.Thesedefinitionscontain asa specialcasethedefinitionsof an affinevectorin3:for,whenthetransformation(4.1)happenstobean orthogonal one with n=3,both (4.21)and (4.22)reduce to (2.11). Remark3.Severalof theequationsaboveshouldnowberewrittenwith varioussubscriptsreplacedbysuperscripts;inparticular,theidentity (4.20)istobe expressed in the form nn L AiCj =L AhCh.(4.23) j= 1h= 1 Remark4.Thecoefficientsoxijoxhwhichappearinthetransformation law(4.21)are,ingeneral,functionsof thevariablesx1,... ,xn,whichare thepositionalcoordinatesof thepointof applicationof thecontravariant vectorA.ThisimpliesthatonecanaddtwocontravariantvectorsAandB if and only if they arelocated at the same point.[For instance, if the points of applicationofthevectorsaredistinct,withcoordinatesxk= xtnand xk=xt2l,respectively,the corresponding transformation equations read Ai =I oxi(xtn) Ahandjji =I oxi(xtz>) Bh. h=ioxhh=ioxh' since the values of the coefficients of Ah and Bh on the right-hand sides do not in general coincide, the sum of the right-hand sides is not linear in Ah+ Bh.] Itis,however,possible tomultiplyboth sidesof (4.21)byascalarfunction (xh)sincethelattersatisfies(4.16),itbeingassumedthatthearguments x1,... ,xnin(xh)alsorefertothepointof application of A.Thustheset of allcontravariantvectorsatapointPof Enconstitutesavectorspaceof dimensionn.Similarobservationsapply,of course,tocovariantvectors. Moreover, it is evident that the addition of co- and contravariant vectors cannot giveriseto a vectorial quantity. As in the case of affine vectors one can formthe products of the respective components of various vectors (provided that these vectors are located at the samepoint),thusobtainingthecomponentsofquantitieswhichwillbe recognized astensors of rank r >1.Again,a clear distinction must be made 44AFFINETENSORALGEBRAINEUCUDEANGEOMETRY betweenco- andcontravariantpropertiesof theentitiesthusconstructed. For instance,giventwocontravariantvectorswithcomponentsAhandBk located atthe same pointPof En,itfollowsfrom(4.21)that the transforms of the products of their components are given by ;PH'= oxiox'AhBk h= Ik= IOXhOXk' whichexemplifiesthetransformationlawof a tensorof type (2,0): 'fi' = oxiox'Thk h=lk=loxh0 ~. (4.24) Similarly (4.22)suggests the transformationlawof a tensor of type (0,2): nnoxh0 ~ sj,=II~ - j~ - ~shk h=lk=luxux (4.25) However,itisalsofeasibletodefinetheso-called"mixed"tensors, possessingboth co- and contravariantproperties.For instance,if weform the products of the components Ahof a contravariant vectorwiththe com-ponentsCkof acovariantvector,itfollowsdirectlyfrom(4.21)and (4.22) that the quantities thus obtained transform according to -- nnoxi oxkh A1C1 =II-;-,;~ - ~ Ack. h= Ik= IUXuX This exemplifiesthe transformationlawof a type (1,1)tensor,namely, -.nnoxi oxkh Vf=II-;-,;~ - ~vk. h=lk=1uxux (4.26) (4.27) Remarks2and4madeaboveinrespectof thedefinitionsofco- and contravariant vectors apply more or lessverbatim to the tensor components whichappear in (4.24),(4.25),and (4.27). Furthermore, by analogy with the theory of affine tensors it is now obvious how one can extend thesetransformation equationstotensorsof arbitrary type.Weshall defer these formaldefinitionsto the nextchapter. Inthesequelweencounternumerousexamplesofthevarioustensors discussed here.However, already at this stage weshould draw attention to a mixed tensor of special importance.Bearing in mind that, bydefinition,the Kroneckerdeltaassumesthenumericalvalues0and1 irrespectiveof the choice of coordinates, and writing (4.5)in the form nnoxi oxk 8j,=II-;-,;~ - ~bhk, h= 1k=1UXUX 2.5DIGRESSION:PARALLELVECTORFIELDSINE,45 weseeby comparison with(4.27)thattheKroneckerdeltais,infact,a type (1,1)tensor.Accordinglywehenceforthdenoteitscomponentsbyc5Z,and thenotationally correctversion of itstransformation law is -.nnaxi axkh bf=II~ ah~ a - 'bk, h=lk=lXX while (4.4)and (4.5)should be written in the forms naxhaxj Ia-j~ ak= c5Z, j=lXX and (4.28) (4.29) (4.30) 2.5DIGRESSION:PARALLELVECTORFIELDSINEnREFERREDTO CURVILINEARCOORDINATES Inann-dimensionalEuclideanspaceEnletusconsidertwocoordinate systemsinwhichthecoordinatesof apointParedenotedbyxiandxh, respectively.It isassumedthatthefirstof theseisarectangular coordinate system, whereas the second is an arbitrary curvilinear system, the correspond-ingtransformationbeingof theform(4.1),forwhichthefunctionsxi(xh) arenow supposedtobe of classC2. It ispossibletoconstructaspecialvectorfieldinEnwhichissuchthat there existsaunique vector at each point of En,thesevectors being parallel andofthesamelength.ThecomponentsJ(iofthisfieldrelativetothe rectangularsystemare obviously constant;andthus,foranydisplacement d ~the conditions (5.1) mustbesatisfied. ThecomponentsXhof thesamevectorfieldrelativetothecurvilinear system are related totheXi according to -~axjh XJ=L...~ ahX, h=IX sothat, corresponding toany displacement dxk,wehave (5.2) (5.3) 46AFFINETENSORALGEBRAINEUCLIDEANGEOMETRY Fromthisitisevidentthatthedifferentialsofavectorfieldarenot,in general, components of a tensor: furthermore, the condition (5.1) for a parallel vector field in En does not imply a corresponding relation of the type dXh=0. Thus relativeto our curvilinear coordinate system, a field of parallel vectors of constantlengthisnotcharacterizedbytherequirementthatthecom-ponents of this fieldbe constant. Thusthequestion arisesastohow suchvector fieldsaretobedescribed inarbitrarycurvilinearcoordinates,andwenowgiveacompleteanalysis of thisproblem.Tothisendwerecallthat,relativetoour rectangularco-ordinate system,the lengthIX I of any vector X of our fieldin Enisgiven by n IXI2 =L xrxi.(5.4) j= I Bymeans of (5.2)thiscan bewritten in termsof our curvilinear coordinate system asfollows: IXIz=IIf o x ~o x ~XhXk. h=Ik=Ij= IOXox (5.5) This suggeststhat weintroduce new quantities ghkdefined by (5.6) so that (5.5)can be written in the form nn IXIz=LL ghkxhxk.(5.7) h=Ik=I Again this demonstratesthe factthat thelength of a contravariant vector at Pisaquadratic formin thecomponentsof thatvector,thecoefficientsghk of thisquadratic formasdefinedby(5.6)being evaluated atP.[In fact,the relation (5.7)isthe generalization ton dimensions of theidentity (1.2.16).] For futurereferencewenotethatthedefinition (5.6)impliesthatdet(ghk) =f.0 byvirtue of our assumptionthatthefunctionaldeterminant (4.2)does notvanish.Thusthesymmetricmatrix(ghk)possessesan inverse,tobede-noted by (ghk),sothat (5.8) Weshall now endeavor toeliminate the second derivativeswhich appear on theright-hand sideof (5.3)bymeansof theghkinthefollowingmanner. First, multiplying (5.3) by oxijox1 and summing over j, at the same time noting 2.5DIGRESSION:PARALLELVECTORFIELDSINE.47 (5.6),weobtain noxi-j - nhnnn02Xjoxihk ox'dX- ghldX+ oxh oxkox'Xdx.(5.9) Second, wedifferentiate (5.6)partially with respectto x1,which yields oghk"(o2xioxioxio2xi) ox1 ox1 oxhoxk+ oxhox1 . (5.10) Thesecond derivativesthusobtained arenotyetexpressedin aformmost suitable forour purposes; accordingly weconsidertwo cyclic permutations of theindicesh,k,I in (5.10),correspondingtowhichthecounterpartsof (5.10)are (5.11) and oglhn(o2xjoxioxio2xj) oxkoxk ox1 oxh+ ox1 oxk oxh. (5.12) In order to reduce (5.9)toan acceptable formweseekthe expression whichappearsin both (5.11)and (5.12).These relationsareadded,and itis foundthattheunwantedtermsmayberemovedbysubtracting(5.10) fromthesum thus obtained.In this way wefindthat (ogk,+ og,h_oghk)= 2 o2xjoxi oxhoxkox1 oxh oxkox1. (5.13) The following notation istherefore introduced (in the firstinstance merely forthe sake of brevity): [hklJ= ! (ogk,og,h_oghk) ,2oxh+oxkox1 , (5.14) thesethree-indexquantitiesbeingcalledtheChristoffelsymbolsof the firstkind of our curvilinear coordinate system.Thus (5.13) can be expressed as no2xjoxj I=[hk,lJ, j=luXuXuX (5.15) 48AFFINETENSORALGEBRAINEUCLIDEANGEOMETRY and (5.9) becomes naxjnnn j ~ lax'dXi =h ~ lgh,dXh+ h ~ lk ~ l[hk,l]Xh dxk.(5.16) Now,if itisassumedthatthevectorfieldXi isparalleland of constant length in En,wemayapplythecriterion (5.1)totheleft-hand sideof(5.16), which yieldsthe condition nnn L 9h1 dXh+LL [hk,l]Xh dxk=0(5.17) h=Ih=Ik=I forsuch a vector fieldin curvilinear coordinates.This condition isexpressed entirelyintermsof quantitieswhichpertaintothisparticularcoordinate system. Inordertowrite(5.17)inaslightlybetterform,wemultiplyitbygi1, the inverse of (5.6),and sum over I,at the same timenoting (5.8).This gives nnn dXi+LLL gi1[hk,l]Xh dxk=0,(5.18) l=lh=lk=l which suggeststhe notation n Ud=Igi'[hk,IJ, (5.19) l=l fortheso-calledChristoffelsymbolsof thesecondkindof ourcurvilinear coordinate system.Accordingly wecan write (5.18)in the form nn dXi+LL {/dXh dxk=0,(5.20) h=lk=I whichistheconditionincurvilinear coordinates forafield of parallelvectors of constant length. For many practical purposes itisadvisableto write(5.20)asasystemof partial differential equations.For any displacement dxkwehave .naxjk dX1 =L -ak dx, k=IX (5.21) whichmay besubstituted in (5.20)to yield and,sincethismusthold forentirely arbitrary displacementsdxk,itfollows 2.5DIGRESSION:PARALLELVECTORFIELDSINE,49 that our condition for the field of parallel vectors of constant length isequiva-lentto (}Xin.h --;--;;- +L hJdX=0, uXh=l (5.22) whichisasystemof n2 partialdifferentialequationsof thefirstordertobe satisfied by thecomponentsof the vector field. RemarkJ.Thisconditionisexpressedentirelyintermsofquantities definedrelativetothecurvilinear coordinatesystem.Itholdsinany curvi-linearcoordinatesystem;accordinglyonewouldinferthattheleft-hand side of (5.22) istensorial. (This will be verified by direct calculation at alater stage under more general circumstances.) Remark2.In theoriginalrectangular coordinate systemthe counterparts ghkof thequantitiesghkaregivenbyghk= bhk=const.,asisimmediately evidentbycomparisonof(5.4)with(5.7).ThustheChristoffelsymbols (5.14)and(5.19)vanishidenticallyinanyrectangularcoordinatesystemin En,and accordingly forsuch systems the condition (5.20) reduces formally to (5.1). Remark3.Asasimpleapplicationof theaboveanalysisletusconsider theequation of astraightlineinEn.Relativetothe rectangular coordinate system a straight line xi=xi(s), where s denotesthe arc length, is character-ized by the condition in which wehave written dx'i ds=O, d-i -ri- X X- ds' (5.23) (5.24) In (5.16), which holds forany vector Xi, let us put Xi=x'i, and divide by ds, observing (5.23)atthe same time.This gives n(dx'hndxk) Igh,-d +I[hk,lJx'h -d=o, h= ISk=IS or,proceeding as before in thetransition from (5.17)to (5.20), d2xinn.dxhdxk -dz+LL {hJd-ds -ds =0. Sh= Ik= I (5.25) Thissystem of nsecond-orderordinary differential equations characterizes thestraight linesof Enrelativeto any curvilinearcoordinatesystem. so AFFINETENSORALGEBRAINEUCLIDEANGEOMETRY PROBLEMS 2.1InE3 arotationthroughanangleIXaboutthex3-axisischaracterizedbythe matrix ( cos IXsin IX0) (ail)=-sin IXcos IX0. 001 Showthat thisisaproper orthogonal transformation. A reflection in the plane x2 cos {3=x1 sin {3is determined by (cos2{3sin2{30) (b11)=sin2{3-cos 2{30. 001 Show that this isan orthogonal transformation whichisnot proper. 2.2ThecomponentsA;of anaffinevectorinE3 havethevalues(t, 0, 0)inthexk coordinate system.If -1 0 0 ~ ) isthe matrixof the (orthogonal) transformation fromx1 to xi,computeA1 2.3Prove that the set of all proper orthogonal transformations in E.formsa group, butthatthesetofallorthogonaltransformationsforwhichtheassociated determinant is- 1 does not form a group. 2.4In E. the quantityt/>=tj>(xh)isan affinescalar.Prove that o2tj>f(ox1 ox1)are the components of an affine tensor of rank 2. 2.5In E., Ah, Tii, andU lkmare the components of affinetensors of rank1,2,and 3, respectively.If Ah Tii=UihJ in one rectangular coordinate system, establish that Ah '1;1=DihJ in any other rectangular coordinate system. 2.6In E.prove that c511 are the components of an affine tensor of rank 2. *2. 7In E. the quantities B11 are the components of an affine tensor of rank 2.Construct twoaffine tensors each of rank4,with components C11k1 and D11k1 forwhich are identities. IIcijk!Bk,=Bij + Bji k= Il= I . IIDi}klBkl=Bij- Bji k= Il= I PROBLEMS51 2.8Show that, in E3, the quantities e11k definedin Problem1.1do not constitute the componentsof anaffinetensorof rank3 unlessthetransformationisproper, whereasthequantitiese11kehlmare the components of anaffinetensorof rank6. Establish that and IeiJJ=0, }=I 3 IeiJk ehlk=c5ihc5 11- c5nc5hJ k=l 33 IIeiJkehJk=2 c5ih k= I}= I 2.9InE3 thecomponentsofthevectorproductC=Ax8oftwononparallel vectors are given by The left-hand sides, regarded as components of an affine vector transform accord-ing to (2.11 ),whichislinearinthe coefficients aJhof the orthogonal transforma-tion.Ontheotherhand,theright-handsides,regardedascomponentsof an affinetensor,transformaccordingto(2.25),whichisquadraticinthea1h.Show how this apparent contradiction can be resolved provided that the transformation is proper. 2.10Theorthonormalbasis{e1}inE3 is"right-handed"inthesensethate1 xe2 =e3,e2 xe3 =e1,ande3 xe1 =e2. If {9 representsanorthonormal basisobtainedfrom{e1}accordingto (1.2), show that 3 flX(2=IA3heh, h=l where A1h denotes the co factors in the determinant a=det(a Jh).Hence deduce that flxf2=+f3or-(3 accordingasa =+ 1 or- 1.(Thisresultshowsthatproperorthogonaltrans-formationspreservethe"right-handedness"ororientationoforthonormal systems.) 2.11If A1 andB1 are the components of twoaffine vectors in E. the angle(}between them isdefined according to cos (}='\''\'1/2. ( ~ = IAJAJ . . . . ~ = IBkBk) Prove that this quantity is an affine invariant. 2.12In E., if the quantities A1 =A1(xh) are the components of an affine vector show that oA1foxl are the components of an affine tensor of rank 2. 2.13In E., the quantities A1 are the components of an affine vectorforwhich 52AFFINETENSORALGEBRAINEUCLIDEANGEOMETRY If Pii=A;Ai show that pijpik=pjk. i=l (a)Pv=G 0 ~ ) If, forn=3 0 0 (b)P, =G 0 ~ ) 1 4 0 findA;, if it exists. *2.14In E., b;i and cuare the components of affine tensors of rank 2.If c;i=- cii and I(ciibik+ ckjb;i)=0 j= I for all cii, show that for n>2 bij=A.!5ij. Determine A.and deduce that it isan affine scalar. 2.15In E.thequantitytjJ=tj>(xh)isascalarunder (4.1).Prove thato2tj>fox;oxi are not the components of a tensor of either type (2,0), (1,1), or (0,2).(Compare with Problem 2.4.) 2.16Letxi,xi denotethe coordinates of anarbitrarypointPof E.referredtotwo distinct rectangular coordinate systems.Anarbitrary curvilinear system inE. is related to the tworectangular systems according to Show that (1.1[ "oxi oxi"axi oxi j ~ loxhoxh=j ~ loxhoxk . Deduce that the quantities (5.6) defined forthe curvilinear coordinate system are independent of the choice of the rectangular coordinate system. 2.17Letx\ xhdenotethe coordinates of anarbitrarypointPof E.referredtotwo arbitrary curvilinear coordinatesystems.These coordinatesarerelatedtoeach otherbythetransformationequationsxh=xh(xk).Letthequantities(5.6) defined for each of these systems be denoted by ghkghkrespectively.Show that ""oxi ox1 ghk=II~ - h~ - kgj,, j=l!=l uxux whichindicatesthatthe gi1 arecomponentsof atype(0, 2)tensor.Deducethe transformation law satisfied by g=det(gi1).Is the latter a tensor? pROBLEMS53 2.18Find explicit expressions for the Christoffel symbols in spherical polar coordinates in 3,andhencewrite down the differential equations satisfiedbystraight lines in 3 in these coordinates. 2.19In a four-dimensional space with coordinates x; (i=1,... , 4),if L}are indepen-dent of position and satisfy 44 l1ij=II h=lk= I where c 00

(,,1): -10 0-1 00 then the transformation 4 Xi= Ij= I iscalled a Lorentz transformation. A setof 4'quantitiesTh, .. h,issaidtoconstitutethecomponentsof aLorentz tensor (or a 4-tensor) of rank rif under a Lorentz transformation they transform according to the law Show that if aLorentz tensor vanishes in a given coordinate system, it vanishes in all other systems related tothe firstby aLorentz transformation.Show that if T;JkSiJkandV;Jarethecomponents of Lorentztensorsof rank3,3,and2,re-spectively,thenT;jk+ sijksijk v,,andI:= II'1abv.bareLorentztensorsof rank3,5,and0,respectively,whereasI:= 1 V..isnottensorial(Synge[3], Davis [1]). 2.20In the notation of Problem 2.19 show that the set of allLorentz transformations formsa group. 0'' )._';-I

?/-- }.. ,_:\' - \ 'i_-3 TENSORANALYSIS ONMANIFOLDS Since it is our objective to present a concrete and gradual development of the theoryoftensors,introducingnewconceptsonlywhentheybecomein-dispensable, the discussion of the previous chapters is carried out against the background of aEuclidean spaceEn.The latter ischaracterized bythefact thatitadmitsCartesian coordinatesystems,eachof whichcoversEncom-pletely.However,someofthemostimportantapplicationsof thetensor calculusareconcernedwithsituationswhichrequirethattheunderlying space be of a far more general nature, such as a curved surface, which may not allowforthe existence of acoordinate systembymeansof whichitcan be covered entirely.Thus wemust now divestourselves of theassumption that theunderlying space isEuclidean; instead, itisnow supposed thatthisrole is played by a so-called differentiable manifold. The concept of a differentiable manifold isa somewhat abstract one: it can be described very roughly asan n-dimensionalspaceX nwhichcan becoveredbyopenneighborhoodson eachof whichcoordinatesystemsmaybedefinedinsuchamannerasto ensurethatpairsof such systems are related to each other bydifferentiable coordinatetransformations ..SmoothcurvesandsurfacesinE3 represent simple examples of differentiablemanifolds. The firstsection of this chapter isdevoted to avery rudimentary descrip-tionof suchmanifolds,which,itishoped,issufficientforallsubsequent requirements.(Afarmoresophisticatedandcompleteformulationis presented in theAppendix.)Again,astheresultof furtherabstraction from thetheory of tensors inEn,tensor fieldsonX n are introduced, and itisim-mediatelyevidentthatthecorrespondingalgebraicprocessesareformally identicalwiththosedescribedpreviously.However,whenoneturnstothe calculusof tensors,oneisimmediatelyconfronted withaformidablediffi.-54 3.1COORDINATETRANSFORMATIONS55 culty which resultsdirectly fromthe factthat the derivatives of tensor com-ponentsare notin generaltensorial.Thusanewtypeof derivative,theso-called covariantderivative,isintroduced forthepurposeof differentiation of tensors.This,however,ispossiblesolely on thestrength of an additional assumption,namely,thatthedifferentiablemanifoldisendowedwithan affine connection. Theprocessof covariantdifferentiationformsthebasisofthetensor calculus.In someaspectsitisverysimilartopartialdifferentiationinthe usualsense; nevertheless,afundamentallyimportantdifferencearisesasan immediateconsequenceofthefactthattheorderofrepeatedcovariant differentiatioQwith respectto distinct coordinates xhand xkisby nomeans immaterial.Infact,thisphenomenonleadsdirectlytotheconceptof the curvaturetensorof Xn(and,alreadyatthisstage,itshouldbeemphasized thatthistensorvanishesidenticallyinanEn)A striking geometricalinter-pretation of this state of affairs is afforded by the behavior of parallel vector fieldson X n. 3.1COORDINATETRANSFORMATIONSONDIFFERENTIABLE MANIFOLDS Inthissectionweshallgiveaveryroughandintuitivedescriptionof the conceptof adifferentiablemanifold.In ordertoattain an adequatedegree of generalityforourgeneraltheorywecannotrestrictourselvestospaces which can be covered completely bya single coordinate system (such asthe n-dimensionalEuclideanspaceEn):simpleexamplesofvariouskindsof surfacesembeddedinE3 indicatethat,ingeneral,nosinglecoordinate systemcanexistwhichcoversagivensurfacecompletely.Thisistrue,for example,of thetwo-dimensionalsphereinE3,and consequentlyonecon-siders certain regions on the sphere which are chosen such that it ispossible to construct coordinate systemson eachof theseregions.This can bedone invariousways;forinstance,relativetoarectangularcoordinatesystem (x1,x2,x3)of E3,one may choose the hemisphere for which x1 >0,on which onecanusex2,x3 ascoordinates,forobviouslythishemispherecanbe mapped onto an open disk on the (x2,x3)-plane. Accordingly this hemisphere isreferredtoasa"coordinate neighborhood,"and similarlythefiveother hemispherescorrespondingtotherestrictionsx1 O,x2 0, x3 1 and Ail= A1i, or(b)ifn>2andAiJ=-A1i, then either Ail =0 or A.=J1=v =0. *3.32If ail=a1ianddet(aiJ)#0,showthatailareconstantsundertransformations fromone coordinate system to another if and only if the transformation is linear. (This result is used in relativity to establish the linearity ofthe Lorentz transforma-tion of Problem 2.19; seeRindler [1]). 3.33If bii1k=0 show that Kmihkbml+ Kmlhkb]m=0. 3.34Relativetoarectangularcoordinatesystemin3 Newton'slawof motionis expressed in the form F=m dvjdt, where F is the force, m the (constant) mass of a particle,andv thevelocity.Expressthislawincurvilinear coordinatesusinga covariant differential definedby means of a suitable connection. PROBLEMS99 (10)(a). 3.35In Xz, if(aij)=.2Icompute r/k 0SinX RelativetothisconnectionavectorfieldXiundergoesparalleldisplacement along thecurve x1 =a fromx2 = 0 tox2 = 2n.If initiallyXi= (1,0)calculate Xi finally. 3.36If Tii is skew-symmetric show that 3.37If the connection issymmetric, prove that (Veblen[1]) K/kilm+ K/imlk+ Km1ikli+ Kk1mJii=0. 3.38If T/ isa type (1,1) tensor fieldshow that Hijk=T{aTfjax'- 1}' aT/jax'+ T/(ai;'!axi- aT/fax1) isatype(1,2)tensor(theNijenhuistensor,Nijenhuis[1]). 3.39Let aiikbe the components of a tensor of type (0, 3).Define I Siik=3!(aiik+ akii+ aiki+ aiik+ akii+ aik), 1 diik=3 (aiik+ akii- aiik- aikJ Show that (a)siik iscompletely symmetric; (b)b1ikiscompletely skew-symmetric; (c)ciik=ciik ciik+ ckii+ ciki= 0; (d)diik= dkii diik+ dkii+ diki= 0. If P1a1ik=SiikP2a1ik=b1ikP3a1ik=ciikP4aiik=d1ikshow that P A(P 8auk)=(P A a1ik)c5 ABforA, B=1,... , 4 (nosummation over A). Hence show that aiikhas aunique decomposition of the form aijk=silk+ bijk+ ciik+ diik. How many independent components has S1ik b1ik c1ikand diik? 3.40Under the conditions of the previous problem, define hiik=!(aiik- aiik+ akii- akii) hik=!(aiik- akii+ aiik- aJki). 100TENSORANALYSISONMANIFOLDS Show that (a)huk=-hiik> huk+ hkii+ hiki=0, (b)fijk=- fkjifijk+ fkij+ Jjki=0. If Qlaijk=SijkQ2aijk=bijk'Q3aijk=hijkQ4aijk=hjkShOWthatQA(QBaijk) =(QAaiik)!5A8,(A,B=1,... , 4).Henceshowthataiikhasauniquedecompo-sition of the form aijk=sijk+ bijk+ hijk+ hjk How many independent components has hiikandhik? [Problems3.39and3.40mayberegardedastwodistinctgeneralizations of the decomposition (2.18)to tensors of type (0,3).] 3.41By differentiating (2.23) with respect to x, obtain (2.25). 3.42Obtain (3.21)from (3.16)byinterchanging xi with _xi. 3.43Obtain (8.6)from(6.10)bysettinglj =r:f>uandusingProblem3.21. 3.44Show that the coefficients of the affine connection obtained by firsttransforming from xi to _xiby _xi=_xi(xi) and then from _xito xi by xi =xi(x1 are identical with thosecoefficientsobtainedbytransformingdirectlyfromxi toxi bymeansof xi =xi(xi(xh)). 3.45InX4,Newton's equations of motion of a particle in a gravitational field character-ized by a scalar fieldrj>=rj>(xi)are a r : ~ > ax (IX=1, 2,3). Show that theseareautoparallel curvesof X 4 if thenonzerocomponents of the connectionarer /4 =a2 arj>jax,whereaisaconstant.Henceshowthatthe onlynonzerocomponentsofthecurvaturetensorareK/4p=a2 a2rj>jaxaxP andthatKu =0isequivalenttoLaplace's equation.Byconsideringaii1kshow that r/k are Christoffel symbols of the second kind with respect to the symmetric tensor field au if and only if X 4 is flat. (The geometry of dynamics has been treated fromatensorial point of viewwiththe aidof asymmetric type (0,2)tensor field in great detail in distinct ways by Synge[1].) 4 ADDITIONALTOPICS FROMTHE TENSORCALCULUS Inthischapterwediscussseveralsomewhatdisjointtopicswhich,despite theirimportance, donotfitnaturallyintothe developmentof theprevious chapter.Animportant extensionof the tensor conceptisrepresented by the so-called relative tensors; although this generalization isafairlysimple one whichmerelyentailssomeminormodificationsof thegeneraltheory,itis indispensable from the point of view of physical applications. This is due to the factthat the integral of ascalar is not itself a scalar; indeed, inorder that an n-foldintegraloveraregionof x. beinvariantunderarbitrary coordinate transformationsitisnecessary and sufficientthatitsintegrand be arelative scalar (also called ascalar density). Some considerable space isalso devoted tothe numericalrelativetensors (of whichtheKroneckerdeltaisaspecialcase);intrinsicallythesearenot particularly interesting, but, aswillbe seeninsubsequent chapters, they are exceedinglyusefulandpowerfulmanipulativetools.Abrief descriptionis alsogivenof normalcoordinates,which,inasense,maybe. regardedas generalizations of spherical polar coordinates of Euclidean geometry.Some-times the application of normal coordinates gives rise to useful simplifications of calculationsinvolvingtensors;however,theiruseisfraughtwithdanger. A verybrief discussionisalsogivenof theso-calledLiederivativeswhich appear frequentlyinthe modern literature.In thepresent context such deri-vatives are defined with the aid of an arbitrary vector field,without reference toanunderlying connection.AnalternativeapproachtotheLie derivative issketched inthe Appendix. 101 102ADDITIONALTOPICSFROMTHETENSORCALCULUS 4.1RELATIVETENSORS Inthissectionweintroduceageneralizationof thetensorconceptwhich occursveryfrequentlyinapplications,particularlywhenintegrationpro-cessesareinvolved(SyngeandSchild[1]).Althoughthisextensionisim-portant, it is not an unduly drastic one; in fact,it will be seen that it is a simple mattertoapplythetechniquesandresultsof thepreviouschaptertothe generalization tobe considered below. Weshall beginwith an example whichissuggested bya type (0,2)tensor fieldahk(xP)definedonourdifferentiablemanifoldx .UnderaclassC2 coordinatetransformation xJ=xj(xh), the transformed components of ahkare givenby axhaxk ajl=axJa.xzahk (1.1) (1.2) According to the usual product rule for determinants one therefore finds that (axh)(axk) det(ajl)=detaxjdetaxldet(ahk). (1.3) Weshall write,forthe sake of brevity, a(xP)=det[ahk(xP)],(1.4) while in accordance with (3.1.6) the Jacobian of the inverse of(l.l) is denoted by J: _d(axh)_a(x1,. ,x") J- eta j- a1")' x(x, ... , x of whichitisassumedthatitispositive.Thus (1.3)becomes a(xm)=Jla(xP). (1.5) (1.6) This result indicates very clearly that the determinant of a type (0, 2)tensor isnot ascalar, nor isitatensorof anytype (r,s),withr,s>0.Infact,itis evidentthat (1.6)exemplifiesatransformationlawwhichdiffersfromthose encountered thus far.Furthermore,if weassume forthe sake of the present argument that a(xP)isnonnegative,it followsfrom (1.6)that this istrue also of a(xm),and accordingly itmay beinferred from (1.6)that (1.7) This isagain a new transformation law; furthertransformation laws may be obtained bytaking differentexponents of ( 1.6). 4.1RELATIVETENSORS103 Clearly the examples (1.6)and (1.7)are nontrivial, and accordingly weare ledto introduce thefollowing. DEFINITION A function1/f(xh)of thecoordinatesof themanifoldX"issaidtorepresenta relativescalar fieldof weightw,if,underthecoordinatetransformation(1.1), thetransform of this functionisgiven by (1.8) Remark.A relative scalar of unit weightisoften called a scalar density. Accordinglythe determinant ( 1.4)isarelative scalar of weight2,whileits square rootisarelative scalarof weight1,asindicatedrespectivelyby(1.6) and (1.7).Furthermore, itisobvious that a scalar (or invariant) as defined in Section3.2 issimply arelative scalar of weight0. The importance of the concept of scalar density isillustrated by the follow-ing consideration. Let us suppose that we are given a closed, simply connected regionGinx., onwhichacontinuous function f(xh)of thecoordinatesis defined,sothat wemay construct then-foldintegral I=Lf(xh) dx1 dx". (1.9) [For instance,anintegralof thistypewould definethemassof amaterial bodywhichoccupiestheregionGiff(xh)denotesthedensityof matter.] According to the usualrule fora change of the independent variables inthe integrand of a multiple integral (Apostol[1]) itisfoundthat the value of the integralcorrespondingto(1.9)inthex-coordinate systemisgivenby i =Jf(xJ)dx1 dx"=Jj(xj)lacx:- ' x:) I dx1 . dx", GG8(x, ... , X) where j(xj) isthe transform off(xh). Bymeans of (1.5)this canbewritten as i={J(xj)r 1 dx1 dx". (1.10) If f(xh)isascalar,thatis,if j(xj)=f(xh),itwouldfollowimmediately from(1.9)and(1.10)that1 -:f.I(unlessJ=1,whichisnotassumedhere). Thustheintegralof ascalarfieldisnot,ingeneral,ascalar.However,itis frequentlyof utmost importance to construct invariant integrals: in fact,the theory of most physical fields,such as the electromagnetic field, or the gravit-ationalfieldof generalrelativity,dependscruciallyontheconstructionof invariant integrals. (Also, the simple example referred to above is indicative of 104 ADDITIONALTOPICSFROMTHETENSORCALCULUS this requirement, forsurely the mass of a material body is independent of the choice of special coordinates.) Ontheotherhand,iftheintegrandf(xh)of (1.9)happenstobeascalar density,itfollowsimmediately from (1.8)that f(xJ)r 1 =f(xh), and if thisissubstituted in(1.10),one obtains {J(xj) dx1 dx"={f(xh) dx1 dx",(1.11) sothattherequiredinvarianceof theintegral(1.9)isindeedattained.We therefore concludethata necessaryandsufficientconditioninorderthatthe integral (1.9)beinvariant underthecoordinate transformation (1.1)isthatthe integrand f(xh)bea scalar density. Thus far we have merely generalized the concept of scalar, and the question arises as to whether a corresponding extension inrespect of tensors of arbit-rarytype(r,s)isfeasible.Again,theexampleof ourtype(0,2)tensorfield provides us with an immediate motivation.Letusdenote the cofactor of the element ahkina= det(ahk)byAhk,sothat, according to (1.3.10),the elements of the inverse matrix(ahk)are givenby (1.12) Butweknow thattheahkconstitute thecomponentsof atype(2,0)tensor. Thus if weapply the transformation lawof the latter, together with (1.6),to (1.12)wesee that a-la-ja-la-j -lj _--lj _2~~hk_2~~Ahk A- aa- Ja axhaxka- Jaxhaxk. (1.13) It followsthatthe cofactorsAhkindet(ahk)do not satisfythetransformation law of a type (2,0)tensor; again the factor J2on the right-hand side of ( 1.13) isresponsibleforthis.Thissituationisanalogousto(1.6),andaccord-ingly we shall refer to quantities which transform inthe manner indicated by ( 1.13)asrelativetype (2,0)tensors of weight2. Theaboveconsiderationsclearlyindicatetheneedforthefollowing general definition. DEFINITION Asetofnr+sfunctionsAh, .. hrk, ... k,(xq)issaid toconstitute thecomponentsof a relative tensor field of type (r,s)and weight won themanifold x., if,underthe coordinatetransformation(1.1),thesefunctionstransformaccordingtothe 4.1RELATIVETENSORS 105 relation ( 1.14) inwhichthe Jacobian Jisdefined by (1.5). Remark1.For the purposes of this definition the weight w is restricted to the valuesO,1,2, .... Remark 2.Relativetensorsofunitweightaregenerallycalledtensor densities.In order to distinguish the usual concept of tensor [w=0 in (1.14)], relativetensorsof zero weightare sometimesreferred to as absolute tensors. Remark3.Thealgebraicoperations describedinSection3.2inrespectof tensorsareeasilyextendedtothecaseof relativetensors: 1.Tworelativetensorsofidenticaltypeandweightmaybeaddedto yield arelativetensor of thesame type and weight. 2.Theproductsof thecomponentsof atype(r ~ 's 1)relativetensorof weightw 1 withthose of atype (r 2,s 2)relative tensor of weightw2 constitute thecomponentsofatype(r1 + r2,s1 + s2)relativetensorofweightw1 + w2; inparticular, when a type (r,s)relative tensor of weight w 1 ismultiplied by a relativescalarof weightw2,atype(r,s)relativetensorof weightw1 + w2 isobtained. 3.Theprocessof contractionappliedtoatype(r,s)relativetensorof weightwyieldsatype(r- 1,s- 1)relativetensorofthesameweight (providedthatr~1,s ~1). 4.Symmetryandskew-symmetrypropertiesofrelativetensorsof arbitrary weightare independent of the choice of the coordinate system. Thesestatementsareimmediateconsequencesofthepropertiesofthe correspondingoperationsdescribedinSection3.2.Furthermore,fromthe linearity of (1.14)in the components of the relative tensor fieldit followsthat ifarelativetensorvanishesinagivencoordinatesystem,thiswillbethe casealsoinallother coordinatesystems.Accordinglyanyrelationwhichis expressedentirelyintermsof relativetensorsisinvariantundercoordinate transformations. Again,theordinarypartialderivativesof arelativetensorfielddonot constitute the components of a relative tensor, and accordingly it is necessary to define once more a process of covariant differentiation. It will be seen that thisrequiresanontrivialmodificationofthemethodintroducedinthe preceding chapter. This situation is best illustrated by the consideration of the gradient of a relative scalar field of weight w. To this end we differentiate (1.8) 106ADDITIONALTOPICSFROMTHETENSORCALCULUS withrespectto xj,which gives alfi- JWaxh81/fJW-laJ axj- axj axh+ waxj 1/J. (1.15) In view of the presence of the second term on the right-hand side it is obvious thatthegradientof ascalar densityof weightwisnotarelativecovariant vector(unlessw=0,whichisnotassumedhere).Inordertoconstructan appropriaterelativetensorwehavetoevaluateexplicitlythepartialderi-vatives of the Jacobian J. Letusdenotethe cofactor of the element 8xh/8x1 of J byq, sothat ( 1.16) But from (3.1.3)wehave (1.17) sothat a-1 IX ck= J axk' (1.18) Now, according totheusualrule forthe differentiationof adeterminant as giveninSection1.3,wehave sothat, by (1.18), This issubstituted in(1.15)toyield alflw[axh81/182xhax1 J axj= 1 axJaxh+ wax/ax1 axh1/1' whichrepresents the explicittransformationlaw of the gradient of 1/J. (1.19) (1.20) Inorder to construct asuitabletensor density,wenow invokethetrans-formationlaw (3.3.21)of the connection coefficients,which is 82xhaxm- axmaxhaxk --.--=r.m ----rp 8xJ8x1 axhJ I8xP8Xj8x1 h k 4.1RELATIVETENSORS107 Inordertobeabletosubstitutethisin(1.20),wemustcontractoverthe indices m and I;with the aid of (1.17)wethus obtain 82xh8xl-Ik 8xh-I8xhk 8xj 8xl8xk=rj I- bp8xj r/k =rj I- 8xj rhk(1.21) This,incidentally,isthetransformationlawof thecontractedconnection coefficients.If this issubstituted in (1.20),itisfoundthat 81/i- w8xh81/Jw- Iw8xhk ,/, 8xj- J8xj8xh+ wJrj 11/J- wJ8xjrhk'l' or,if weapply (1.8)to the second term onthe right-hand side, 8ljl-I - - w8xh[81/JkJ 8xj- wrji!/J- J8xj8xh- wrhk!/J' fromwhich it isimmediately evidentthatthe quantities defined by 81/1k 1/J,h=8xh- wrh k!/J, (1.22) (1.23) constitutethecomponentsof atype(0,1)relativetensorof weightw.It is thereforeobviousthat,whenthecovariantderivativeof arelativescalaris constructed,anadditionalterminvolvingthecontractedconnectioncoef-ficientshasto be introduced. Bymeansof (1.21)theargumentgivenaboveiseasilyextendedtotype (r,s)relative tensor fieldsof weightw,and itisfoundthat the covariantderi-vative of such a fieldmust be defined as follows: s ~rmAhjrrhAhjr - L...,lpkhlp-lmlp+l"""ls- wk hl,ls (1.24) fl=l which obviously reduces to the definition (3.5. 7) of the covariant derivative of a type (r,s)tensor fieldwhenever w= 0. Forexample,thecovariantderivativeof arelativecontravariantvector fieldof weight wis .8Aj.hh. A(k=8xk+ r/kA- wrk hAl. (1.25) In particular, if we contract over the indicesj and kin this relation, we obtain 108ADDITIONALTOPICSFROMTHETENSORCALCULUS Thusforarelativecontravariantvectorfieldof unitweightthecontracted covariant derivative issimply the sum of the corresponding ordinary partial derivatives: .oAj Afj =oxj (w=1).( 1.27) Inanalogywiththecorrespondingconceptofelementaryvectoranalysis we regard A{j ( ~ o rany value of w) as the divergence of the relative contravariant vector fieldA1 Since the contracted connection coefficients obviously play an important role whenever tensor densities are considered, let us evaluate these coefficients explicitlyforthecasewhentheconnectionisderiveddirectlyfromanon-singularclassC1 symmetrictype(0,2)tensorfieldahk(x1)aswasdonein Section3.5.From (3.5.15)and (3.5.20)wehave,fortheChristoffelsymbols which definesuch a connection, Contracting over jand k inthisrelation, weobtain (a)j- ~lj(oajloalh- oahj) r h j- 2 aoxh+oxjox1 But asa result of the postulated symmetry of ahjwehave lj oalh- jloa jh- lj oahj aoxj- aox1- aox1 , so that (1.29)reducesto ~j- ~jl oajz- ~-lAjz oajz hj-2aoxh-2aoxh' (1.28) ( 1.29) ( 1.30) where,inthe laststep,wehaveused (1.12).Accordingly (1.30)becomes (a)1oa1001oJa r h}j= 2 a-1oxk= 2 oxh (Ina)= oxh (Inja) =Jaoxh,(1.31) inwhich a denotesIdet(ahj)l. Thisisthe required expression forthe contracted Christoffelsymbols. Incidentally, sinceJa isascalardensityof unitweightinviewof (1.7), it followsfrom (1.23)that r:.oJa.r:. (y u)lh=oxh- rh}jy' a (1.32) 4.2THENUMERICALRELATIVETENSORS 109 foranarbitraryconnection.However,ifwechoosethespecialconnection (1.28)itfollowsimmediately from (1.31)that (ja)lh =0,(1.33) whichisa natural counterpart of theidentities (3.5.26)and (3.5.27). 4.2THENUMERICALRELATIVETENSORS In this section we are concerned with the so-called numerical relative tensors whichareoftenextremelyusefulincomplicatedmanipulativeprocesses involving tensors (Veblen [2]). We have already encountered an example of a numericaltensor, namely,the Kronecker delta.This tensor isaspecial case of oneofthemostimportantnumericaltensors,theso-calledgeneralized Kroneckerdelta.ThegeneralizedKroneckerdelta,denotedby possesses an equal number of sub- and superscripts (r) and is defined in terms of an rxrdeterminant asfollows: (2.1) Clearly theKronecker deltaisaspecialcase of this definitionwhichcor-responds to thevaluer=1.With r=2 in(2.1)weseethat Ingeneral isthesumof r!terms,eachof whichistheproductof r Kronecker deltas.Since,aswehaveseen,the ordinary Kronecker deltaisa type (1,1)tensor, it followsimmediately that the generalized Kronecker delta (2.1)isatype (r,r)tensor. In(2.1)wenoticethattheindicesj1,j2,... ,j,correspondtothe1st, 2nd, ... , rth rows of the determinant, whereas h1,h2, ,h, correspond to the 1st,2nd,... ,rthcolumnsofthedeterminant.Inviewof thefactthatthe sign of a determinant is changed by interchanging two rows (or two columns) weimmediately have the followingimportant properties: 5i_l" ..= _ birikih!r )1Jr)1Jr' 5itir__ 5itir itih ... jk'"ir- it .. jk'ih .. 'ir' (2.2) foralldistincth,k,1 h,kr;thatis,thegeneralizedKroneckerdelta :::}.isskew-symmetric under interchange of any two of the indices i 1 i, 110ADDITIONALTOPICSFROMTHETENSORCALCULUS and isskew-symmetricunderinterchangeof any twoof theindices j 1 j,. Clearlythen,if anytwoof theindicesi1 i,(or j1 j,) coincide,thecor-responding vanishesidentically.If r>n,thenatleasttwoofthe indicesi1 i, mustcoincide, in which case wehavetheusefulresult ifr >n.(2.3) To demonstratetheapplicationof (2.3),andto gainsomemanipulative skill involving (2.1),webrieflyconsider two examples, which willbeusedin the sequel. EXAMPLE1CURVATURETENSORINX2 For n=2 wehave, from (2.3)and (2.1), WhenthelatterismultipliedbyKh1ijthecurvaturetensorof X 2 defined by (3.6.8),accountbeing taken of (3.8.1 ),wefind 0=

+

+ 2Kh\r By virtue of (3.8.13),(2.4)can be expressed inthe form forn=2; that is,inX 2 theRicci tensorcompletely determinesthe curvature tensor. EXAMPLE2TENSORSOFTYPE(2,2) INX3 (2.4) (2.5) Consider atensor of type (2,2),Hijhk,which has the followingproperties: Hijhk= - Hjihk= - H;\h, Hij-0 hj- . (2.6) (Specificexamplesoftensorswiththesepropertiesarediscussedlater.) Wenowwishtoevaluatethetensor

Expandingaboutthe firstrow (the index k)weseethat, by (2.6), _ Hij uijtukl- uijut1- uijtul Each of the generalized Kronecker deltas on the right-hand side of the latter are now expanded about the firstrow (the indexI)so that, by (2.6), Hij_ uijtukl- uijtu- uijuP which finallyyields _4H's uijtukl- tu (2.7) 4.2THENUMERICALRELATIVETENSORS111 Since,forn=3,(2.3)implies wesee from (2.7)thatif H;\1 satisfies (2.6)then forn=3.(2.8) In order to obtain other important properties of the generalized Kronecker delta, we now return to (2.1) and, by virtue of (2.3), we restrict ourselves to the casern.Letusexpand(2.1)intermsof theelementsof itslastcolumn andthecofactorsthereof,thelatterbeing(r- 1)x(r- 1)determinants, whichareagainexpressedbymeansofthegeneralizedKroneckerdelta. One thus obtains (jh)r- (jjr(jhjr-1(jjr-1(jhjr-2jr+ (jjr-2(jhjr-3jr-1jr h1hr- hrhthr-1- hrht'hr-1h,- hthr-1 + + ( -1y+

We now contract over the indices j,, h,, obtaining. However, by virtue of (2.2),the latter reducesto (jhjr-dr= (n- r+1)(5hjr-1. hthr-1Jrhthr-1 (2.9) (2.10) Inthisformulawenowcontractover j,_1,h,_1,applyingtheformulato itself forthe case when r isreduced to r- 1.This gives (jhjr-2jr- dr=(n- r+ 2)(n- r+1)(5hjr-2. ht .. hr-2)r-1Jrht .. hr-2 If this process isrepeated, weobtain the followinggeneral identity: ....(n- s)!.. {Jll .. ')s}s+l'"}r- {Jll" .. )s ht"hsis+t"ir- (n- r)!hths' (2.11) which isfrequentlyused.Anequivalent form of(2.11)isthe following: (2.12) In particular, whent=r,wehave (j/1fr=_n_'_ -. 11}r(n- r)! (2.13) Asafurtherexample,letusconsiderthesumbf:rkAjhk,whereA jhkisa type (0,3)tensor whichissupposedto beskew-symmetricinallits indices. 112ADDITIONALTOPICSFROMTHETENSORCALCULUS By (2.9)(with r=3)wehave 'h'k UistA jhk=bfsA jht- bis A jtk+ UzsAthk =3!Alst In fact,forany integer rn,it iseasily seenthat (jh "-J'rA ...=r' A hthrl1J2 ... Jrh1h2 .. hr' (2.14) provided that Ah .. ;'risskew-symmetric in all its indices.In a similar way we also have (2.15) provided that Bh1 .. hrisskew-symmetric inallits indices. We shall now derive another important result associated with the general-izedKronecker delta.LetT;1 ... ;rbe a type (0,r)tensor fieldsothat - -1oxh1oxhrk 7;1"';r(x)=oxi1... oxirT,.1"'hr(x). We differentiate this equation with respectto x_ir+ 1 to find oi;1'"iro2xh"0Xh1 ... oxh" -1 OXh" + 1 oxhrT. oxir+1= 1'';:1oxir+1oxi"oxi1oxi-10Xi"+1oxirh1"'hr 0Xh1 oxhr oxhr+ 1 oT.

oxi1oxiroxir+ 1oxhr+ 1 ' (2.16) whichclearlyindicatesthatoT,.1 ... h)oxhr+ 1 arenottensors,inaccordwith Section 3.5.However, the generalized Kronecker delta enables us to associate with the latter a quantity which istensorial. This isachieved inthe following way.If wemultiply(2.16)by8i.l"'ir+ 1 andnotethato2xh"/oxir+ 1 oxi"is Jt .. Jr+ 1 symmetricini, + 1 il'forf.1=1,... , r whereas 11 isskew-symmetricin i, + 1 il'forf.1=1,... , r wethus find ..oT...oxh1 oxhr+ 1 oT. (}lF .. lr+t l5'.tlr+t--. Jt .. Jr+laxlr+lJt .. ]r+laxllaxlr+laxhr+l (2.17) Since

isatype (r+1,r+1)tensor wealso have ..oxh1oxhr+ 1oxk1oxkr+ 1 8'1"'1r+1__ ,,, ___ (jh1"'hr+1__ ,,, __ h .. jr+1oii1oxir+1- k1"kr+1oxhox/r+1' which,ifsubstitutedin(2.17)establishesthat:::t: oi;1 ... ;)oxir+ 1 isa (0,r+ 1)tensor.We willreturnto thisresultin Section 5.2. We return once more to (2.1) in order to introduce two additional numerical tensors,thepermutationsymbols(whicharesometimesreferredtoasthe 4.2THENUMERICALRELATIVETENSORS 113 Levi-Civita symbols or alternating symbols). In (2.1)we set r=nand define the permutation symbols eh, .. hnehj"by and(2.18) respectively.We draw attention to the factthateach of these quantitieshas exactlynindices,wherenisthedimensionofx .From(2.18)and(2.2), eh, .. h"and eJ" .. j"areeachskew-symmetricinalltheirindicesand willthus vanish if any of the indices coincide.Furthermore, from (2.18)and (2.1) so that wehave e12 ...=b ~ L :=1, e12=b ~ L :=1, { + 1if h1 h. isan evenpermutation of 1,2,... , n; eh,h"=eh, .. h"=-1if h1 h. isan odd permutation of 1, 2,... , n; 0otherwise. (2.19) The terminology permutation symbolisthus justified. Wenow wishtoestablish that (2.20) To thisend consider the quantity (2.21) whichisclearlyskew-symmetricinboth setsof indices.Consequently,the onlypossiblenonzerocomponentsof A ~ ~ : : twilloccurwhen j1 j. and h1 h. are eachapermutationof 1,2,... ,n.However,from(2.19),(2.21) and (2.1)itiseasily seenthat A ~ ~ : : : :=0. Wehave thus shownthat which, by (2.21)establishes (2.20). From (2.20)and (2.13)we alsohave (2.22) If weconsiderthedeterminantofanarbitrarysetofn2 quantitiesa;j expressedintheform(1.3.2),weseethatthesummationtakentogether 114 ADDITIONALTOPICSFROMTHETENSORCALCULUS withthefactor( -1)" isverycloselyrelatedto(2.19).Infact(1.3.2)canbe expressed inthe form det(a. )=i'hj"a. a.a . l)1)12}2"Jn Byvirtueof (2.19)itiseasilyseenthatthislastexpressioncanbewritten inthe equivalent form In asimilar way,if b ~and ci1 are each sets of n2 quantities wehave e..det(bi.)=e.. bb. . Vi" lt .. lnJJt .. Jn11ln' and If wemultiply (2.24)by e;, ... ;",noting (2.22)and (2.20),we find .1.... det(b'.)= - b'1 ... , ~b!1 b!n J'Jt .. Jn11ln' n. (2.23) (2.24) (2.25) (2.26) (2.27) which isan expression forthe determinant of ( b ~ )interms of the generalized Kronecker delta. Wenowwishtodeterminethetensorcharacterofthepermutation symbols.Ina new x-coordinate system (2.18)becomes oxhoxhox1" 0Xh1 0Xh2 oxh" thh .. jn= 8J1'12 ... .J.nn- c)k, .. kn - 0Xk 10Xk2 oxk"ox1ox2...ox"h, ... hn' (2.28) inwhichwesubstituteontheright-handsidefrom(2.20),notingatthe same time that, by virtue of (2.24) (2.29) where J denotes the Jacobian (1.5).Thus (2.28)reducesto (2.30) whichdemonstratesthatthepermutationsymbolsek, .. knconstitutethe componentsof atype(n,0)relativetensorof weight+ 1,thatis,atype (n,0) tensor density. Similarly,itmaybeshownthatthepermutationsymboleh, .. h"constitute the components of atype (0,n)relativetensorof weight-1. 4.2THENUMERICALRELATIVETENSORS 115 Weshallnowderivesomeimportantresultsconcerningdeterminants. Again,letusconsideran(nxn)matrixwitha= Thesumof all (1x1)principal minors issimply the trace, denoted by a)AW+ 1>A"w, (c)v(dl>)=where 1>,Q are p-forms, wisaq-fonn, and c1,c2 are constants. 5.18Let 1>be ap-form, 1>=a,1 ..dx'1 A Adx',and letA' be the components of a type (1,0) tensor.Define the (p- 1)-form Aim=__ 1_ iea . .Ahdx''AAdx'- (p- 1)!qlp-lh)t .. )p and _:!Jet>=0forp=0. Show that (a).iJ (:ijt>)=0; forpz1 (b).iJ(t>A'I')= (:ijt>)A'I'+ ( -1)"1>A.iJ'I'where 'I' isaq-form; (c).:i.J(Al>)=A(:ijl>); (d)d(.iJl>)+.iJ(dl>)=Act>. 5.19In the notation of Problem 5.18define eli=a,, ... ,.(dx'- A''dt)A A(dx'- A' dt). Show that eli=1>- (.iJ 1>)Adt. 5.20If Ah=Ah(xi) are the components of a type (1,0) vector field in x. show that the partial differential equation PROBLEMS179 has(n- 1)independentsolutionst/>1, ,tj>"-1Byconsideringthechangeof coordinatesx'=tJ>',i=1,... ,n- 1,x"=fwherefisanyfunctionof x'for whichIox'fox1 I #0 provethat there alwaysexistsa coordinate system in which all the components of Ahexcept one are zero. *5.21Show that condition (4.51)isequivalent to each of the following (a)There exist!-forms e)forwhich dw'=e) 1\wi; (b)There exists a!-form A.forwhich d(w1 AAw")=A.Aw1 AAw". *5.22Provethat(4.7)iscompletelyintegrableifandonlyifthereexistfunctions A}, vhfor which [Hint:substitute (4.13) in (4.6)and noteand (4.22).] 5.23If w=A, dx' show that there exist IX,tjJforwhich otJ> A.=IX---, 'ox' if and only if dwAw =0. 5.24If w1 =A, dx', w2 =B1 dx1 show that there exist IX,{3,y,15,tjJand t/Jforwhich otJ>ot/J A,=IX---; + f3---:, ox'ox' otJ>ot/J B,=y---: + 15-----:, ox'ox' ifandonlyifdw"Aw1 Aw2 =0(IX=1,2).Expressthislatterconditionin terms of A1 and B1. 5.25Prove that otJ>otJ> y--x-=0 oxoy otJ>otJ>.ot/> x- -y-+smz-=0 oxoyoz is completely integrableand hastwofunctionallyindependent solutions. 5.26In aparticular X4,w/, definedby (2.20),aregivenby w/ = + + where A.,J1are functions of x1 alone and A.=dA.fdx1Compute Q/ and Ql, defined by (6.3)and (6.5),respectively.Hence show that K/hk=0 if and only if 2 jl- A.Ji+ Ji2 =0. 80THECALCULUSOFDIFFERENTIALFORMS 5.27If Q=A,ik dx'AdxiAdxkshowthatthereexist1-forms w1,w2,w3 forwhich Q=w1 Aw2 Aw3 if and only if assumingwithoutlossof generalitythatA,ikarecompletelyskew-symmetric. Prove that this isan identity if n =4. 5.28If Qisa 2-form and w1 and w2 are1-forms show that QAW1 Aw2 =Q if and only if there exist1-formsn1 and n2 forwhich Q=1t1 AW1 + 1tzAw2 6 INVARIANTPROBLEMS INTHECALCULUS OFVARIATIONS A mostfascinatingphenomenonistheimpactof theconcept of invariance under coordinate transformations onthe calculusof variations.Thisisdue tothefactthatwhenoneassumesthattheactionintegralof avariational principleisaninvariant(asisusuallydoneinphysicalfieldtheories),itis necessaryto stipulatethattheintegrandbeascalar density.Theprofound implications of this state of affairs were clearly recognized by Hilbert [1], and itistheobjectiveof thischaptertodescribethesemattersinsomedetail. Sinceitisnot assumed thatthe readerisfamiliarwith the calculus of varia-tions, some of the elementary aspects of the latter are developed ab initio with theaidof tensormethodsandthecalculusof forms.Moreover,withthe exceptionofthefirstpartofSection6.2,thefairlyrecentapproachof Caratheodory tothe calculus of variations isintroduced inpreference to the more widely known classical techniques based on the theory of the firstand secondvariations.Inthissensethischapteristoberegardedasanendin itself.However, it should be pointed out that forthe purposes of Riemannian geometry as presented inChapter 7,only an acquaintance with the contents of Sections6.1and6.2isrequired,whereasthematerialof Chapter8is dependenton Sections 6.1,6.2,6.5,and 6.7. Particular emphasis isplaced on thetheorems of Noether, which are con-cerned with the implications of the assumption that the fundamental integ