PrefaceThisbookcontainsthesolutionsofalltheexercisesofmybook:PrinciplesofTensorCalculus.Thesesolutionsaresufficientlysimplifiedanddetailedforthebenefitofreadersofalllevelsparticularlythoseatintroductorylevels.TahaSochiLondon,September2018

TableofContentsPrefaceNomenclatureChapter1PreliminariesChapter2Spaces,CoordinateSystemsandTransformationsChapter3TensorsChapter4SpecialTensorsChapter5TensorDifferentiationChapter6DifferentialOperationsChapter7TensorsinApplicationAuthorNotesFootnotes

NomenclatureInthefollowinglist,wedefinethecommonsymbols,notationsandabbreviationswhichareusedinthebookasaquickreferenceforthereader.∇ nabladifferentialoperator∇;and∇; covariantandcontravariantdifferentialoperators

∇f gradientofscalarf∇⋅A divergenceoftensorA∇ × A curloftensorA∇2,∂ii,∇ii Laplacianoperator

∇v,∂ivj velocitygradienttensor,(subscript) partialderivativewithrespecttofollowingindex(es);(subscript) covariantderivativewithrespecttofollowingindex(es)hat(e.g.Âi,Êi) physicalrepresentationornormalizedvectorbar(e.g.ũi,Ãi) transformedquantity

○ innerorouterproductoperator⊥ perpendicularto1D,2D,3D,nD one-,two-,three-,n-dimensionalδ ⁄ δt absolutederivativeoperatorwithrespecttot∂iand∇i partialderivativeoperatorwithrespecttoithvariable∂;i covariantderivativeoperatorwithrespecttoithvariable[ij, k] Christoffelsymbolof1stkindA areaB,Bij Fingerstraintensor

B − 1,Bij − 1 Cauchystraintensor

C curveCn ofclassnd,di displacementvectordet determinantofmatrix

diag[⋯] diagonalmatrixwithembraceddiagonalelementsdr differentialofpositionvectords lengthofinfinitesimalelementofcurvedσ areaofinfinitesimalelementofsurfacedτ volumeofinfinitesimalelementofspaceei ithvectoroforthonormalvectorset(usuallyCartesianbasis

set)er, eθ, eφ basisvectorsofsphericalcoordinatesystemerr, erθ, ⋯, eφφ unitdyadsofsphericalcoordinatesystemeρ, eφ, ez basisvectorsofcylindricalcoordinatesystemeρρ, eρφ, ⋯, ezz unitdyadsofcylindricalcoordinatesystemE,Eij firstdisplacementgradienttensor

Ei,Ei ithcovariantandcontravariantbasisvectors

ℰi ithorthonormalizedcovariantbasisvectorEq./Eqs. Equation/Equationsg determinantofcovariantmetrictensorg metrictensorgij,gij,gji covariant,contravariantandmixedmetrictensororits

componentsg11, g12, ⋯gnn coefficientsofcovariantmetrictensor

g11, g12, ⋯gnn coefficientsofcontravariantmetrictensorhi scalefactorforithcoordinateiff ifandonlyifJ Jacobianoftransformationbetweentwocoordinate

systemsJ Jacobianmatrixoftransformationbetweentwocoordinate

systemsJ − 1 inverseJacobianmatrixoftransformationL lengthofcurven,ni normalvectortosurfaceP pointP(n, k) k-permutationsofnobjects

qi ithcoordinateoforthogonalcoordinatesystemqi ithunitbasisvectoroforthogonalcoordinatesystemr positionvectorℛ RiccicurvaturescalarRij,Rij Riccicurvaturetensorof1stand2ndkind

Rijkl,Rijkl Riemann-Christoffelcurvaturetensorof1stand2ndkind

r, θ, φ coordinatesofsphericalcoordinatesystemS surfaceS,Sij rateofstraintensorS̃,S̃ij vorticitytensort timeT(superscript) transpositionofmatrixT,Ti tractionvectortr traceofmatrixui ithcoordinateofgeneralcoordinatesystemv,vi velocityvectorV volumew weightofrelativetensorxi,xi ithCartesiancoordinate

x’i,xi ithCartesiancoordinateofparticleatpastandpresenttimes

x, y, z coordinatesof3Dspace(mainlyCartesian)γ,γij infinitesimalstraintensorγ̇ rateofstraintensorΓkij Christoffelsymbolof2ndkind

δ Kroneckerdeltatensorδij,δij,δji covariant,contravariantandmixedordinaryKronecker

deltaδijkl,δijklmn,δi1…in

j1…jn

generalizedKroneckerdeltain2D,3DandnDspace

Δ,Δij seconddisplacementgradienttensorcovariantrelativepermutationtensorin2D,3DandnD

ϵij,ϵijk,ϵi1…in space

ϵij,ϵijk,ϵi1…in contravariantrelativepermutationtensorin2D,3DandnDspace

εij,εijk,εi1…in covariantabsolutepermutationtensorin2D,3DandnDspace

εij,εijk,εi1…in contravariantabsolutepermutationtensorin2D,3DandnDspace

ρ, φ coordinatesofplanepolarcoordinatesystemρ, φ, z coordinatesofcylindricalcoordinatesystemσ,σij stresstensorω vorticitytensorΩ regionofspaceNote:duetotherestrictionsontheavailabilityandvisibilityofsymbolsinthemobiformat,aswellassimilarformattingissues,weshoulddrawtheattentionoftheebookreaderstothefollowingpoints:1.Barsoversymbols,whichareusedintheprintedversion,werereplacedbytildes.However,forconveniencewekeptusingtheterms“barred”and“unbarred”inthetexttorefertothesymbolswithandwithouttildes.2.Thesquarerootsymbolinmobiis√()wheretheargumentiscontainedinsidetheparentheses.Forexample,thesquarerootofgissymbolizedas√(g).3.Inthemobiformat,superscriptsareautomaticallydisplayedbeforesubscriptsunlesscertainmeasuresaretakentoforcetheoppositewhichmaydistortthelookofthesymbolandmaynotevenbetherequiredformatwhenthesuperscriptsandsubscriptsshouldbesidebysidewhichisnotpossibleinthemobitextandliveequations.Therefore,forconvenienceandaestheticreasonsweonlyforcedtherequiredorderofthesubscriptsandsuperscriptsorusedimagedsymbolswhenitisnecessary;otherwiseweleftthesymbolstobedisplayedaccordingtothemobichoicealthoughthismaynotbeideallikedisplayingtheChristoffelsymbolsofthesecondkindas:Γijkorthegeneralized

Kroneckerdeltaas:δi1…inj1…jninsteadoftheirnormallookas: and

.4.Duetothedifficultyofconvertingtheordinaryintegralsymbol(i.e.\int)correctlytothemobiformatweusethefollowingintegralsymbol⨏(i.e.\fint)assubstituteinthetextualmathematicalexpressions.Inbrief,alltheintegralsymbolsinthisbookrepresentordinaryintegrals.5.Somesymbolsinthemobiversionarenotthesameasinthepaperversion.ThereaderthereforeshouldconsulttheNomenclatureofthegivenversionforclarification.

Chapter1Preliminaries1. Differentiatebetweenthesymbolsusedtolabelscalars,vectorsandtensors

ofrank > 1.Answer(seeFootnote1in§8↓):Scalars:non-indexedlowercaselightfaceitalicLatinletters(e.g.fandh)areusedtolabelscalars.

Vectors:non-indexedloweroruppercaseboldfacenon-italicLatinletters(e.g.aandA)areusedtolabelvectorsinsymbolicnotationwiththeexceptionofthebasisvectorswhereindexedboldfaceloweroruppercasenon-italicsymbols(e.g.e1andEi)areused.

Tensorsofrank > 1:non-indexeduppercaseboldfacenon-italicLatinletters(e.g.AandB)areusedtolabeltensorsofrank > 1insymbolicnotation.

IndexedlightfaceitalicLatinsymbols(e.g.aiandBjki)areusedtodenotetensorsofrank > 0(i.e.vectorsandtensorsofrank > 1)intheirexplicittensorform,i.e.indexnotation.

2. WhatthecommaandsemicoloninAjk, iandAk;imean?Answer:Thecommameanspartialderivativewithrespecttothevariablewhoseindexfollowsthecomma(i.e.theithvariableinAjk, i),whilesemicolonmeanscovariantderivativewithrespecttothevariablewhoseindexfollowsthesemicolon(i.e.theithvariableinAk;i).

3. Statethesummationconventionandexplainitsconditions.Towhattypeofindicesthisconventionapplies?Answer:Accordingtothesummationconvention,dummyindicesimplysummationovertheirrange.Moreclearly,atwice-repeatedvariable(i.e.notnumeric)indexinasingleterm(whetherthetwice-repeatedindexoccursinonetensororintwotensors)impliesasumoftermsequalinnumbertotherangeoftherepeatedindex.Hence,ina2Dspacewehave:Baa = B11 + B22

whileina3Dspacewehave:CaDa = C1D1 + C2D2 + C3D3

4. Whatisthenumberofcomponentsofarank-3tensorina4Dspace?Answer:Thenumberofcomponentsofarank-rtensorinannDspaceisgivenbynr.Hence,thenumberofcomponentsis43 = 64.

5. AsymbollikeBjkimaybeusedtorepresenttensororitscomponents.Whatisthedifferencebetweenthesetworepresentations?Dotherulesofindicesapplytobothrepresentationsornot?Justifyyouranswer.Answer:Thedifferenceisthatwhenthesesymbolsrepresenttensorstheyshouldbetreatedastensorsandhencetheyobeytherulesoftensors(e.g.thetransformationrulesandtherulesofindices),whilewhentheyrepresentcomponentstheyarelikescalarsandhencetheyareordinarynumbersorvariables(e.g.realnumbers).Forexample,whenBjkirepresentsatensoritiswrongtowriteBjki + CwhereCisascalar,butthisiscorrectwhenBjkirepresentsacomponent.Asindicated,therulesofindicesapplytotensorsbutnottotheircomponents.

6. Whatisthemeaningofthefollowingsymbols:∇∂j∂kk∇2∂φh, jkAi;n∂n∇;kCi;kmAnswer:∇:nabladifferentialvectoroperator.

∂j:partialderivativeoperatorwithrespecttothejthvariable.

∂kk:LaplaciandifferentialscalaroperatorinCartesianform.

∇2:Laplaciandifferentialscalaroperator.

∂φ:partialderivativewithrespecttothevariableφ.

h, jk:secondorderpartialderivativewithrespecttothevariablesindexedbyjandk.

Ai;n:covariantderivativeofthecontravariantvectorAiwithrespecttothevariableindexedbyn.

∂n:contravariantderivativewithrespecttothevariableindexedbyn.

∇;k:contravariantderivativewithrespecttothevariableindexedbyk.

Ci;km:secondordercovariantderivativeofthecovariantvectorCiwithrespecttothevariablesindexedbykandm.

7. Whatisthedifferencebetweensymbolicnotationandindicialnotation?Forwhattypeoftensorsthesenotationsareused?Whataretheothernamesgiventothesetypesofnotation?Answer:Thesymbolicnotationisageometricallyorientednotationwithnoreferencetoaparticularcoordinatesystemandhenceitisintrinsicallyinvarianttothechoiceofcoordinatesystem,whiletheindicialnotationtakesanalgebraicformbasedoncomponentsidentifiedbyindicesandreferredtoaparticularsetofbasisvectorsofagivencoordinatesystemandhencethenotationissuggestiveofanunderlyingcoordinatesystem.Also,thesymbolicnotationisusuallyidentifiedbyusingboldfacenon-italicsymbols,likeaandB,whiletheindicialnotationisidentifiedbyusinglightfaceindexeditalicsymbolssuchasaiandBij.Thesenotationsareusedfornon-scalartensorsandhencetheybelongtotensorsofrank > 0.Othernamesforsymbolicnotationareindex-freenotation,directnotation,andGibbsnotation.Othernamesforindicialnotationareindexnotation,componentnotation,andtensornotation.

8. “Thecharacteristicpropertyoftensorsisthattheysatisfytheprincipleofinvarianceundercertaincoordinatetransformations”.Doesthismeanthatthecomponentsoftensorsareconstant?Whythisprincipleisveryimportantinphysicalsciences?

Answer:No.Theprincipleofinvarianceisabouttheinvarianceoftheformandnotabouttheinvarianceorconstancyofthevaluesoftheindividualcomponents(seeFootnote2in§8↓).Thisprincipleisveryimportantinphysicalsciencesbecausethelawsofscienceshouldsatisfytheprincipleofform-invariancewhentheyaretransformedacrosscoordinatesystemsandframesofreference.Thisisbecauseforthelawsofsciencetobeusefulandofcommonvalue,theyshouldbeindependentofthecoordinatesystems,framesofreferenceandobservers.

9. Stateandexplainallthenotationsusedtorepresenttensorsofallranks(rank-0,rank-1,rank-2,etc.).Whataretheadvantagesanddisadvantagesofusingeachoneofthesenotations?Answer:Regardingrank-0tensors(i.e.scalars),theyhaveonlyonewayoflabelingwhichiscommonlynon-indexedlightfaceitalicLatinletters(e.g.f)orGreekletters(e.g.φ).Regardingtensorsofrank > 0(i.e.vectorsandhigherranktensors),theyhavesymbolicnotationandindicialnotationwhichareexplainedinapreviousquestion(seeExercise71↑).Thesymbolicnotationisofgeometricnaturewithnoreferencetoaparticularcoordinatesystem,whiletheindicialnotationisofalgebraicnaturewithanindicationtoanunderlyingcoordinatesystemandbasistensors.Non-indexedboldfacestraightsymbolsareusuallyusedtorepresentsymbolicnotation,whileindexedlightfaceitalicsymbolsareusuallyusedtorepresentindicialnotation.Symbolicnotationisusedingeneralrepresentationwhileindicialnotationisusedinspecificrepresentations,formulationsandcalculations.Regardingtheadvantagesanddisadvantages,symbolicnotationismoregeneralandsuccinctandeasiertoreadthanindicialnotation,whileindicialnotationismorespecificandinformative.Indicialnotationmaybesusceptibletosomeconfusionsincethesamesymbol(likeAi)maybeusedtorepresentthecomponentsaswellasthetensoritselfbutfromthisveryperspectiveitismoreflexibleandversatile.Indicialnotationmayalsobemoresusceptibletoerrorinwritingandtypesettingduetothepresenceofindicesandmayalsorequiremoreoverheadinthisregardsincewritingandtypesettingindexedsymbolsinalegibleformusuallyrequireextraeffortespeciallywhenusingsimpleeditors,forexample,althoughbold-facing(orusingsimilarnotationaltechniqueslikeunderliningorusingover-arrows)alsorequiresadditionaleffort.Inbrief,symbolicnotationisrecommendedforgeneralrepresentationwhileindicialnotationshouldbeusedinspecific

representationthatrequirestherevelationoftheunderlyingstructureandindicationofthecoordinatesystemorreferenceframeandbasisvectorssuchasduringformulationandcalculation.Forexample,whenwetalkaboutatensorthatcanbecovariantorcontravariantormixedorwewanttotalkaboutatensorwhosevariancetypeisirrelevantinthatcontext,thenitismoreappropriatetousesymbolicnotationlikeAbecauseitisgeneralandcanrepresentanyvariancetype,butifwetalkspecificallyaboutthepropertiesandtherulesthatapplyspecificallytooneofthesevariancetypes(suchascovarianttype)orweintendtousethetensorsymbolinexplicittensorformulation,calculationanddevelopmentofanalyticalargumentsandproofsthenitismoreappropriate(andmayevenbenecessary)touseindicialnotationlikeAiforthattensor.

10. Statethecontinuityconditionthatshouldbemetiftheequality:∂i∂j = ∂j∂iistobecorrect.Answer:Thecontinuityconditionmeansthatthefunctionanditsfirstandsecondpartialderivativesdoexistandtheyarecontinuousintheirdomain.

11. Explainthedifferencebetweenfreeandboundtensorindices.Also,statetherulesthatgoverneachoneofthesetypesofindexintensorterms,expressionsandequalities.Answer:Thedifferencescanbesummarizedasfollows:(a)Freeindexoccursonlyonceinatensorterm,whilebound(ordummy)indexoccurstwiceinatensorterm.(b)Thesummationconventionappliestoboundindicesbutnottofreeindices.(c)Freeindiceshaveextendedpresenceinalltermsoftensorexpressionsandequalities,whileboundindicesarerestrictedtotheirtermsandhencetheycanoccuronlyinsometermsoftensorexpressionsandequalities.(d)Boundindicescanbereplacedinindividualterms(aslongasthenewlabelisnotusedinthatterm)butfreeindicescannotalthoughfreeindicescanbereplacedinalltermsifthenewlabelisnotinuseinthatcontext.(e)Whenboundindicesarepresentinmorethanonetermoftensorexpressionsandequalitiestheycanbenameddifferentlyineachtermbutfreeindicesshouldbenameduniformlyinallterms.(f)Freeindicescountintensorrankandorderbutboundindicescountonlyintensororder.(g)Boundindicescanbepresentinscalarquantity(whenallindicesarecontracted)butfreeindicescannot.Therulesoffreeindices:

(A)Eachtermshouldhavethesamenumberoffreeindices.(B)Afreeindexshouldhavethesamevariancetypeinallterms.(C)Eachtermshouldhavethesamesetoffreeindices,e.g.alltermsshouldhavei, j, kandhenceitisnotallowedtohaveonetermwithi, j, ksetandanothertermwithi, j, nset.(D)Thefreeindicesshouldhavethesamearrangementinallterms,e.g.ijklinalltermsandnotijklinsometermsandikjlinotherterms.(E)Eachindexshouldhavethesamerange(i.e.spacedimensionality)inallterms,andhencetheindexiinAi + Biexpressionshouldhaveidenticalrangeinbothterms.Therulesofboundindices:(i)Theyareusuallysubjecttothesummationconvention.(ii)Theygenerallyshouldbeofoppositevariancetype(i.e.onecovariantandonecontravariant)exceptinorthonormalCartesiansystemswheretheycanhavethesamevariancetype.(iii)Theycanbenamedindependentlyineachterm.(iv)Theydonotcontributetothetensorrankbuttheycontributetothetensororder.

12. Explainthedifferencebetweentheorderandtherankoftensorsandlinkthistothefreeanddummyindices.Answer:Theorderrepresentsthetotalnumberofindicesincludingdummyindices,whiletherankrepresentsthenumberoffreeindicesonly.Accordingly,freeindicescountintensorrankandorderbutdummyindicescountonlyintensororder.

13. Whatisthedifferencebetweencovariant,contravariantandmixedtypetensors?Giveexamplesforeach.Answer:Inbrief:(a)Covarianttensorshaveonlysubscriptindices.Contravarianttensorshaveonlysuperscriptindices.Mixedtensorshavebothsubscriptandsuperscriptindices.(b)Covariantandcontravarianttensorsareofrank > 0(i.e.vectorsandhigherranktensorsandhencetheyexcludeonlyscalars),whilemixedtensorsareofrank > 1(andhencetheyexcludescalarsandvectors).(c)Covarianttensorsareassociatedwithcontravariantbasisvectors.Contravarianttensorsareassociatedwithcovariantbasisvectors.Mixedtensorsareassociatedwithbothcovariantandcontravariantbasisvectors(i.e.theircovariant/contravariantindicescorrespondtocontravariant/covariantbasisvectors).

Examples:Ai,Bij,Cijkarecovarianttensors.Ai,Bij,Cijkarecontravarianttensors.Aij,Bijk,Cmnijkaremixedtensors.

14. Whatisthemeaningof“unit”and“zero”tensors?Whatisthecharacteristicfeatureofthesetensorswithregardtothevalueoftheircomponents?Answer:Unittensorisatensorwhoseallcomponentsarezeroexceptthosewithidenticalvaluesofallindiceswhichareassignedthevalue1.Zerotensorisatensorwhoseallcomponentsarezero.Thecharacteristicfeatureofthesetensorsisthatalltheircomponentsareconstant(i.e.0and1forunittensorand0forzerotensor).

15. Whatisthemeaningof“orthonormalvectorset”and“orthonormalcoordinatesystem”?Stateanyrelevantmathematicalcondition.Answer:Orthonormalvectorsetmeansasetofvectorswhicharemutuallyorthogonalandeachoneisofunitlength.Theorthonormalityofavectorsetmaybeexpressedmathematicallybythefollowingdotproductequations:Vi⋅Vj = δijorVi⋅Vj = δijwheretheindexedδistheKroneckerdeltasymbolandtheindexedVsymbolizesavectorinthesetwhileiandjarerangingoverthedimensionoftheunderlyingspace.Orthonormalcoordinatesystemmeansacoordinatesystemwhosebasisvectorsetisorthonormalatallpointsofthespacewherethesystemisdefined.

16. Whatistherulethatgovernsthepairofdummyindicesinvolvedinsummationregardingtheirvariancetypeingeneralcoordinatesystems?Whichtypeofcoordinatesystemisexemptofthisruleandwhy?Answer:Theruleisthattheindicesshouldbeofoppositevariancetype(i.e.onecovariantandtheothercontravariant).ThecoordinatesystemthatisexemptfromthisruleistheorthonormalCartesianbecausethecovariantandcontravarianttypesofthissystemareidentical.

17. Statealltherulesthatgoverntheindicialstructureoftensorsinvolvedintensorexpressionsandequalities(rank,setoffreeindices,variancetype,orderofindicesandlabeling).Answer:Therulesare:(a)Eachtermshouldhavethesamenumberoffreeindices,i.e.thesamerank.(b)Afreeindexshouldhavethesamevariancetypeinallterms.

(c)DummyindicesshouldbeofoppositevariancetypeexceptinorthonormalCartesiansystems.(d)Eachtermshouldhavethesamesetoffreeindices,e.g.alltermsshouldhavei, j, kandhenceitisnotallowedtohaveonetermwithi, j, ksetandanothertermwithi, j, nset.(e)Thefreeindicesshouldhavethesamearrangement,e.g.ifirstksecondandmthird.(f)Eachindexshouldhavethesamerange(i.e.spacedimensionality)inallterms.(g)Dummyandfreeindicesshouldhavedistinctlabeling,i.e.nodummyindexcansharethesamelabelwithafreeindex.

18. Howmanyequalitiesthatthefollowingequationcontainsassuminga4Dspace:Bki = Cki?Writealltheseequalitiesexplicitly,i.e.B11 = C11,B21 = C21,etc.Answer:Sixteen.Theyare:B11 = C11B12 = C12B13 = C13B14 = C14B21 = C21B22 = C22B23 = C23B24 = C24B31 = C31B32 = C32B33 = C33B34 = C34B41 = C41B42 = C42B43 = C43B44 = C44

19. Whichofthefollowingtensorexpressionsislegitimateandwhichisnot,givingdetailedexplanationineachcase?Aki − Bi

Caa + Dnm − Bbba + BScdjcdk + FabjabkAnswer:First:illegitimatebecausethenumberoffreeindicesisdifferent,i.e.thetwotensorsareofdifferentrank.

Second:illegitimatebecausetheinvolvedtensorsareofdifferentrank,i.e.CaaandBbbareofrank-0whileDnmisofrank-2.

Third:legitimatebecauseaandBarescalars.

Fourth:legitimatebecauseScdjcdkandFabjabkhaveidenticalindicialstructurethatfollowsalltherulesoffreeandboundindicesintensorexpressions.Thedifferenceinthedummyindices(i.e.canddinScdjcdkandaandbinFabjabk)doesnotmatterbecauseboundindicesarerestrictedtotheirownterm.

20. Whichofthefollowingtensorequalitiesislegitimateandwhichisnot,givingdetailedexplanationineachcase?A.in = Bn.iD = Scc + Nabba3a + 2b = JaaBmk = CkmBj = 3c − DjAnswer:First:illegitimatebecausethearrangementoftheindicesisdifferent,i.e.ininA.inandniinBn.i.

Second:legitimatebecausealltheinvolvedtensorsinthisequalityareofrank-0.

Third:legitimatebecausealltheinvolvedtensorsareofrank-0.

Fourth:illegitimatebecausethecorrespondingindicesareofdifferentvariancetype.

Fifth:illegitimatebecausetheinvolvedtensorsareofdifferentrank,i.e.BjandDjareofrank-1whilecisofrank-0.

21. Explainwhytheindicialstructure(rank,setoffreeindices,variancetypeandorderofindices)oftensorsinvolvedintensorexpressionsandequalitiesareimportantreferringinyourexplanationtothevectorbasissettowhichthetensorsarereferred.Alsoexplainwhytheserulesarenotobservedintheexpressionsandequalitiesoftensorcomponents.Answer:Thereasonisthattheindicialnotationoftensorsisbasedonaparticularsetofbasisvectorsandhencethecharacteristicsoftheindicialstructure(i.e.rank,setoffreeindices,variancetypeandorderofindices)haveparticularsignificancesincetheyhavecertainassociationandrepresentationofthebasissetanditscharacteristics.Forexample,thetensorAijmeansAijEiEjwhilethetensorBijkmeansBijkEiEjEkandhenceAij + BijkandAij = BijkareincorrectbecauseAijEiEj + BijkEiEjEkandAijEiEj = BijkEiEjEkaremeaningless.Therulesofindicialstructurearenotobservedintheexpressionsandequalitiesoftensorcomponentsbecausethesecomponentsarescalarsinnatureandhencetheseexpressionsandequalitiesdonotrefertothebasisvectors.Forexample,ifthevalueofthecomponentBijofthetensorBis10andthevalueofthecomponentCijofthetensorCisalso10thenitismeaningfulandusefultostateBij = Cij.Similarly,whenwewriteϵij = ϵijwemeanthecorrespondingcomponentsofthetensorsϵijandϵijhaveidenticalvalues,e.g.ϵ12 = ϵ12 = 1andϵ11 = ϵ11 = 0whichislegitimateandcorrect.

22. Whyfreeindicesshouldbenameduniformlyinalltermsoftensorexpressionsandequalitieswhiledummyindicescanbenamedineachtermindependently?Answer:Thereasonisthatfreeindicesrefertothecommonbasisvectorsetandhencetheyhavereachbeyondtheirownindividualterms(i.e.theyrepresentacommonreferenceacrossalltermsofthetensorexpressionsandequalities),whileeachdummyindexrepresentsasuminitsowntermwithnoreachorpresenceintoothertermsandhencedummyindicescanbenamedindependentlyineachterm.

23. Whataretherulesthatshouldbeobservedwhenreplacingthesymbolofafreeindexwithanothersymbol?Whataboutreplacingthesymbolsofdummyindices?Answer:Therulesaboutreplacingthesymbolofafreeindexare:

(a)Thereplacementshouldbeuniformandthoroughwithinthegivencontextandhenceitshouldtakeplaceoveralltheoccurrencesofthereplacedindexinthatcontextandnotonlyoversometermsorexpressionsorequalities.(b)Thenewindexshouldnotbeinusealreadyasalabeltoanotherfreeorboundindexinthatcontext.(c)Alltheindicialstructuralaspectsoftheoldindex(variancetype,position,etc.)shouldbeinheritedbythenewindex.Inbrief,thechangeshouldberestrictedtonamingandshouldnottouchanyotheraspectofthereplacedindex.Regardingthereplacementofthesymbolsofdummyindices,theycanbereplacedbyanothersymbolwhichisnotpresentasafreeordummyindexintheirtermaslongasthereisnoconfusionwithasimilarsymbolinthatcontext.

24. Whyingeneralwehave:∂iAj ≠ Aj∂i?Whatarethesituationsunderwhichthefollowingequalityisvalid:∂iAj = Aj∂i?Answer:Thereasonisthat∂iAjandAj∂ihavedifferentmeaningandmathematicalsignificanceandimplicationbecause∂iAjmeansthat∂iisoperatingonAjwhileAj∂imeansthat∂iisoperatingonsomethingelseandAjjustmultipliestheresultofthisoperation.Theequality∂iAj = Aj∂iholdsidenticallywhenAj = 0.Italsoholdsinmanyotherspecialcases.Forexample,itholdswhenAjandtheoperandof∂iontherighthandsideareconstant(ormoregenerallytheyareindependentoftheithvariableandhencethepartialderivativesarezero).

25. Whatisthedifferencebetweentheorderofatensorandtheorderofitsindices?Answer:Theorderofatensorisanindicatorofthetotalnumberoftensorindices,whiletheorderofitsindicesrepresentsthearrangementoftheseindices(i.e.whichisfirst,whichissecond,etc.).

26. InwhichcaseAijkisequaltoAikj?WhataboutAijkandAikj?Answer:AijkisequaltoAikjwhenwehaveasymmetryinthejandkindices.AijkandAikjareequalwhenwehaveanunderlyingorthonormalCartesiancoordinatesystemplusasymmetryinthejandkindices(seeFootnote3in§8↓).

27. Whataretherank,orderanddimensionofthetensorAijkina3Dspace?

WhataboutthescalarfandthetensorAabmabjnfromthesameperspectives?Answer:TherankofAijkis3,itsorderis3anditsdimensionis3.Therankoffis0,itsorderis0anditsdimensionis3.TherankofAabmabjnis3,itsorderis7anditsdimensionis3.

28. WhatistheorderofindicesinAjik?Insertadotinthissymboltomaketheordermoreexplicit.Answer:Theorderisjik.OninsertingadotbetweenjandkwegetAj.ikwhichismoreexplicitabouttheorderofindices.

29. Whytheorderofindicesofmixedtensorsmaynotbeclarifiedbyusingspacesorinsertingdots?Answer:Somereasonsare:●Theorderofindicesisirrelevantinthegivencontext,e.g.anyordercanachievetheintendedpurpose.●Theorderisclearfromotherindicatorsinthegivencontext.●TheorderisindicatedimplicitlybythealphabeticalorderoftheselectedindicesandhenceAjiforinstancemeansifirstandjsecond.

30. Whatisthemeaningof“tensorfield”?IsAiatensorfieldconsideringthespatialdependencyofAiandthemeaningof“tensor”?Answer:Tensorfieldisatensorthatisdefinedoveranextendedandcontinuousregion(orregions)ofthespaceoroverthewholespace.Yes,thevectorAishouldbeatensorfieldwhenitisdefinedoveranextendedregionofthespacewhere“tensor”hereisusedinitsgeneralsensethatincludesvectors.

Chapter2Spaces,CoordinateSystemsandTransformations1. Givebriefdefinitionstothefollowingterms:Riemannianspace,coordinate

systemandmetrictensor.Answer:Riemannianspaceisamanifoldcharacterizedbytheexistenceofasymmetricrank-2tensorcalledthemetrictensorthatisdefinedoverthewholemanifold.Coordinatesystemisanabstractmathematicaldeviceofgeometricnaturethatisusedbyanobservertoidentifythepositionofpointsandobjectsanddescribeeventsinagivenspaceormanifold.Metrictensorisarank-2symmetricabsolutenon-singulartensorthatisassociatedwithagivenRiemannianspace.Themetrictensorcontainsvitalinformationabouttheessentialgeometricpropertiesofthespace

2. Discussthemainfunctionsofthemetrictensorinagivenspace.Howmanytypesthemetrictensorcanhave?Answer:Thefunctionsofthemetrictensorinclude:●Identifyingthegeometricpropertiesofthespace.●Raisingandloweringindicesandhencefacilitatingthetransformationbetweenthecovariantandcontravarianttypes.Typesofthemetrictensor:covariant,contravariantandmixed.

3. Whatisthemeaningof“flat”and“curved”space?Givemathematicalconditionsforthespacetobeflatintermsofthelengthofaninfinitesimalelementofarcandintermsofthemetrictensor.Whytheseconditionsshouldbeglobalforthespacetobeflat?Answer:Flatspaceisaspacetowhichacoordinatesystemwhosemetrictensorcanbecastintoadiagonalformwithallthediagonalentriesbeing + 1or − 1doesexist,whilecurvedspaceisaspacetowhichsuchacoordinatesystemdoesnotexist.ThemathematicalconditionforannDspacetobeflatintermsofthelengthofaninfinitesimalelementofarcdsisgivenby:(ds)2 = 

ζ1(du1)2 + ζ2(du2)2 + … + ζn(dun)2 = 

Σni = 1ζi(dui)2

wheretheindexedζare±1whiletheindexeduarethecoordinatesofthespace.ThemathematicalconditionforannDspacetobeflatintermsofthemetrictensorisgivenby:gij = ±1(i = j)

gij = 0(i ≠ j)

wheregijaretheelementsofthemetrictensor.Whenwedescribeaspacetobeflatwemeangloballyflat(otherwisewedescribeitaslocallyflat)andhencetheseconditionsshouldbeglobalforthespacetobeflatinaglobalsense.

4. Givecommonexamplesofflatandcurvedspacesofdifferentdimensionsjustifyingineachcasewhythespaceisflatorcurved.Answer:●Planeisanexampleofa2Dflatspacebecauseitcanbecoordinatedbya2DCartesiansystemwithadiagonalmetrictensorwhosealldiagonalelementsare + 1.●OrdinaryEuclideanspaceisanexampleofa3Dflatspacebecauseitcanbecoordinatedbya3DCartesiansystemwithadiagonalmetrictensorwhosealldiagonalelementsare + 1.●MinkowskispacetimemanifoldthatunderliesthemechanicsofLorentztransformationsisanexampleofa4Dflatspacebecauseitcanbecoordinatedbya4Dcoordinatesystemwithadiagonalmetrictensorwhosealldiagonalelementsare±1.●Sphereandellipsoidareexamplesof2Dcurvedspacebecausetheycannotbecoordinatedbyasystemwithadiagonalmetrictensorwhosediagonalelementsare±1.●Examplesofcurvedspacesofhigherdimensionalitycanbefoundinmathematicsandsometheoriesofmodernphysicswhereabstractcurvedspacesareusedtoconceptualizeandquantifymathematicalandphysicaltheories.Forexample,inthegeneraltheoryofrelativitycurved4Dspaces(representingthespacetimeofthephysicalworld)areusedtoformulatea

geometrictheoryofgravity.5. Explainwhyall1DspacesareEuclidean.

Answer:Thereasonisthatanycurvecanbemappedisometricallytoastraightlinewherebotharenaturallyparameterizedbyarclength.Thismeansthatanycurvatureofa1Dspacedoesnotbelongtothespaceitselfbuttotheembeddingspace,i.e.thecurvatureisextrinsicratherthanintrinsic.

6. Giveexamplesofspaceswithconstantcurvatureandspaceswithvariablecurvature.Answer:Plane,sphereandBeltramipseudo-sphereareexamplesof2Dspaceswithconstantcurvature(wherethecurvatureofplaneis0,thecurvatureofsphereis1 ⁄ r2withrbeingitsradiusandthecurvatureofBeltramipseudo-sphereis − 1 ⁄ ρ2withρbeingthepseudo-radiusofthepseudo-sphere),whileellipsoid,torusandellipticandhyperbolicparaboloidsareexamplesof2Dspaceswithvariablecurvature.

7. StateSchurtheoremoutliningitssignificance.Answer:SchurtheoremindifferentialgeometryassertsthatiftheRiemann-ChristoffelcurvaturetensorateachpointofannDspace(n > 2)isafunctionofthecoordinatesonly,thenthecurvatureisconstantalloverthespace.AnexampleofitspracticalsignificanceisthatbyperformingasimpletestonthedependencyoftheRiemann-Christoffelcurvaturetensorandfindingthecurvatureonasinglepointwewillhaveinformationaboutthecurvatureofthespaceatagloballevel.Thetheoryalsohasotherimportanttheoreticalsignificanceandimplications.

8. Whatistheconditionforaspacetobeintrinsicallyflatandextrinsicallyflat?Answer:AspaceisintrinsicallyflatifftheRiemann-Christoffelcurvaturetensorvanishesidenticallyoverthespace,anditisextrinsically(aswellasintrinsically)flatiffthecurvaturetensorvanishesidenticallyoverthewholespace.

9. WhatisthecommonmethodofinvestigatingtheRiemanniangeometryofacurvedmanifold?Answer:ThecommonmethodforinvestigatingtheRiemanniangeometryofacurvedmanifoldistoembedthemanifoldinaEuclideanspaceofhigherdimensionalityandinspectthepropertiesofthemanifoldfromthisperspective.

10. Givebriefdefinitionstocoordinatecurvesandcoordinatesurfacesoutliningtheirrelationstothebasisvectorsets.Howmanyindependent

coordinatecurvesandcoordinatesurfaceswehaveateachpointofa3Dspacewithavalidcoordinatesystem?Answer:Coordinatecurvesarecurvesalongwhichexactlyonecoordinatevarieswhilealltheothercoordinatesareconstant,whilecoordinatesurfacesaresurfacesoverwhichexactlyonecoordinateisconstantwhilealltheothercoordinatesvary.Thecovariantbasisvectorsaretangentvectorstothecoordinatecurves,whilethecontravariantbasisvectorsaregradientofthespacecoordinatesandhencetheyareperpendiculartothecoordinatesurfaces.Weshouldhave3independentcoordinatecurvesand3independentcoordinatesurfacesateachpointofa3Dspacewithavalidcoordinatesystem.

11. Whyacoordinatesystemisneededintensorformulations?Answer:Coordinatesystemsareneededintensorcalculustodefinenon-scalartensorsinaspecificformandidentifytheircomponentsinreferencetothebasissetofthesystem.

12. Listthemaintypesofcoordinatesystemoutliningtheirrelationstoeachother.Answer:Coordinatesystemscanbeclassifiedfromdifferentperspectives.●Forexample,theycanbeclassifiedasrectilinearcoordinatesystemswhicharecharacterizedbythepropertythatalltheircoordinatecurvesarestraightlinesandalltheircoordinatesurfacesareplanes,andcurvilinearcoordinatesystemswhicharecharacterizedbythepropertythatatleastsomeoftheircoordinatecurvesarenotstraightlinesandsomeoftheircoordinatesurfacesarenotplanes.●Theycanalsobeclassifiedasorthogonalcoordinatesystemswhicharecharacterizedbyhavingmutuallyperpendicularcoordinatecurvesandcoordinatesurfacesateachpointintheirspace,andnon-orthogonalwhicharenotso.●Theycanalsobeclassifiedashomogeneouswhenthemetrictensoroftheirspaceistheunitytensor,andnon-homogeneousotherwise.●TheymayalsobeclassifiedindividuallyfromtheperspectiveoftheirowncharacteristicsandhencewehaveCartesian,cylindrical,sphericalaswellasmanyothertypesofcoordinatesystems(e.g.parabolicandparaboliccylindrical)whosepropertiesarethoroughlyinvestigatedinmathematicaltexts.

13. “Thecoordinatesofasystemcanhavethesamephysicaldimensionordifferentphysicaldimensions”.Giveanexampleforeach.Answer:ThecoordinatesofCartesiansystemshavethesamephysical

dimension(i.e.length),whilethecoordinatesofcylindricalsystemshavedifferentphysicaldimensions(i.e.lengthforρandzandangleforφwhichisdimensionless).

14. Provethatsphericalcoordinatesystemsareorthogonal.Answer:Orthogonalsystemsarecharacterizedbyhavingmutuallyperpendicularbasisvectors,andhenceallweneedforestablishingthisproofistoshowthat:er⋅eθ = 0

er⋅eφ = 0

eθ⋅eφ = 0

ThebasisvectorsforsphericalcoordinatesystemsaregiveninorthonormalCartesianformbythefollowingequations:er = sinθcosφi + sinθsinφj + cosθk

eθ = cosθcosφi + cosθsinφj − sinθk

eφ =  − sinφi + cosφj

wherei, j, karetheCartesianunitbasisvectors.Accordingly,wehave:er⋅eθ = 

sinθcosφcosθcosφ + sinθsinφcosθsinφ − cosθsinθ = 

sinθcosθcos2φ + sinθcosθsin2φ − cosθsinθ = 

sinθcosθ(cos2φ + sin2φ) − cosθsinθ = 

sinθcosθ − cosθsinθ = 

0

er⋅eφ = 

 − sinθsinφcosφ + sinθsinφcosφ = 

0

eθ⋅eφ = 

 − cosθcosφsinφ + cosθcosφsinφ = 

015. Whatisthedifferencebetweenrectilinearandcurvilinearcoordinate

systems?Answer:Allthecoordinatecurvesofrectilinearcoordinatesystemsarestraightlinesandalltheircoordinatesurfacesareplanes,whilethecoordinatecurvesandcoordinatesurfacesofcurvilinearsystemsarenotsoandhenceatleastsomeoftheircoordinatecurvesarenotstraightlinesandsomeoftheircoordinatesurfacesarenotplanes.Consequently,thebasisvectorsofrectilinearsystemsareconstantwhilethebasisvectorsofcurvilinearsystemsarevariableingeneralsincetheirdirectionor/andmagnitudedependonthepositioninthespaceandhencetheyarecoordinatedependent.

16. Giveexamplesofcommoncurvilinearcoordinatesystemsexplainingwhytheyarecurvilinear.Answer:Themostcommonexamplesarethecylindricalandsphericalcoordinatesystems.Thecylindricalcoordinatesystemsarecurvilinearbecausetheρ, φ, zcoordinatecurvesarestraightlines,circlesandstraightlinesrespectively(andhencesomeoftheircoordinatecurvesarenotstraightlines),whiletheρ, φ, zcoordinatesurfacesarecylinders,semi-planesandplanesrespectively(andhencesomeoftheircoordinatesurfacesarenotplanes).Thesphericalcoordinatesystemsarecurvilinearbecausether, θ, φcoordinatecurvesarestraightlines,semi-circlesandcirclesrespectively(andhencesomeoftheircoordinatecurvesarenotstraightlines),whilether, θ, φcoordinatesurfacesarespheres,conesandsemi-planesrespectively(andhencesomeoftheircoordinatesurfacesarenotplanes).

17. Giveanexampleofacommonlyusedcurvilinearcoordinatesystemwithsomeofitscoordinatecurvesbeingstraightlines.Answer:Anexampleisthecylindricalcoordinatesystemwhoseρandz

coordinatecurvesarestraightlines.18. Definebrieflytheterms“orthogonal”and“homogeneous”coordinate

system.Answer:Orthogonalcoordinatesystemisasystemwhosecoordinatecurves,aswellasitscoordinatesurfaces,aremutuallyperpendicularateachpointinthespace.Accordingly,thevectorsofitscovariantbasissetandthevectorsofitscontravariantbasissetaremutuallyorthogonaleverywhereinthespace.Homogeneouscoordinatesystemisasystemassociatedwiththeunitytensorasthemetricofitsunderlyingspace.

19. Giveexamplesofrectilinearandcurvilinearorthogonalcoordinatesystems.Answer:OrthonormalCartesiansystemsareexamplesofrectilinearorthogonalcoordinatesystems,whilecylindricalandsphericalcoordinatesystemsareexamplesofcurvilinearorthogonalcoordinatesystems.

20. Whatistheconditionofacoordinatesystemtobeorthogonalintermsoftheformofitsmetrictensor?Explainwhythisisso.Answer:Thenecessaryandsufficientconditionforacoordinatesystemtobeorthogonalisthatitsmetrictensorisdiagonal.Thiscanbeinferredfromthedefinitionofthecomponentsofthemetrictensorasthedotproductsofthebasisvectorssincethedotproductinvolvingtwodifferentvectors(i.e.Ei⋅EjorEi⋅Ejwithi ≠ j)willvanishifthebasisvectors,whethercovariantorcontravariant,aremutuallyperpendicular.Astheconditioni ≠ jisassociatedwiththenon-diagonalcomponentsofthemetrictensorthenthismeansthatallthenon-diagonalcomponentsarezeroandhencethetensorisdiagonal.

21. Whatisthemathematicalconditionforacoordinatesystemtobehomogeneous?Answer:Themathematicalconditionforacoordinatesystemtobehomogeneousmaybegivenintermsofthemetrictensorthatassociatesthesystemby:gij =  + 1(i = j)

gij = 0(i ≠ j)

wheregijaretheelementsofthemetrictensor.22. Howcanwehomogenizeanon-homogeneouscoordinatesystemofaflat

space?Answer:Acoordinatesystemofaflatspacecanbehomogenizedby

allowingthecoordinatestobeimaginary.Thisisdonebyredefiningthecoordinatesas:Ui = √(ζi)ui

whereζi = ±1andwithnosumoveri.ThenewcoordinatesUiarerealwhenζi = 1andimaginarywhenζi =  − 1.

23. Giveexamplesofhomogeneousandnon-homogeneouscoordinatesystems.Answer:OrthonormalCartesiansystemsareexamplesofhomogeneouscoordinatesystems,whilecylindricalandsphericalsystemsareexamplesofnon-homogeneouscoordinatesystems.

24. Giveanexampleofanon-homogeneouscoordinatesystemthatcanbehomogenized.Answer:ThecoordinatesystemoftheMinkowskispacetime(whichisthespaceofthemechanicsofLorentztransformationswhosemetricmaybegivenbydiag[ − 1,  + 1,  + 1,  + 1]ordiag[ + 1,  − 1,  − 1,  − 1])isanexampleofanon-homogeneouscoordinatesystemthatcanbehomogenizedbyallowingthetemporalcoordinate(forthefirstformofthemetric)orthespatialcoordinates(forthesecondformofthemetric)tobeimaginary.

25. Describebrieflythetransformationofspacesandcoordinatesystemsstatingrelevantmathematicalrelations.Answer:AtransformationfromannDspacetoanothernDspaceisacorrelationthatmapsapointfromthefirstspace(original)toapointinthesecondspace(transformed)whereeachpointintheoriginalandtransformedspacesisidentifiedbynindependentcoordinates.Thetransformationofcoordinatesmaybeexpressedmathematicallybythefollowingrelation:ũi = ũi(u1, u2, …, un)wheretheunbarredandbarredindexedurepresentthecoordinatesoftheoriginalandtransformedspacesandi = 1, 2, …, nwithnbeingthedimensionofthespaces.

26. What“injectivetransformation”means?Isitnecessarythatsuchatransformationhasaninverse?Answer:Injectivetransformationmeansone-to-one.Itisnotnecessarythatsuchatransformationhasaninverseunlessitissurjective(i.e.onto)aswell.

27. WritetheJacobianmatrixJofatransformationbetweentwonDspaceswhosecoordinatesarelabeledasuiandũiwherei = 1, ⋯, n.Answer:TheJacobianmatrixofsuchatransformationisgivenby:

28. StatethepatternoftherowandcolumnindicesoftheJacobianmatrixinrelationtotheindicesofthecoordinatesofthetwospaces.Answer:Thepatternisthattheindicesofuinthenumeratorprovidetheindicesfortherowswhiletheindicesofũinthedenominatorprovidetheindicesforthecolumns.Thisindexingpatternmaybeinterchanged.

29. WhatisthedifferencebetweentheJacobianmatrixandtheJacobianandwhatistherelationbetweenthem?Answer:ThedifferenceisthattheJacobianmatrixisamatrixwhiletheJacobianisadeterminant.TherelationbetweentheJacobianmatrixandtheJacobianisthattheJacobianisthedeterminantoftheJacobianmatrixandthiscanbeexpressedmathematicallyas:J = det(J)whereJandJaretheJacobianandtheJacobianmatrixrespectively.

30. WhatistherelationbetweentheJacobianofagiventransformationandtheJacobianofitsinverse?Writeamathematicalformularepresentingthisrelation.Answer:TheJacobianoftheinversetransformationisthereciprocaloftheJacobianoftheoriginaltransformation.Thisrelationcanbeexpressedmathematicallyas:J̃ = 1 ⁄ JwhereJ̃istheJacobianoftheinversetransformationandJistheJacobianoftheoriginaltransformation.

31. Isthelabelingoftwocoordinatesystems(e.g.barredandunbarred)involvedinatransformationrelationessentialorarbitrary?Hence,discussifthelabelingofcoordinatesintheJacobianmatrixcanbeinterchanged.Answer:Labelingonecoordinatesystemasbarredandtheotheras

unbarredisamatterofchoiceandconvenienceandhenceitisratherarbitrary.Accordingly,thelabelingofcoordinatesintheJacobianmatrixasbarredandunbarredcanbeinterchangedandconsequentlytheJacobianmaybenotatedasunbarredoverbarredorbarredoverunbarred.However,althoughthisisapurenotationalmatterthearbitrarinesssometimespropagateseventotheterminologywherethe“Jacobian”isusedinreferencetotheoppositetransformation.Therefore,insteadofhaving“Jacobian”and“inverseJacobian”weprefertohave“Jacobianoftheoriginaltransformation”and“Jacobianoftheinversetransformation”.

32. UsingthetransformationequationsbetweentheCartesianandcylindricalcoordinatesystems,findtheJacobianmatrixofthetransformationbetweenthesesystems,i.e.CartesiantocylindricalandcylindricaltoCartesian.Answer:TheequationsofcoordinatetransformationbetweentheCartesianandcylindricalsystemsaregivenby:Cartesiantocylindrical:ρ = √(x2 + y2)

φ = arctan(y ⁄ x)

z = z

CylindricaltoCartesian:x = ρcosφ

y = ρsinφ

z = z

FortheJacobianmatrixofthetransformationfromCartesiantocylindricalwehave(seeFootnote4in§8↓):∂ρ ⁄ ∂x = x ⁄ √(x2 + y2)

∂ρ ⁄ ∂y = y ⁄ √(x2 + y2)

∂ρ ⁄ ∂z = 0

∂φ ⁄ ∂x =  − y ⁄ (x2 + y2)

∂φ ⁄ ∂y = x ⁄ (x2 + y2)

∂φ ⁄ ∂z = 0

∂z ⁄ ∂x = 0

∂z ⁄ ∂y = 0

∂z ⁄ ∂z = 1

Therefore,theJacobianmatrixforthistransformationis:

FortheJacobianmatrixofthetransformationfromcylindricaltoCartesianwehave:∂x ⁄ ∂ρ = cosφ

∂x ⁄ ∂φ =  − ρsinφ

∂x ⁄ ∂z = 0

∂y ⁄ ∂ρ = sinφ

∂y ⁄ ∂φ = ρcosφ

∂y ⁄ ∂z = 0

∂z ⁄ ∂ρ = 0

∂z ⁄ ∂φ = 0

∂z ⁄ ∂z = 1

Therefore,theJacobianmatrixforthistransformationis:

Note:asverificationtest,wecalculateinthefollowingtheJacobiansofthetwotransformations.TheJacobianofthetransformationfromCartesiantocylindricalis:J(x, y, z → ρ, φ, z) = 

[x ⁄ √(x2 + y2)] × [x ⁄ (x2 + y2)] − [y ⁄ √(x2 + y2)] × [ − y ⁄ (x2 + y2)] = 

[x2 + y2] ⁄ [√(x2 + y2)(x2 + y2)] = 

1 ⁄ √(x2 + y2) = 

1 ⁄ ρ

whiletheJacobianofthetransformationfromcylindricaltoCartesianis:J(ρ, φ, z → x, y, z) = 

ρcos2φ + ρsin2φ = 

ρ

Hence:J(x, y, z → ρ, φ, z) = 1 ⁄ J(ρ, φ, z → x, y, z)asitshouldbe.

33. RepeatQuestion322↑forthespherical,insteadofcylindrical,systemtofindtheJacobianthistime.Answer:TheequationsofcoordinatetransformationbetweentheCartesianandsphericalsystemsaregivenby:Cartesiantospherical:r = √(x2 + y2 + z2)

θ = arccos[z ⁄ √(x2 + y2 + z2)]

φ = arctan(y ⁄ x)

SphericaltoCartesian:x = rsinθcosφ

y = rsinθsinφ

z = rcosθ

FortheJacobianmatrixofthetransformationfromCartesiantosphericalwehave:∂r ⁄ ∂x = x ⁄ √(x2 + y2 + z2)

∂r ⁄ ∂y = y ⁄ √(x2 + y2 + z2)

∂r ⁄ ∂z = z ⁄ √(x2 + y2 + z2)

∂θ ⁄ ∂x = [zx] ⁄ [(x2 + y2 + z2)√(x2 + y2)]

∂θ ⁄ ∂y = [zy] ⁄ [(x2 + y2 + z2)√(x2 + y2)]

∂θ ⁄ ∂z =  − √(x2 + y2) ⁄ (x2 + y2 + z2)

∂φ ⁄ ∂x =  − y ⁄ (x2 + y2)

∂φ ⁄ ∂y = x ⁄ (x2 + y2)

∂φ ⁄ ∂z = 0

Therefore,theJacobianforthistransformationis(seeFootnote5in§8↓):

 = [x2 ⁄ {r3√(x2 + y2)}] + [y2 ⁄ {r3√(x2 + y2)}] + [(z2x2 + z2y2) ⁄ {r3√(x2 + y2)(x2 + y2)}]

 = [x2 ⁄ {r3√(x2 + y2)}] + [y2 ⁄ {r3√(x2 + y2)}] + [z2 ⁄ {r3√(x2 + y2)}]

 = [x2 + y2 + z2] ⁄ [r3√(x2 + y2)]

 = r2 ⁄ [r3√(x2 + y2)]

 = 1 ⁄ [r√(x2 + y2)]

 = 1 ⁄ [r2{√(x2 + y2) ⁄ r}]

 = 1 ⁄ [r2sinθ]

FortheJacobianmatrixofthetransformationfromsphericaltoCartesianwehave:∂x ⁄ ∂r = sinθcosφ

∂x ⁄ ∂θ = rcosθcosφ

∂x ⁄ ∂φ =  − rsinθsinφ

∂y ⁄ ∂r = sinθsinφ

∂y ⁄ ∂θ = rcosθsinφ

∂y ⁄ ∂φ = rsinθcosφ

∂z ⁄ ∂r = cosθ

∂z ⁄ ∂θ =  − rsinθ

∂z ⁄ ∂φ = 0

Therefore,theJacobianforthistransformationis:

 = sinθcosφ(rsinθcosφrsinθ) + rcosθcosφ(rsinθcosφcosθ) − rsinθsinφ( − sinθsinφrsinθ − rcosθsinφcosθ)

 = r2sin3θcos2φ + r2cos2θsinθcos2φ + r2sin3θsin2φ + r2cos2θsinθsin2φ

 = r2sinθ(sin2θcos2φ + cos2θcos2φ + sin2θsin2φ + cos2θsin2φ)

 = r2sinθ(cos2φ[sin2θ + cos2θ] + sin2φ[sin2θ + cos2θ])

 = r2sinθ(cos2φ + sin2φ)

 = r2sinθ

Aswesee,wehave:J(x, y, z → r, θ, φ) = 1 ⁄ J(r, θ, φ → x, y, z)asitshouldbe.

34. Giveasimpledefinitionofadmissiblecoordinatetransformation.Answer:Anadmissiblecoordinatetransformationisamappingrepresentedbyasufficientlydifferentiablesetofequationsanditisinvertiblebyhavinganon-vanishingJacobian.

35. WhatisthemeaningoftheCncontinuitycondition?Answer:TheCncontinuityconditionmeansthatthefunctionandallitsfirstnpartialderivativesdoexistandtheyarecontinuousovertheirdomain.

36. What“invariant”objectorpropertymeans?Givesomeillustratingexamples.Answer:Anobjectorpropertyisdescribedasinvariantifitdoesnotchangeundercertainadmissiblecoordinatetransformations.Forexample,inclassicalmechanicsthevalueofmassisinvariantundertheGalileantransformationsoftimeandspacecoordinatessincethevalueofmassisanintrinsicpropertyofthemassiveobjectandhenceitisindependentoftheobserver.Similarly,inthemechanicsofLorentztransformationsMaxwell’sequationsareinvariantundertheLorentztransformationsofspacetimecoordinatessincetheformoftheseequationsdoesnotchangeunderthesetransformations.

37. Whatisthemeaningof“compositionoftransformations”?Stateamathematicalrelationrepresentingsuchacomposition.Answer:Compositionoftransformationsmeansasuccessionoftransformationswheretheoutputofonetransformationistakenasaninputtothenexttransformation.Thismaybeexpressedmathematicallyas:Tc(O) = TmTm − 1⋯T2T1(O)wherethetransformationsTi(i = 1, 2, ⋯, m)arecomposedtoproducethecompositetransformationTcandOisanobjectthatistransformedbythesetransformations.Intheaboveequation,theoutputofthetransformationT1

istakenasaninputtothetransformationT2andsoonuntilfinallytheoutputofthetransformationTm − 1istakenasaninputtothetransformationTmtoobtainthecompositetransformationTc.

38. WhatistheJacobianofacompositetransformationintermsoftheJacobiansofthesimpletransformationsthatmakethecompositetransformation?WriteamathematicalrelationthatlinksalltheseJacobians.Answer:TheJacobianofthecompositetransformationistheproductoftheJacobiansoftheindividualtransformationswhichthecompositionismadeof.Thiscanbeexpressedmathematically(inreferencetotheequationofthepreviousquestion)as:Jc = JmJm − 1⋯J2J1whereJcistheJacobianofthecompositetransformationTcandJi(i = 1, 2, ⋯, m)istheJacobianoftheTitransformation.

39. “Thecollectionofalladmissiblecoordinatetransformationswithnon-vanishingJacobianformagroup”.Whatthismeans?Stateyouranswerinmathematicalanddescriptiveforms.Answer:Thismeansthattheyaregroupinthetechnicalsenseofthistermaccordingtothegrouptheory,andhencetheysatisfythepropertiesofclosure,associativity,identityandinverse.Mathematically,ifwehaveasetoftransformationsT1, T2, ⋯definedonacertaindomainthentheyshouldsatisfythefollowingconditions:●Closure,i.e.ifTiandTjareanytwotransformationsinthisgroupthentheircompositionTc = TiTjisalsoatransformationinthegroup.●Associativity,i.e.ifTi, Tj, Tkareanythreetransformationsinthisgroupthenweshouldhave:Ti(TjTk) = (TiTj)Tk●Identity,i.e.thereisasingletransformationTIinthegroupsuchthat:TITm = TmTI = TmwhereTmisanytransformationinthegroup.●Inverse,i.e.foranytransformationTminthegroupthereisexactlyonetransformationTm − 1(whichiscalledtheinverseofTm)suchthat:TmTm − 1 = Tm − 1Tm = TIwhereTIistheidentitytransformation.

40. Isthetransformationofcoordinatesacommutativeoperation(seeFootnote6in§8↓)?Justifyyouranswerbyanexample.Answer:ThetransformationofcoordinatesisnotcommutativeingeneralandhencewemayhaveTiTj ≠ TjTi.Anexampleofthisisthecompositionoftworotationswhoseoutcomedependsontheorderoftherotations,asexplainedanddemonstratedinthetext.

41. AtransformationT3withaJacobianJ3isacompositetransformation,i.e.T3 = T2T1wherethetransformationsT1andT2haveJacobiansJ1andJ2.WhatistherelationbetweenJ1,J2andJ3?Answer:Therelationis:J3 = J2J1

42. Twotransformations,R1andR2,arerelatedby:R1R2 = IwhereIistheidentitytransformation.WhatistherelationbetweentheJacobiansofR1andR2?Whatweshouldcallthesetransformations?Answer:TheJacobianoftheidentitytransformationis1.Therefore,therelationbetweentheJacobiansofR1andR2is:J1J2 = 1whereJ1andJ2aretheJacobiansofR1andR2respectively.ThismeansthatJ1andJ2arereciprocalofeachother.Weshouldcalleachoneofthesetransformationstheinverseoftheothertransformation.

43. Discussthetransformationofonesetofbasisvectorsofagivencoordinatesystemtoanothersetofoppositevariancetypeofthatsystemandtherelationofthistothemetrictensor.Answer:Thecovariantbasisvectorsaretransformedtothecontravariantbasisvectorsbythecontravariantmetrictensorofthesystem,whilethecontravariantbasisvectorsaretransformedtothecovariantbasisvectorsbythecovariantmetrictensorofthesystem.Thiscanbeexpressedmathematicallyas:Ei = gijEj

Ei = gijEj

whereEi, Ejarecovariantbasisvectors,Ei, Ejarecontravariantbasisvectors,gijisthecovariantmetrictensorofthesystemandgijisthecontravariantmetrictensorofthesystem.

44. Discussthetransformationofonesetofbasisvectorsofagivencoordinate

systemtoanothersetofthesamevariancetypeofanothercoordinatesystem.Answer:Thetransformationofthebasissetsofthesamevariancetypebetweentwocoordinatesystems(unbarredandbarred)isgivenbythefollowingrelations:Ei = (∂ũj ⁄ ∂ui)Ẽj

Ẽi = (∂uj ⁄ ∂ũi)Ej

Ei = (∂ui ⁄ ∂ũj)Ẽj

Ẽi = (∂ũi ⁄ ∂uj)Ej

wheretheindexeduandũrepresentthecoordinatesintheunbarredandbarredsystems,whiletheindexedEandẼarethebasisvectorsoftherelevantvariancetypeintheunbarredandbarredsystems.

45. DiscussandcomparetheresultsofQuestion432↑andQuestion442↑.Also,comparethemathematicalformulationthatshouldapplyineachcase.Answer:WhileinQuestion432↑wearetransformingabasissetofthesamesystemfromonevariancetypetoanothervariancetype,inQuestion442↑wearetransformingabasissetofthesamevariancetypefromonesystemtoanothersystem.Aswesee,theformerisfacilitatedbythemetrictensorofthesystemwhilethelatterisfacilitatedbytheJacobianmatrixbetweenthetwosystems.

46. Defineproperandimpropercoordinatetransformations.Answer:Propertransformationsarethosetransformationsthatpreservethehandedness(right-orleft-handed)ofthecoordinatesystemsuchasrotation,whileimpropertransformationsarethosetransformationsthatreversethehandednessofthecoordinatesystemsuchasreflection.

47. Whatisthedifferencebetweenpositiveandnegativeorthogonaltransformations?Answer:Positivetransformationsconsistsolelyoftranslationandrotationwhilenegativetransformationsincludereflectionbyanoddnumberofaxesreversal.Accordingly,positivetransformationscanbedecomposedintoaninfinitenumberofcontinuouslyvaryinginfinitesimalpositivetransformationseachoneofwhichemulatesanidentitytransformationwhilenegativetransformationscannot.

48. Givedetaileddefinitionsofcoordinatecurvesandcoordinatesurfacesof

3Dspacesdiscussingtherelationbetweenthem.Answer:Thecoordinatecurvesarethecurvesalongwhichexactlyonecoordinatevarieswhiletheothercoordinatesareheldconstant,whilethecoordinatesurfacesarethesurfacesoverwhichallcoordinatesvaryexceptonewhichisheldconstant.Accordingly,theithcoordinatecurveisthecurvealongwhichonlytheithcoordinatevarieswhiletheithcoordinatesurfaceisthesurfaceoverwhichonlytheithcoordinateisconstant.In3Dspace,thecoordinatecurvesrepresentthecurvesofmutualintersectionofthecoordinatesurfaces.

49. Foreachoneofthefollowingcoordinatesystems,whatistheshapeofthecoordinatecurvesandcoordinatesurfaces:Cartesian,cylindricalandspherical?Answer:Cartesian:allcoordinatecurvesarestraightlinesandallcoordinatesurfacesareplanes.

Cylindrical:theρ,φandzcoordinatecurvesarestraightlines,circles,andstraightlinesrespectively,whiletheρ,φandzcoordinatesurfacesarecylinders,semi-planesandplanesrespectively.

Spherical:ther,θandφcoordinatecurvesarestraightlines,semi-circles,andcirclesrespectively,whilether,θandφcoordinatesurfacesarespheres,conesandsemi-planesrespectively.

50. Makeasimpleplotrepresentingtheφcoordinatecurvewiththeρandzcoordinatesurfacesofacylindricalcoordinatesystem.Answer:TheplotshouldlooksomewhatlikeFigure1↓.

Figure1 Theφcoordinatecurve(CC)withtheρandzcoordinatesurfaces(CS).

51. Makeasimpleplotrepresentingthercoordinatecurvewiththeθandφcoordinatesurfacesofasphericalcoordinatesystem.Answer:TheplotshouldlooksomewhatlikeFigure2↓.

Figure2 Thercoordinatecurve(CC)withtheθandφcoordinatesurfaces(CS).

52. Define“scalefactors”ofacoordinatesystemandoutlinetheirsignificance.Answer:Thescalefactorsofagivencoordinatesystemarefactorsrequiredtomultiplythecoordinatedifferentialstoobtainthedistancestraversedduringachangeinthecoordinateofthatmagnitude.Forexample,incylindricalsystemsthescalefactorρisthefactorthatmultipliesthesecondcoordinateφtoobtainthedistancedtraversedinthespacebyagivenchangeinthiscoordinateΔφ,i.e.d = ρΔφ.Similarly,insphericalsystemsthescalefactorristhefactorthatmultipliesthesecondcoordinateθtoobtainthedistancedtraversedinthespacebyagivenchangeinthiscoordinateΔθ,i.e.d = rΔθ.Thesignificanceofthescalefactorsisthatthey

transformthecoordinatesofthesystemtolengthswhicharetherealphysicaldimensionsofthespaceandhencetheyfacilitatethecalculationoflengths,areasandvolumesaswellasanyotherphysicalvariablesthatdependonlengthsanddistances.

53. Givethescalefactorsofthefollowingcoordinatesystems:orthonormalCartesian,cylindricalandspherical.Answer:Cartesian(x, y, z):hx = hy = hz = 1.

Cylindrical(ρ, φ, z):hρ = hz = 1andhφ = ρ.

Spherical(r, θ, φ):hr = 1,hθ = randhφ = rsinθ.54. Define,mathematicallyandinwords,thecovariantandcontravariantbasis

vectorsetsexplaininganysymbolsinvolvedinthesedefinitions.Answer:Thecovariantbasisvectorsarethetangentvectorstothecoordinatecurves,whilethecontravariantbasisvectorsarethegradientofthespacecoordinatesandhencetheyareperpendiculartothecoordinatesurfaces.Mathematically,thecovariantandcontravariantbasisvectorsaredefinedrespectivelyby:Ei = ∂r ⁄ ∂ui

Ei = ∇ui

whereristhepositionvectorinCartesiancoordinates(x1, …, xn),uirepresentsgeneralcoordinates,andi = 1, ⋯, nwithnbeingthespacedimension.

55. Whatistherelationofthecovariantandcontravariantbasisvectorsetswiththecoordinatecurvesandcoordinatesurfacesofagivencoordinatesystem?Makeasimplesketchrepresentingthisrelationforageneralcurvilinearcoordinatesystemina3Dspace.Answer:Asstatedearlier,thecovariantbasisvectorsaretangentstothecoordinatecurves,whilethecontravariantbasisvectorsareperpendiculartothecoordinatesurfaces.ThesketchshouldlooksomethinglikeFigure3↓.

Figure3 Thecovariantandcontravariantbasisvectorsofageneralcurvilinearcoordinatesystemandtheassociatedcoordinatecurves(CC)andcoordinatesurfaces(CS)atagivenpointPina3Dspace.

56. Thecovariantandcontravariantcomponentsofvectorscanbetransformedonetotheother.How?Stateyouranswerinamathematicalform.Answer:ThecovariantcomponentsofavectorAareobtainedfromthecontravariantcomponentsofAbyusingthecovariantmetrictensorgij(orloweringoperator),thatis:Ai = gijAj

Similarly,ThecontravariantcomponentsofavectorBareobtainedfromthecovariantcomponentsofBbyusingthecontravariantmetrictensorgij(orraisingoperator),thatis:Bi = gijBj

57. Whatisthesignificanceofthefollowingrelations?Ei⋅Ej = δij

Ei⋅Ej = δijAnswer:Thesignificanceoftheserelationsisthatthecovariantandcontravariantbasissetsarereciprocalbasissystems.

58. Writedownthemathematicalrelationsthatcorrelatethebasisvectorstothecomponentsofthemetrictensorintheircovariantandcontravariantforms.Answer:Ei⋅Ej = gij

Ei⋅Ej = gij

whereEi, Ejarecovariantbasisvectors,Ei, Ejarecontravariantbasisvectors,gijisthecovariantmetrictensorandgijisthecontravariantmetrictensor.

59. UsingtheequationEi = [Ej × Ek] ⁄ [Ei⋅(Ej × Ek)]showthatifEi, Ej, EkformarighthandedorthonormalsystemthenEi = Ei.RepeatthequestionusingthistimetheequationEi = [Ej × Ek] ⁄ [Ei⋅(Ej × Ek)]whereEi, Ej, Ekformarighthandedorthonormalsystem.Hence,concludethatwhenthecovariantorcontravariantbasisvectorsetisorthonormalthenthecovariantandcontravariantcomponentsofagiventensorareidentical.Answer:IfEi, Ej, EkformarighthandedorthonormalsystemthenEi⋅(Ej × Ek)(whichisthevolumeoftheparallelepipedformedbythevectorsEi, Ej, Ek)equals1andhencewehave:Ei = Ej × Ek

However,sinceEi, Ej, Ekformarighthandedorthonormalsystemthenweshouldalsohave:Ei = Ej × EkOncomparingthelasttwoequationsweconcludethatEi = Ei.Thesecondpartofthequestioncanbeansweredsimilarlybyjustexchangingthevariancetypeoftheinvolvedvectorsinthefirstpart.Now,sincethecovariantandcontravariantbasisvectorsetsareidentical

whenthecovariantorcontravariantbasisvectorsetisorthonormalthenbytheprincipleofinvarianceoftensorsthecovariantandcontravariantcomponentsofagiventensorshouldalsobeidenticalbecauseotherwisethetensorwillvarydependingontheemployedbasisset.

60. Statethemathematicalrelationsbetweentheoriginalandtransformed(i.e.unbarredandbarred)basisvectorsetsintheircovariantandcontravariantformsunderadmissiblecoordinatetransformations.Answer:Theserelationsaregivenby:Ei = (∂ũj ⁄ ∂ui)Ẽj

Ẽi = (∂uj ⁄ ∂ũi)Ej

Ei = (∂ui ⁄ ∂ũj)Ẽj

Ẽi = (∂ũi ⁄ ∂uj)Ej

wheretheindexeduandũrepresentthecoordinatesintheunbarredandbarredsystems,whiletheindexedEandẼarethebasisvectorsintheunbarredandbarredsystems.

61. Correct,ifnecessary,thefollowingequationsexplainingallthesymbolsinvolved:E1⋅( E2 ×  E3) = 1 ⁄ √(g)

E1⋅( E2 ×  E3) = √(g)

Answer:Thecorrectrelationsare:E1⋅( E2 ×  E3) = √(g)

E1⋅( E2 ×  E3) = 1 ⁄ √(g)

whereE1,  E2,  E3arethecovariantbasisvectors,E1,  E2,  E3arethecontravariantbasisvectorsandgisthedeterminantofthecovariantmetrictensorwhilethedotandcrossrepresentthedotproductandcrossproductoperationsofvectors.

62. Obtaintherelation:g = J2fromtherelation:JTJ = [gij]givingfullexplanationofeachstep.Answer:Westartfromthegivenrelation(whichisadefinition):

[gij] = JTJwhere[gij]isthematrixrepresentingthecovariantmetrictensorandJandJTaretheJacobianmatrixanditstransposewhiletheproductontherightisamatrixproduct.Bytakingthedeterminantofbothsidesofthisequationweobtain:g = JTJwhereg, JT, Jarethedeterminantsofthecorrespondingentities.Thelastrelationisjustifiedbythewellknownruleoflinearalgebrathatthedeterminantofaproductofmatrices(i.e.JTJ)isequaltotheproductofthedeterminantsofthesematrices(i.e.JTJ).Now,accordingtoanotherruleoflinearalgebrawhichstatesthatthedeterminantofamatrixisequaltothedeterminantofitstransposewehaveJT = Jandhencethelastequation(i.e.g = JTJ)becomes:g = J2asrequired.

63. Statethreeconsequencesofhavingmutuallyorthogonalcontravariantbasisvectorsateachpointinthespacejustifyingtheseconsequences.Answer:Thefollowingareexamplesoftheseconsequences(otherconsequencesgiveninthetextcanalsobequoted):(a)Thecovariantbasisvectorsshouldalsobemutuallyorthogonal,i.e.Ei⋅Ej = 0wheni ≠ j.Thereasonisthatthecorrespondingvectorsofeachbasissetareinthesamedirectionduetothefactthatthetangentvectortotheithcoordinatecurveandthegradientvectoroftheithcoordinatesurfaceatagivenpointinthespacehavethesamedirectionandhenceifthevectorsofoneset(i.e.covariantorcontravariant)aremutuallyorthogonalthenthevectorsoftheothersetshouldalsobemutuallyorthogonal.(b)Thecovariantandcontravariantmetrictensorsarediagonalwithnon-vanishingdiagonalelements,thatis:gij = 0gij = 0 (i ≠ j)

gii ≠ 0gii ≠ 0 (nosumoni)

Thereasonisthatfromtherelationsgij = Ei⋅Ejandgij = Ei⋅Ejwecanseethatthedotproduct(andhencetheelementofthemetrictensor)iszerowhentheindicesaredifferentduetothemutualorthogonalityofthebasisvectors.Moreover,thedotproduct(andhencetheelementofthemetric

tensor)shouldbenon-zerowhentheindicesareidenticalbecausethebasisvectorscannotvanishattheregularpointsofthespacesincethetangenttothecoordinatecurveandthegradienttothecoordinatesurfacedoexistandtheycannotbezero.(c)Thediagonalelementsofthecovariantmetrictensorandthecorrespondingelementsofthecontravariantmetrictensorarereciprocalsofeachother,thatis:gii = 1 ⁄ gii(nosummationoni)Thereasonisthatsincethecovariantandcontravariantmetrictensorsareinversesofeachotherandtheyarediagonalwithnon-vanishingdiagonalelements(asestablishedinthepreviouspoint)thentheircorrespondingdiagonalelementsshouldbereciprocalofeachother(asprovedinlinearalgebra).

64. Discusstherelationshipbetweentheconceptsofspace,coordinatesystemandmetrictensor.Answer:Therequiredanswertothisquestionshouldbenomorethanashortsummaryofsection2.7(RelationshipbetweenSpace,CoordinatesandMetric)ofthebook.Theessenceofthissummaryistheneedofanyabstractspaceforacoordinatesystemtoidentifyitspointsanddescribeitspropertiesandthiswillleadtotheemergenceofthemetrictensorasamathematicalentitythatidentifiesandcharacterizesthegeometricpropertiesofthespacelocallyandglobally.

Chapter3Tensors1. Define“covariant”and“contravariant”tensorsfromtheperspectiveoftheir

notationandtheirtransformationrules.Answer:Covarianttensorsarenotatedwithsubscriptindiceswhilecontravarianttensorsarenotatedwithsuperscriptindices.Acovarianttensorofrank-mistransformedaccordingtothefollowingrule:Ãij⋯m = (∂up ⁄ ∂ũi)(∂uq ⁄ ∂ũj)⋯(∂ur ⁄ ∂ũm)Apq⋯r

whileacontravarianttensorofrank-mistransformedaccordingtothefollowingrule:B̃ij⋯m = (∂ũi ⁄ ∂up)(∂ũj ⁄ ∂uq)⋯(∂ũm ⁄ ∂ur)Bpq⋯r

2. Writethetransformationrelationsforcovariantandcontravariantvectorsandforcovariant,contravariantandmixedrank-2tensorsbetweendifferentcoordinatesystems.Answer:Thetransformationrelationsforcovariantandcontravariantvectorsare:Ãi = (∂uj ⁄ ∂ũi)Aj

B̃i = (∂ũi ⁄ ∂uj)Bj

Thetransformationrelationsforcovariant,contravariantandmixedrank-2tensorsare:Ãij = (∂up ⁄ ∂ũi)(∂uq ⁄ ∂ũj)Apq

B̃ij = (∂ũi ⁄ ∂up)(∂ũj ⁄ ∂uq)Bpq

C̃ij = (∂up ⁄ ∂ũi)(∂ũj ⁄ ∂uq)Cpq

3. Statethepracticalrulesforwritingthetransformationrelationsoftensorsbetweendifferentcoordinatesystems.Answer:Thepracticalrulescanbesummarizedasfollowswherewetransformfromunbarredsystemtobarredsystem(usingAijkasanexampleofatensorthatwetransform):

●Writethesymbolofthetransformedtensoronthelefthandsideofthetransformationequationandthesymboloftheoriginaltensorontherighthandside:Ã = A●Indextheoriginaltensorwithitsoriginalindicesandindexthetransformedtensorwithdifferentindicesnotingthatitsindicialstructureshouldbesimilartotheindicialstructureoftheoriginaltensor:Ãlmn = Aijk

●Insertanumberofpartialdifferentialoperatorsontherighthandsideequaltothenumberoffreeindices:Ãlmn = (∂u ⁄ ∂u)(∂u ⁄ ∂u)(∂u ⁄ ∂u)Aijk

●Indexthecoordinatesofthetransformedtensorinthenumeratorordenominatorintheseoperatorsaccordingtotheorderoftheindicesinthetensorwheretheseindicesareinthesameposition(upperorlower)astheirpositioninthetensor:Ãlmn = (∂ul ⁄ ∂u)(∂u ⁄ ∂um)(∂un ⁄ ∂u)Aijk

●Becausethetransformedtensorisbarredthenitscoordinatesshouldalsobebarred:Ãlmn = (∂ũl ⁄ ∂u)(∂u ⁄ ∂ũm)(∂ũn ⁄ ∂u)Aijk

●Indexthecoordinatesoftheoriginaltensorinthenumeratorordenominatorintheseoperatorsaccordingtotheorderoftheindicesinthetensorwheretheseindicesareintheoppositeposition(upperorlower)totheirpositioninthetensor:Ãlmn = (∂ũl ⁄ ∂ui)(∂uj ⁄ ∂ũm)(∂ũn ⁄ ∂uk)Aijk

4. Whataretheraisingandloweringoperatorsandhowtheyprovidethelinkbetweenthecovariantandcontravarianttypes?Answer:Theraisingoperatoristhecontravariantmetrictensorwhiletheloweringoperatoristhecovariantmetrictensor.Theraisingoperatorcanchangecovariantindicestocontravariantindiceswhiletheloweringoperatorcanchangecontravariantindicestocovariantindices.Sincetheseoperatorscanchangeatensorfromonevariancetypetoanother,theyprovidealinkbetweenthedifferentvariancetypesofthetensorandhencetheyfacilitatethetransformationbetweendifferentbasissetsofagivencoordinatesystem.

5. Aisatensoroftype(m, n)andBisatensoroftype(p, q, w).Whatthismeans?Writethesetensorsintheirindicialform.Answer:ItmeansthatAisatensorwithmcontravariantindicesandn

covariantindices,andBisatensorwithpcontravariantindices,qcovariantindicesandweightw.Inindicialform,thesetensorsshouldbewrittenasAi1⋯im

j1⋯jnandBi1⋯ipj1⋯jq.

6. Writethefollowingequationsinfulltensornotationandexplaintheirsignificance:Ei = ∂r ⁄ ∂ui

Ei = ∇uiAnswer:Ei = (∂xj ⁄ ∂ui)ej

Ei = (∂ui ⁄ ∂xj)ej

whereEiandEiarecovariantandcontravariantgeneralbasisvectors,xjareCartesiancoordinates,uiaregeneralcoordinatesandejareCartesianbasisvectors.Thesignificanceoftheseequationsisthatthecovariantbasisvectorsaretangentstothecoordinatecurveswhilethecontravariantbasisvectorsaregradientstothecoordinatesurfaces.

7. Writetheorthonormalizedformofthecovariantbasisvectorsina2Dgeneralcoordinatesystem.Verifythatthesevectorsareactuallyorthonormal.Answer:Theyare:ℰ1 =  E1 ⁄ | E1| =  E1 ⁄ √(g11)

ℰ2 = (g11 E2 − g12 E1) ⁄ √(g11g)

whereℰ1andℰ2areorthonormalizedcovariantbasisvectors,E1andE2aregeneralcovariantbasisvectors,theindexedgarecoefficientsofthecovariantmetrictensorandgisitsdeterminant.Verification:ℰ1⋅ ℰ1 = 

(E1 ⁄ | E1|)⋅( E1 ⁄ | E1|) = 

(E1⋅ E1) ⁄ | E1|2 = 

|E1|2 ⁄ | E1|2 = 

1

ℰ2⋅ ℰ2 = 

[(g11 E2 − g12 E1) ⁄ √(g11g)]⋅[(g11 E2 − g12 E1) ⁄ √(g11g)] = 

[(g11 E2 − g12 E1)⋅(g11 E2 − g12 E1)] ⁄ [g11g] = 

(g11g11 E2⋅ E2 − g11g12 E2⋅ E1 − g12g11 E1⋅ E2 + g12g12 E1⋅ E1) ⁄ (g11g) = 

(g11g11g22 − g11g12g21 − g12g11g12 + g12g12g11) ⁄ (g11g) = 

(g11g11g22 − g11g12g21) ⁄ (g11g) = 

[g11(g11g22 − g12g21)] ⁄ [g11g] = 

(g11g) ⁄ (g11g) = 

1

ℰ1⋅ ℰ2 = 

[E1 ⁄ √(g11)]⋅[(g11 E2 − g12 E1) ⁄ √(g11g)] = 

[E1⋅(g11 E2 − g12 E1)] ⁄ [g11√(g)] = 

[g11 E1⋅ E2 − g12 E1⋅ E1] ⁄ [g11√(g)] = 

[g11g12 − g12g11] ⁄ [g11√(g)] = 

0

Hence,thevectorsℰ1andℰ2areorthonormal(i.e.theyareorthogonaltoeachotheraccordingtothethirddotproductandofunitlengthaccordingtothefirstandseconddotproducts).

8. Whythefollowingrelationsarelabeledasthereciprocityrelations?Ei⋅Ej = δij

Ei⋅Ej = δij

Answer:Becausetheserelationsexpressthefactthatthecovariantandcontravariantbasisvectorsarereciprocalsystems.Itisnoteworthythattwosetsofvectorsofthesamenumberofelements(e.g.V1,  V2, ⋯ VnandW1,  W2, ⋯ Wn)(seeFootnote7in§8↓)aredescribedasreciprocaltoeachotheriftheysatisfythefollowingrelations:Vi⋅Wj = 1(i = j)

Vi⋅Wj = 0(i ≠ j)9. ThecomponentsofthetensorsA,BandCaregivenby:Aikj,BjnmqandCk

li.Writethesetensorsintheirfullnotationthatincludestheirbasistensors.Answer:A = AikjEiEkEj

B = BjnmqEjEnEmEq

C = CkliEkElEi

10. A,BandCaretensorsofrank-2,rank-3andrank-4respectivelyinagivencoordinatesystem.Writethecomponentsofthesetensorswithrespecttothefollowingbasistensors:,EiEn,EiEkEmandEjEiEkEn.Answer:A = AinEiEn

B = BikmEiEkEm

C = CjiknEjEiEkEn

11. What“dyad”means?Writeallthenineunitdyadsassociatedwiththe

doubledirectionsofrank-2tensorsina3DspacewitharectangularCartesiancoordinatesystem(i.e.e1 e1⋯ e3 e3).Answer:Dyadisarank-2tensorobtainedbythedirectmultiplicationoftwovectors.Thenineunitdyadsare:e1 e1e1 e2e1 e3e2 e1e2 e2e2 e3e3 e1e3 e2e3 e3

12. MakeasimplesketchoftheninedyadsofExercise113↑.Answer:ThesketchshouldlooklikeFigure4↓.

Figure4 Thenineunitdyadsassociatedwiththedoubledirectionsofrank-2tensorsina3DspacewitharectangularCartesiancoordinatesystem.

13. Comparetrueandpseudovectorsmakingacleardistinctionbetweenthetwowithasimpleillustratingplot.Generalizethistotensorsofanyrank.Answer:Truevectorstransforminvariantlyundercoordinatetransformationsandhencetheykeeptheirdirection,whilepseudovectorsdonottransforminvariantlyunderimproperorthogonaltransformationswhichinvolveinversionofcoordinateaxesthroughtheoriginofcoordinateswithachangeofsystemhandednesssincetheyacquireaminussignundersuchtransformationsandhencetheyreversetheirdirection.The

plotshouldlooklikeFigure5↓whereweseeatruevectorvthatkeepsitsdirectioninthespacefollowingareflectionofthecoordinatesystemthroughtheoriginofcoordinatesandapseudovectorpthatreversesitsdirectionfollowingthisoperation.Togeneralizethesepropertiestotensorsofanyrank,wesimplyreplacethereversalofthesingledirectionintheabovedefinitionoftrueandpseudovectorswiththechangeofmultidirectionsthatassociatetensorsofhigherranks.

Figure5 Thebehaviorofatruevector(vandV)andapseudovector(pandP)followingareflectionofthecoordinatesystemintheoriginofcoordinates.

14. Justifythefollowingstatement:“Thetermsofconsistenttensorexpressionsandequationsshouldbeuniformintheirtrueandpseudotype”.Answer:Letassumethatwehavetensorexpressions/equationswithmixedterms(i.e.sometrueandsomepseudo)andwetransformtheseexpressions/equationsbyanimproperorthogonaltransformation,thenthetruetermswilltransforminvariantlywhilethepseudotermswillnot,andthisdoesnotmakesensebecausetheseexpressions/equationswillbehaveneitherastruetensorsnoraspseudotensorsandhencetheyhaveindeterminatestate.

15. Whatisthecurlofapseudovectorfromtheperspectiveoftrue/pseudoqualification?Answer:Itshouldbeatruevector.

16. Defineabsoluteandrelativetensorsstatinganynecessarymathematicalrelations.

Answer:AbsolutetensorsarethosetensorsthathavenoJacobianfactorintheirtransformationequation,whilerelativetensorshavesuchafactor.Inmathematicalterms,thetransformationequationofthesetwotypesoftensoris:Ãij…k

lm…n = |∂x ⁄ ∂x̃|w(∂x̃i ⁄ ∂xa)(∂x̃j ⁄ ∂xb)⋯(∂x̃k ⁄ ∂xc)(∂xd ⁄ ∂x̃l)(∂xe ⁄ ∂x̃m)⋯(∂xf ⁄ ∂x̃n)Aab…c

de…f

wherew = 0forabsolutetensorandw ≠ 0forrelativetensor.17. WhatistheweightoftheproductofAandBwhereAisatensoroftype

(1, 2, 2)andBisatensoroftype(0, 3,  − 1)?Answer:TheweightofAis2andtheweightofBis − 1andhencetheweightoftheirproductisthesumoftheirweights,i.e.w = 2 − 1 = 1.

18. Showthatthedeterminantofarank-2absolutetensorAisarelativescalarandfindtheweightinthecaseofAbeingcovariantandinthecaseofAbeingcontravariant.Answer:Thetransformationequationofarank-2absolutecovarianttensorisgivenby:Ãij = (∂up ⁄ ∂ũi)(∂uq ⁄ ∂ũj)ApqOntakingthedeterminantofbothsidesweobtain(seeFootnote8in§8↓):|Ãij| = 

|∂up ⁄ ∂ũi||∂uq ⁄ ∂ũj||Apq| = 

J2|Apq|

andhencethedeterminantofarank-2absolutecovarianttensorAisarelativescalarofweight2(seeFootnote9in§8↓).

Similarly,thetransformationequationofarank-2absolutecontravarianttensorisgivenby:Ãij = (∂ũi ⁄ ∂up)(∂ũj ⁄ ∂uq)Apq

Ontakingthedeterminantofbothsidesweobtain:|Ãij| = 

|∂ũi ⁄ ∂up||∂ũj ⁄ ∂uq||Apq| = 

(J − 1)2|Apq| = 

J − 2|Apq|

andhencethedeterminantofarank-2absolutecontravarianttensorAisarelativescalarofweight − 2.

19. Whythetensortermsoftensorexpressionsandequalitiesshouldhavethesameweight?Answer:Theweightisacharacteristicpropertyofthetensoranditstransformationrulesandhenceforconsistencyandhomogeneitythetermsoftensorexpressionsandequalitiesshouldhavethesameweight.Wecanrepeatourpreviousargumentaboutthenecessityofconsistencyoftermswithregardtotheirtrueandpseudonatureandhenceifwetransformatensorexpressionorequalitywhosetermshavedifferentweightsthenwewillnothaveaconsistenttransformationrulebecausedifferenttermsrequiredifferentweights.

20. What“isotropic”and“anisotropic”tensormean?Answer:Isotropictensorsarethosetensorswhosecomponentsdonotchangeunderproperrotationaltransformations,whileanisotropictensorsarethosetensorswhosecomponentsdochangeundersuchtransformations.

21. Giveanexampleofanisotropicrank-2tensorandanotherexampleofananisotropicrank-3tensor.Answer:TheKroneckerdeltaisanexampleofanisotropicrank-2tensor,whilethepiezoelectricmodulitensorisanexampleofananisotropicrank-3tensor.

22. Whatisthesignificanceofthefactthatthezerotensorofallranksandalldimensionsisisotropicwithregardtotheinvarianceoftensorsundercoordinatetransformations?Answer:Thesignificanceisthatifthecomponentsofatensorvanishinaparticularcoordinatesystemthentheywillvanishinallproperlyandimproperlyrotatedcoordinatesystems(seeFootnote10in§8↓).Asaresult,ifthecomponentsoftwotensorsareequalinaparticularcoordinatesystemthentheyshouldbeequalinallothersystemssincethetensoroftheirdifferenceisazerotensorandhenceitisinvariant.Thisleadstotheconclusionthatidentitiesandequalitiesoftensorsareinvariantundercoordinatetransformations.

23. Define“symmetric”and“anti-symmetric”tensor.Whyscalarsandvectorsarenotqualifiedtobesymmetricoranti-symmetric?

Answer:Symmetrictensorsarethosetensorswhosecomponentvaluesdonotchangeundercertainexchangeofindices,whileanti-symmetrictensorsarethosetensorswhosecomponentvaluesreversetheirsign(i.e.identicalmagnitudewithoppositesign)undercertainexchangeofindices.Accordingly,ifAijisasymmetrictensorandBijisananti-symmetrictensor,thentheircomponentswillsatisfythefollowingrelations:Aji =  + Aij

Bji =  − Bij

Becausesymmetryandanti-symmetryrequireatleasttwoindicestohaveanexchange,thenscalarswithnoindexandvectorswithjustoneindexcannotbesymmetricoranti-symmetricsincenoexchangeofindicescanbeimagined.

24. Writethesymmetricandanti-symmetricpartsofthetensorAij.Answer:Symmetricpart:A(ij) = (Aij + Aji) ⁄ 2

Anti-symmetricpart:A[ij] = (Aij − Aji) ⁄ 2

25. Writethesymmetrizationandanti-symmetrizationformulaeforarank-ntensorAi1…in.Answer:Symmetrizationformula:A(i1…in) = [1 ⁄ (n!)](Σevenpermutationsofi's + Σoddpermutationsofi's)

Anti-symmetrizationformula:A[i1…in] = [1 ⁄ (n!)](Σevenpermutationsofi's − Σoddpermutationsofi's)

26. Symmetrizeandanti-symmetrizethetensorAijklwithrespecttoitssecondandfourthindices.Answer:Ai(j)k(l) = (Aijkl + Ailkj) ⁄ 2

Ai[j]k[l] = (Aijkl − Ailkj) ⁄ 227. Writethetwomathematicalconditionsforarank-ntensorAi1…intobe

totallysymmetricandtotallyanti-symmetric.

Answer:Theconditionsare:Totallysymmetric:Ai1…in = A(i1…in)

Totallyanti-symmetric:Ai1…in = A[i1…in]

Similarly:Totallysymmetric:A[i1…in] = 0

Totallyanti-symmetric:A(i1…in) = 0

28. ThetensorAijkistotallysymmetric.Howmanydistinctcomponentsithasina3Dspace?Answer:Ina3Dspace,Aijkhas33 = 27componentswhichcorrespondtoallthepossiblepermutations(includingtherepetitiveones)ofthenumbers1,2,3.However,duetothetotalsymmetrytheorderofthenumbersisirrelevantandhencewhatweneedistoextractallthepossiblecombinations(includingtherepetitiveones)ofthenumbers1,2,3,thatis:

Combinationswithonlyonedistinctnumber:111,222,333.

Combinationswithonlytwodistinctnumbers:112,113,122,223,133,233.

Combinationswiththreedistinctnumbers:123.

Accordingly,Aijkshouldhave10distinctcomponents(seeFootnote11in§8↓).

29. ThetensorBijkistotallyanti-symmetric.Howmanyidenticallyvanishingcomponentsithasina3Dspace?Howmanydistinctnon-identicallyvanishingcomponentsithasina3Dspace?Answer:Ina3Dspace,Bijkhas33 = 27componentswhichcorrespondtoallthepossiblepermutations(includingtherepetitiveones)ofthenumbers1,2,3.Now,sinceitistotallyanti-symmetricthenanypermutationwithtwoidenticalnumbersshouldbezero.So,allpermutationsexceptthe

permutationsofthecombination123shouldbezero.Becausethecombination123hassixpermutationsthenweshouldhave27 − 6 = 21identicallyvanishingcomponentsand6non-identicallyvanishingcomponents.These6non-identicallyvanishingcomponentscorrespondto3evenpermutationsof123and3oddpermutationsof123.Thecomponentsofthe3evenpermutationsof123shouldbeidenticalandthecomponentsofthe3oddpermutationsof123shouldbeidentical.Hence,Bijkshouldhavetwodistinctnon-identicallyvanishingcomponentswherethesetwodifferonlyinsign.However,since“distinct”insuchcontextsmeans“independent”thenweshouldhaveonlyonedistinctnon-identicallyvanishingcomponentbecausethecomponentscorrespondingtotheevenandoddpermutationsdifferonlyinsign.Alltheseresultscanbeobtainedfrominspectingtherank-3permutationtensor.

30. Givenumericorsymbolicexamplesofarank-2symmetrictensorandarank-2skew-symmetrictensorina4Dspace.Countthenumberofindependentnon-identicallyvanishingcomponentsineachcase.Answer:Symmetric:

Thenumberofindependentnon-identicallyvanishingcomponentsis[n(n + 1)] ⁄ 2 = 10.Skew-symmetric:

Thenumberofindependentnon-identicallyvanishingcomponentsis[n(n − 1)] ⁄ 2 = 6.

31. Writetheformulaforthenumberofindependentcomponentsofarank-2symmetrictensor,andtheformulaforthenumberofindependentnon-zerocomponentsofarank-2anti-symmetrictensorinnDspace.Answer:Theformulaeare:Ns = [n(n + 1)] ⁄ 2

Na = [n(n − 1)] ⁄ 2

whereNsisthenumberofindependentcomponentsofarank-2symmetrictensorandNaisthenumberofindependentnon-zero(i.e.non-identicallyvanishing)componentsofarank-2anti-symmetrictensor.

32. Explainwhytheentriescorrespondingtoidenticalanti-symmetricindicesshouldvanishidentically.Answer:Becauseanexchangeoftwoidenticalindices,whichidentifiesthesameentry,shouldchangethesignoftheentryduetoanti-symmetryandhencetheentryshouldbeequaltoitsnegativeandthiscanbetrueonlyiftheentryisidenticallyzero.

33. Whytheindiceswhoseexchangedefinesthesymmetryandanti-symmetryrelationsshouldbeofthesamevariancetype?Answer:Becausethecovariantandcontravariantindicescorrespondtodifferentbasissets(i.e.contravariantandcovariantrespectively)andhenceevenifthevaluesofthecomponentssatisfythepropertyofsymmetryoranti-symmetryitdoesnotleadtothesymmetryoranti-symmetryofthetensorduetotheinvolvementofdifferentbasissetsbetweenwhichnosymmetryoranti-symmetrycanbedefinedsensibly.Forexample,lethave

arank-2mixedtensorAwhosecomponentvaluessatisfytherelationAij = Aji.However,sincethetensorcomponentsAijmeanAijEiEjwhileAjimeanAjiEjEithennosymmetryoranti-symmetrycanbedefinedforthetensorbecausenosymmetryoranti-symmetrycanbedefinedsensiblyforEiEjandEjEiduetothefacttheyinvolvevectorsofoppositevariancetype.Thisisunlikethesituationwitharank-2covarianttensorBforexamplebecauseeventhoughthecomponentBijmeansBijEiEjwhilethecomponentBjimeansBjiEjEisymmetryoranti-symmetrycanbedefinedforthetensorsincesymmetryoranti-symmetrycanbedefinedfortheuniquebasisset,i.e.EiEjandEjEicanbeseenasasymmetricbasistensorsincebothbasisvectorsbelongtothesamebasisset.Toputitinmoresimpleterms,whenthetwoindicesareofthesamevariancetypethenexchangingtheindicesisachievedbyjustexchangingtheirvaluesandhencewehaveBijEiEjandBjiEjEiaresymmetricoranti-symmetricinasensibleway,butwhenthetwoindicesareofdifferentvariancetypethenexchangingtheindicesisnotsufficienttoachievesymmetryoranti-symmetrysincetheorderofthetwoindicescannotbechanged(duetotheirdifferentvariancetype)becausewhenweexchangetheindicesofAijEiEjwegetAjiEjEi(ratherthanAijEiEjorAjiEjEi)andhencesensiblesymmetryoranti-symmetrycannotbeachieved.

34. Discussthesignificanceofthefactthatthesymmetryandanti-symmetrycharacteristicofatensorisinvariantundercoordinatetransformationsandlinkthistotheinvarianceofthezerotensor.Answer:Thesignificanceisthatbyknowingthatatensorissymmetricoranti-symmetricorneitherinonesystemweknowthatitissymmetricoranti-symmetricorneitherinallothersystemswithnoextraefforttoestablishitsstatusfromthisperspective.Thelinkbetweenthisinvariancepropertyandtheinvarianceofthezerotensoristhatthedifferencebetweenatensoranditssymmetricassociateisthezerotensorwhilethesumofatensoranditsanti-symmetricassociateisthezerotensorandbecausethezerotensorisinvariantacrossallcoordinatesystemsthenthedifferenceorsumshouldalsobeinvariantacrossallcoordinatesystemsandthisleadstotheconclusionthatthesymmetricoranti-symmetricpropertyofatensorshouldbeinvariantacrossallcoordinatesystems(seeFootnote12in§8↓).

35. VerifytherelationAijBij = 0,whereAijisasymmetrictensorandBijisananti-symmetrictensor,bywritingthesuminfullassuminga3Dspace.

Answer:Wewritethesuminamatrix-likeform,thatis:AijBij = A11B11 + A12B12 + A13B13 + A21B21 + A22B22 + A23B23 + A31B31 + A32B32 + A33B33

Now,becauseAijissymmetricandBijisanti-symmetricthenthecorrespondingtermsacrossthemaindiagonal(e.g.A13B13versusA31B31

withA13 = A31andB13 =  − B31)shouldbeequalinmagnitudeandoppositeinsign,thatis:AijBij = A11B11 − A21B21 − A31B31 + A21B21 + A22B22 − A32B32 + A31B31 + A32B32 + A33B33

thatis:AijBij = A11B11 + A22B22 + A33B33

Finally,becauseBijisanti-symmetricthenB11 = B22 = B33 = 0andhencewehave:AijBij = 0 + 0 + 0 = 0asrequired.

36. Classifythecommontensoroperationswithrespecttothenumberoftensorsinvolvedintheseoperations.Answer:Additionandsubtraction,multiplicationoftensorbyscalar,tensormultiplicationandinnerproductinvolvetwotensors(consideringscalarasrank-0tensor).Permutationinvolvesonetensor.Contractioncaninvolveonetensorortwotensors.

37. Whichofthefollowingoperationsarecommutative,associativeordistributivewhenthesepropertiesapply:algebraicaddition,algebraicsubtraction,multiplicationbyascalar,outermultiplication,andinnermultiplication?Answer:Algebraicadditioniscommutativeandassociative.

Algebraicsubtractionisneithercommutativenorassociative.

Multiplicationbyascalariscommutative,associativeanddistributiveoveralgebraicadditionandalgebraicsubtraction.

Outermultiplicationisnotcommutativebutitisdistributiveoveralgebraicadditionandalgebraicsubtraction.

Innermultiplicationisgenerallylikeoutermultiplicationinthisrespect(withsomeexceptionssuchasthecommutativityofmultiplicationoftwovectors).

38. ForQuestion373↑,writealltherequiredmathematicalrelationsthatdescribethoseproperties.Answer:Algebraicaddition:A + B = B + A

(A + B) + C = A + (B + C)

Algebraicsubtraction:A − B ≠ B − A

(A − B) − C ≠ A − (B − C)

Multiplicationbyascalar:aA = Aa

a(bA) = (ab)A

a(A±B) = aA±aB

Outermultiplication:AB ≠ BA

A(B±C) = AB±AC

Innermultiplication:A⋅B ≠ B⋅A

A⋅(B±C) = A⋅B±A⋅C39. Thetensorsinvolvedintensoraddition,subtractionorequalityshouldbe

compatibleintheirtypes.Giveallthedetailsaboutthese“types”.Answer:Theinvolvedtensorsshouldhavethesamerank,thesamespacedimension,thesameindicialstructure(e.g.samesetoffreeindices,samevariancetypeofeachindexandsameorderofindices),andthesamebasisvectorset(i.e.theset,whethercovariantorcontravariant,belongstothesamecoordinatesystem).Theyshouldalsobeofthesametrue/pseudotypeandhavethesameweightwwhetherw = 0(absolute)orw ≠ 0(relative)(seeFootnote13in§8↓).

40. Whatisthemeaningofmultiplyingatensorbyascalarintermsofthecomponentsofthetensor?Answer:Multiplyingatensorbyascalarmeansmultiplyingeachcomponentofthetensorbythatscalarandhenceitisasimpleuniformscalingofthetensor(assumingthescalarisnotzero).

41. Atensoroftype(m1, n1, w1)ismultipliedbyanothertensoroftype(m2, n2, w2).Whatisthetype,therankandtheweightoftheproduct?Answer:Thetypeis(m1 + m2, n1 + n2, w1 + w2),therankism1 + m2 + n1 + n2andtheweightisw1 + w2.

42. Wehavetwotensors:A = AijEiEjandB = BklEkEl.WealsohaveC = ABandD = BA.UsethepropertiesoftensoroperationstoobtainthefullexpressionofCandDintermsoftheircomponentsandbasistensors(i.e.C = AB = ⋯etc.).Answer:C = 

AB = 

AijEiEjBklEkEl = 

AijBklEiEjEkEl

D = 

BA = 

BklEkElAijEiEj = 

BklAijEkElEiEj

43. Explainwhytensormultiplication,unlikeordinarymultiplicationofscalars,isnotcommutativeconsideringthebasistensorstowhichthetensorsarereferred.Answer:Referringtothepreviousexercise,weseethattheorderofthebasisvectorsdependsontheorderofthemultipliedtensorsandsincetheorderofthebasisvectorsissignificantindeterminingthebasistensor(e.g.EjEk ≠ EkEj)andcannotbechangedorreversedarbitrarily,thentensormultiplicationisnotcommutative.Thisisunlikeordinarymultiplicationofscalarssincescalarsarenotassociatedwithbasisvectors.

44. Thedirectproductofvectorsaandbisab.Editthefollowingequationbyaddingasimplenotationtomakeitcorrectwithoutchangingtheorder:ab = ba.Answer:Wesimplyconvertthistoanequationofinnerproducebyaddingadotbetweenthevectors,thatis:a⋅b = b⋅aThejustificationisthatwhilethedirectproductofvectorsisnotcommutative,theinnerproductofvectorsiscommutative.

45. Whatisthedifferenceinnotationbetweenmatrixmultiplicationandtensormultiplicationoftwotensors,AandB,whenwewriteAB?Answer:InmatrixnotationthematrixmultiplicationABrepresentsaninnerproductoperation,whileintensornotationthetensormultiplicationABrepresentsanouterproductoperation.

46. Definethecontractionoperationoftensors.Whythisoperationcannotbeconductedonscalarsandvectors?Answer:Intensorcalculus,contractionoftensorsmeansmakingtwofreeindicesofagiventensororoftwotensorsinvolvedinouterproductidenticalbyunifyingtheirsymbols.Thiswillthenbefollowedbyperformingsummationovertheserepeatedindices.Forexample,whenwecontracttherank-2tensorAjiinnDspaceweobtain:Aii = A11 + A22 + ⋯ + Ann

Similarly,whenwecontractthejandkindicesoftheouterproductAijBkinnDspaceweobtain:AijBj = Ai1B1 + Ai2B2 + ⋯ + AinBn

Fromtheabovedefinition,weseethatcontractionrequirestwoindicesandhenceitcannotbeconductedonscalarsandvectorssincescalarshavenoindexandvectorshaveonlyoneindex.

47. Inreferencetogeneralcoordinatesystems,asinglecontractionoperationisconductedonatensoroftype(m, n, w)wherem, n > 0.Whatistherank,thetypeandtheweightofthecontractedtensor?Answer:Therankism + n − 2,thetypeis(m − 1, n − 1, w)andtheweightisw.

48. Whatistheconditionthatshouldbesatisfiedbythetwotensorindicesinvolvedinacontractionoperationassumingageneralcoordinatesystem?WhatabouttensorsinorthonormalCartesiansystems?Answer:Ingeneralcoordinatesystemsthetwotensorindicesthatareinvolvedinacontractionoperationshouldbeofoppositevariancetype,i.e.onecovariantandonecontravariant.However,inorthonormalCartesiansystemsthevariancetypeisirrelevant,sincethereisnodifferencebetweencovariantandcontravarianttypesinthesesystems,andhencecontractioncantakeplacebetweenanytwoindicesinthesametensorterm.

49. Howmanyindividualcontractionoperationscantakeplaceinatensoroftype(m, n, w)inageneralcoordinatesystem?Explainwhy.Answer:Ingeneralcoordinatesystems,m × nindividualcontractionoperationscantakeplaceinthistensorbecausewecanhaveonecontractionoperationforeachcombinationofoneupperindexandonelowerindex.

50. Howmanyindividualcontractionoperationscantakeplaceinarank-ntensorinanorthonormalCartesiancoordinatesystem?Explainwhy.Answer:InorthonormalCartesiansystems,arank-ntensorcanhave[n(n − 1)] ⁄ 2possibleindividualcontractionoperations.Thereasonisthatinthesesystemsthereisnodifferencebetweencovariantandcontravarianttypesandhenceeachoneofthenindicescanbecontractedwitheachoneoftheremaining(n − 1)indicesandthereforeweshouldhaven(n − 1)possibleindividualcontractionoperations.However,becausetheoperationofcontractionisindependentoftheorderofthetwocontractedindices,sincecontractingiwithjisthesameascontractingjwithi,thenthenumbern(n − 1)shouldbereducedtohalfbydividingby2andhencewehave[n(n − 1)] ⁄ 2distinctcontractionoperations.

51. ListallthepossiblesinglecontractionoperationsthatcantakeplaceinthetensorAijklm.

Answer:Assumingageneralcoordinatesystem,wehave:AijkimAijkliAijkjmAijkljAijkkmAijklk

52. ListallthepossibledoublecontractionoperationsthatcantakeplaceinthetensorAijkmn.Answer:Assumingageneralcoordinatesystem,wehave:AijijnAijimjAijkijAijjinAijjmiAijkji

53. Giveexamplesofcontractionoperationfrommatrixalgebra.Answer:Takingthetraceofasquarematrixisanexampleofacontractionoperationonasingletensor,whilematrixproductisanexampleofanoperationthatcontainsacontractionoperationbetweentwotensors.

54. Showthatcontractingarank-ntensorresultsinarank-(n − 2)tensor.Answer:Thenumberoffreeindicesinarank-ntensorisn.Onconductingacontractionoperationonsuchatensor2freeindiceswillbeconsumedsincetheywillbecomeboundindicesandhencethecontractedtensorwillbecomearank-(n − 2)tensor.

55. Discussinnermultiplicationoftensorsasanoperationcomposedoftwomoresimpleoperations.Answer:Innermultiplicationconsistsofanoperationofoutermultiplicationontwonon-scalartensorsfollowedbyacontractionoperationontwoindicesoftheresultantproduct.Hence,innermultiplicationcanbeseenasacompositionoftwomoresimpleoperations:outermultiplicationandcontraction.

56. Givecommonexamplesofinnerproductoperationfromlinearalgebraandvectorcalculus.Answer:Anexampleofinnerproductoperationfromlinearalgebrais

matrixmultiplication.Anexampleofinnerproductoperationfromvectorcalculusisthedotproductoftwovectors.

57. Whyinnerproductoperationisnotcommutativeingeneral?Answer:Innerproductoperationisnotcommutativeingeneralbecausethebasisvectorsofthebasistensoroftheproductarenotcommutative.Forexample,theinnerproductoperationofA = AijEiEjwithB = BklEkElwhichinvolvestheindicesjandkis:A⋅B = 

(AijEiEj)⋅(BklEkEl) = 

AijBkl(EiEj)⋅(EkEl) = 

AijBklEi(Ej⋅Ek)El = 

AijBklEiδjkEl = 

AijBjlEiEl

whiletheinnerproductoperationofBwithAwhichinvolvesthesameindicesis:B⋅A = 

(BklEkEl)⋅(AijEiEj) = 

BklAij(EkEl)⋅(EiEj) = 

BklAijEl(Ek⋅Ej)Ei = 

BklAijElδjkEi = 

BjlAijElEi

Now,sinceEiEl ≠ ElEithenthetwooperationsaredifferentandhenceinnerproductoperationisnotcommutative.

58. CompletethefollowingequationswhereAandBarerank-2tensorsof

oppositevariancetype:A:B = ?A⋅⋅B = ?Answer:Ifweusethesametensorsofthepreviousquestionthenwehave:A:B = 

(AijEiEj):(BklEkEl) = AijBkl(Ei⋅Ek)(Ej⋅El) = 

AijBklδikδjl = 

AijBij

A⋅⋅B = 

(AijEiEj)⋅⋅(BklEkEl) = 

AijBkl(Ei⋅El)(Ej⋅Ek) = 

AijBklδilδjk = 

AijBji

59. Writeab:cdincomponentformassuminga3DCartesiansystem.Repeatthiswithab⋅⋅cd.Answer:Wehave:ab:cd = 

(a⋅c)(b⋅d) = 

(a1c1 + a2c2 + a3c3)(b1d1 + b2d2 + b3d3) = 

a1c1b1d1 + a2c2b1d1 + a3c3b1d1 + a1c1b2d2 + a2c2b2d2 + a3c3b2d2 + a1c1b3d3 + a2c2b3d3 + a3c3b3d3

ab⋅⋅cd = 

(a⋅d)(b⋅c) = 

(a1d1 + a2d2 + a3d3)(b1c1 + b2c2 + b3c3) = 

a1d1b1c1 + a2d2b1c1 + a3d3b1c1 + a1d1b2c2 + a2d2b2c2 + a3d3b2c2 + a1d1b3c3 + a2d2b3c3 + a3d3b3c3

60. Whytheoperationofinnermultiplicationoftensorsresultsinatensor?Answer:Becauseinnerproductoperationissynthesizedfromanouterproductoperationfollowedbyacontractionoperationandboththeseoperationsontensorsproducetensors.So,ifwestartfromtwotensorsandsubjectthemtooutermultiplicationweobtainatensor,andwhenwesubjectthistensortocontractionwewillalsogetatensor(whichisthefinalresultoftheinnermultiplication).Accordingly,theoperationofinnermultiplicationoftensorsshouldproduceatensor.

61. Wehave:A = AiEi,B = BjEjandC = CmnEmEn.Findthefollowingtensorproducts:AB,ACandBC.Answer:AB = AiBjEiEj

AC = AiCmnEiEmEn

BC = BjCmnEjEmEn

62. ReferringtothetensorsinQuestion613↑,findthefollowingdotproducts:B⋅B,C⋅AandC⋅B.Answer:B⋅B = 

(BjEj)⋅(BkEk) = 

BjBk(Ej⋅Ek) = 

BjBkδjk = 

BjBj

C⋅A = 

(CmnEmEn)⋅(AiEi) = 

CmnAiEm(En⋅Ei) = 

CmnAiEmδni = 

CmiAiEm

C⋅B = 

(CmnEmEn)⋅(BjEj) = 

CmnBj(Em⋅Ej)En = 

CmnBjδmjEn = 

CjnBjEn

63. Definepermutationoftensorsgivinganexampleofthisoperationfrommatrixalgebra.Answer:Permutationoftensorsistheoperationofexchangingthepositionoftwofreeindices.Anexampleofthisoperationfrommatrixalgebraistakingthetransposeofamatrixbyexchangingitsrowsandcolumns.

64. Statethequotientruleoftensorsinwordsandinaformalmathematicalform.Answer:Theessenceofthequotientruleoftensorsisthatiftheinnerproductofasuspectedtensorbyaknowntensorisatensorthenthesuspectisatensor.Mathematically,ifAisasuspectedtensorandBandCareknowntensorsandwehave:A⋅B = CthenAisatensor.

65. Whythequotientruleisusuallyusedintensortestsinsteadofapplyingthetransformationrules?

Answer:Becauseusingthequotientruleisgenerallymoreconvenientandrequireslessworkthanapplyingthetransformationrules.Moreover,insomecasesthequotientruledoesnotrequireactualworkwhenweknow,fromourpastexperience,thatarelationliketheoneseeninthepreviousquestionisalreadysatisfiedbyourtensorssoallweneedfromthequotientruleistodrawtheconclusionthatthesuspectedtensorisatensorindeed.

66. Outlinethesimilaritiesanddifferencesbetweenthethreemainformsoftensorrepresentation,i.e.covariant,contravariantandphysical.Answer:Somevalidpointsare:●Alltheseformsarelegitimaterepresentationsoftensorsandhenceanytensorcanberepresentedcovariantly,contravariantlyorphysicallywithoutaffectingthepropertiesofthetensor,i.e.allthesedifferentformsrepresentthesametensor.●Thecomponentsofthecovariantandcontravariantformsmayhavedifferentphysicaldimensions(orevendimensionless),butthecomponentsofthephysicalformhavethesamephysicaldimension.Thisalsoappliestothebasisvectorsoftheserepresentations.●Thebasisvectorsinthephysicalrepresentationarenormalizedanddimensionless,andthismaynotbethecaseinthecovariantandcontravariantrepresentations.

67. Define,mathematically,thephysicalbasisvectors,ÊiandÊi,intermsofthecovariantandcontravariantbasisvectors,EiandEi.Answer:Wehave:Êi = Ei ⁄ |Ei|andÊi = Ei ⁄ |Ei|

where|Ei|and|Ei|arethemagnitudesofEiandEirespectively.68. Correct,ifnecessary,thefollowingrelation:Âiknjm = [(hihjhn) ⁄ (hkhm)]Aiknjm

(nosumonanyindex)whereAisatensorinanorthogonalcoordinatesystem.Answer:Thecorrectformofthisrelationis:Âiknjm = [(hihkhn) ⁄ (hjhm)]Aiknjm

withnosumonanyindex.69. Whythenormalizedcovariant,contravariantandphysicalbasisvectorsare

identicalinorthogonalcoordinatesystems?

Answer:Becauseinorthogonalcoordinatesystemsthecorrespondingcovariantandcontravariantbasisvectorshavethesamedirectionandhencewhentheyarenormalizedtheybecomeidentical.So,thenormalizedbasisvectorsetsofalltheserepresentationsarethesame.

70. Whatisthephysicalsignificanceofbeingabletotransformonetypeoftensorstoothertypesaswellastransformingbetweendifferentcoordinatesystems?Answer:Thephysicalsignificanceisthatthetensorasamathematicalandphysicalentitydoesnotchangebytransformingfromonetypetoanother(e.g.covarianttocontravariant)andhenceitpossessesthesameproperties(i.e.itisinvariantinthissense)regardlessofthetypebywhichitisrepresented.Moreover,thetensorisinvariantacrossallcoordinatessystemsandhenceitsrealpropertieswillnotchangebytransformingfromonesystemtoanother.Thisinvariantnatureoftensorsmakesthemveryvaluabletoolinformulatingthelawsofphysicswhichshouldbeinvariantacrossallrepresentationsandacrossallcoordinatesystems.

71. Whythephysicalrepresentationoftensorsisusuallypreferredinthescientificapplicationsoftensorcalculus?Answer:Becausethephysicalrepresentationoftensorsisastandardizedformwhereallthecomponentshavethesamephysicaldimensionwhileallthebasisvectorsarenormalizedanddimensionless.Thisuniformityfacilitatesthemanagementandcomprehensionoftensorsinthescientificapplicationsoftensorcalculus.

72. Giveafewcommonexamplesofphysicalrepresentationoftensorsinmathematicalandscientificapplications.Answer:ManyexamplescanbefoundinfluidorcontinuummechanicsorgeneralrelativityforinstancewhereCartesianorcylindricalorsphericalcoordinatesystemswithnormalizeddimensionlessbasisvectorsareusedtorepresentandformulatetensors(e.g.stresstensor)inphysicalform(referto§7↓inthebook).

73. Whatistheadvantageofrepresentingthephysicalcomponentsofatensor(e.g.A)bythesymbolofthetensorwithsubscriptsdenotingthecoordinatesoftheemployedcoordinatesystem,e.g.(Ar, Aθ, Aφ)insphericalcoordinatesystems?Answer:Oneadvantageisthattheemployedcoordinatesystemtowhichthetensorisreferredcanbeeasilyinferredfromthenotation.Anotheradvantageisthatthecoordinatestowhichthecomponentsarereferredwillnotbeconfusediftheorderofcoordinatesisunclearoritissusceptibleto

change,unlikeusingnumbersforexample(e.g.A1, A2, A3)tolabelthecoordinates.

Chapter4SpecialTensors1. WhatisspecialabouttheKroneckerdelta,thepermutationandthemetric

tensorsandwhytheydeservespecialattention?Answer:Someofthespecialcharacteristicsofthesetensorsthatqualifythemforspecialattentionare:●Theyarepartofthetheoryoftensorcalculusitselfandhencetheyarepresentalmosteverywhereintensorcalculusanditsapplications.●Theyenterinessentialdefinitionsandoperationsoftensorcalculus.●Theyhaveverydistinctmathematicalpropertiesthatdistinguishthemfromothertensors(refertothebookfordetails).

2. Givedetaileddefinition,inwordsandinsymbols,oftheordinaryKroneckerdeltatensorinannDspace.Answer:TheordinaryKroneckerdeltatensor,alsoknownastheunittensor,isarank-2numeric,absolute,symmetric,constant,isotropictensorinalldimensions.Itisdefinedinitscovariantformas:δij = 1(i = j)

δij = 0(i ≠ j)

wherei, j = 1, 2, …nwithnbeingthespacedimension,andhenceitcanbeconsideredastheidentitytensorormatrix.Theabovedefinitionsimilarlyappliestothecontravariantandmixedformsofthistensor(i.e.δijandδij).

3. Listanddiscussallthemaincharacteristics(e.g.symmetry)oftheordinaryKroneckerdeltatensor.Answer:Themaincharacteristicsofthistensorare:●Itisarank-2tensorandhenceitpossessesn2componentsinannDspace.●Itisnumerictensorandhencethevaluesofitscomponentsare1and0inanycoordinatesystem.●Thevalueofanyparticularcomponent(e.g.δ12)ofthistensoristhesameinanycoordinatesystemandhenceitisconstanttensorinthissense.●Itscomponentshaveidenticalvaluesinallvariancetypes,i.e.δij = δij = δij.

●Itissymmetrictensorforbothvariancetypesandhenceδij = δjiandδij = δji.●Itisabsolutetensorandhenceitsweightwiszero.●Itpossessestheabovepropertiesinanyspacedimensionandinanycoordinatesystem.

4. WritethematrixthatrepresentstheordinaryKroneckerdeltatensorina4Dspace.Answer:

5. Doweviolatetherulesoftensorindiceswhenwewrite:δij = δij = δij = δij?

Answer:No,becausetheseequalitiesbelongtothevaluesofthecomponentsoftheKroneckerdeltatensorandhencetheyareessentiallyscalarequalitiesandnottensorequalities.Wemayclaimthattheindicesintheseequalitiesarelabelstotheindividualcomponentsratherthantensorindicesthatcanvaryovertheirrange.

6. Explainthefollowingstatement:“TheordinaryKroneckerdeltatensorisconservedunderallproperandimpropercoordinatetransformations”.Whatistherelationbetweenthisandthepropertyofisotropyofthistensor?Answer:ThisstatementmeansthatthecomponentsoftheordinaryKroneckerdeltatensorareconstantandhencetheydonotchangeunderproperorimpropercoordinatetransformations.Beingconservedinthissenseisstrongerthanbeingisotropicbecausetheformerappliestobothproperandimpropertransformationswhilethelatterappliesbydefinitiononlytopropertransformations.

7. Listanddiscussallthemaincharacteristics(e.g.anti-symmetry)ofthepermutationtensor.Answer:Themaincharacteristicsofthepermutationtensorare:●Itisnumerictensorandhencethevaluesofitscomponentsare − 1, 1

and0inallcoordinatesystems.●Thevalueofanyparticularcomponent(e.g.ϵ312)ofthistensoristhesameinanycoordinatesystemandhenceitisconstanttensorinthissense.●Itisrelativetensorofweight − 1foritscovariantformand + 1foritscontravariantform.●Itisisotropictensorsinceitscomponentsareconservedunderpropertransformations.●Itistotallyanti-symmetricineachpairofitsindicesforbothvariancetypes,i.e.itchangessignonswappinganytwoofitsindices.●Itispseudotensorsinceitacquiresaminussignunderimproperorthogonaltransformationofcoordinates.●Thepermutationtensorofanyrankhasonlyoneindependentnon-vanishingcomponentbecauseallthenon-zerocomponentsofthistensorareeither + 1or − 1.●Therank-npermutationtensorpossessesn!non-zerocomponentswhichisthenumberofthenon-repetitivepermutationsofitsindices.●TherankandthedimensionofthepermutationtensorareidenticalandhenceinannDspaceitisofrank-nandthereforeithasnncomponents.

8. Whataretheothernamesusedtolabelthepermutationtensor?Answer:ThepermutationtensorisalsoknownastheLevi-Civitatensor,theanti-symmetrictensorandthealternatingtensor.

9. Whytherankandthedimensionofthepermutationtensorarethesame?Accordingly,whatisthenumberofcomponentsoftherank-2,rank-3andrank-4permutationtensors?Answer:Therearetwomaindefiningpropertiesofthepermutationtensor:(a)Itisveryabstractmathematicalentitythathasnoassociationwithanyphysicalpropertyofthespaceandhenceifitisassociatedwithanypropertyofthespaceitshouldbethedimensionalityofthespace.(b)Itisaboutpermutingoralternatingsomethingandhenceitshouldbeaboutpermutingoralternatingsomethingrelatedtothedimensionalityofthespace.Accordingly,ifeachindependentdimensionofthespaceisidentifiedbyanindependentfreeindexthenthepermutationtensor(whosefunctionistopermutetheseindicesthatrefertothedimensionsofthespace)ofannDspaceshouldhavenfreeindicesandhenceitisofrank-n,i.e.therankandthedimensionofthepermutationtensorarethesame,asclaimed.Thenumberofcomponentsofarank-rtensorinannDspaceisnrandhencethenumberofcomponentsofthepermutationtensorinannDspace

isnn.Therefore,thenumberofcomponentsoftherank-2,rank-3andrank-4permutationtensorsarerespectively:22 = 4,33 = 27and44 = 256.

10. Whythepermutationtensorofanyrankhasonlyoneindependentnon-vanishingcomponent?Answer:Bydefinition,thecomponentsofthepermutationtensorofanyrankareeither0or + 1or − 1.So,ifweexcludethevanishingcomponents(i.e.0)thenalltheremainingnon-vanishingcomponentsareofunitymagnitude(i.e.either + 1or − 1)andhencewehaveonlyoneindependentnon-vanishingcomponent(i.e.allthesecomponentscanbeobtainedfromasinglenon-vanishingcomponentbyatmostareversalofsignandhencetheyarenotindependent).

11. Provethattherank-npermutationtensorpossessesn!non-zerocomponents.Answer:Bydefinition,allthecomponentsofthepermutationtensorwithrepetitiveindicesarezerowhileallthecomponentsofthepermutationtensorwithnon-repetitiveindicesarenon-zero(i.e.either + 1or − 1).Therefore,thenon-zerocomponentsofthepermutationtensorofanyrankarerestrictedtothenon-repetitivepermutationsoftheindices.Now,forrank-npermutationtensorwehavendistinctindices;moreoverthenumberofnon-repetitivepermutationsofnobjectsisn!(i.e.npossibleselectionsofndistinctobjectstimesn − 1possibleselectionsoftheremainingn − 1objects⋯times2times1).Hence,thenumberofnon-zerocomponentsoftherank-npermutationtensorisn!.

12. Whythepermutationtensoristotallyanti-symmetric?Answer:Bydefinition,theexchangeofanytwoindicesofthepermutationtensorleadstoreversalofsignandhenceitistotallyanti-symmetricaccordingtothedefinitionoftotallyanti-symmetrictensor.

13. Givetheinductivemathematicaldefinitionofthecomponentsofthepermutationtensorofrank-n.Answer:Thecomponentsoftherank-npermutationtensoraredefinedinductivelyas:ϵi1…in = ϵi1…in =  + 1(i1, …, inevenpermutationof1, …, n)

ϵi1…in = ϵi1…in =  − 1(i1, …, inoddpermutationof1, …, n)

ϵi1…in = ϵi1…in = 0 (repeatedindex)

14. Statethemostsimpleanalyticalmathematicaldefinitionofthecomponentsofthepermutationtensorofrank-n.

Answer:Itis:ϵa1⋯an = ϵa1⋯an =  ∏1 ≤ i < j ≤ nsgn(aj − ai)wheresgnisthesignfunction.

15. Makeasketchofthearrayrepresentingtherank-3permutationtensorwherethenodesofthearrayaremarkedwiththesymbolsandvaluesofthecomponentsofthistensor.Answer:ThesketchshouldlooklikeFigure6↓.

Figure6 Graphicalillustrationoftherank-3permutationtensorϵijkwheretheblackcirclesrepresentthe0components,thebluesquaresrepresentthe1componentsandtheredtrianglesrepresentthe − 1components.

16. Define,mathematically,therank-ncovariantandcontravariantabsolutepermutationtensors,εi1…inandεi1…in.Answer:Theyare:Covariant:εi1…in = √(g)ϵi1…in

Contravariant:εi1…in = [1 ⁄ √(g)]ϵi1…in

wheretheindexedϵandεarerespectivelytherelativeandabsolutepermutationtensorsofthegiventype,andgisthedeterminantofthecovariantmetrictensorgij.

17. Showthatϵijkisarelativetensorofweight − 1andϵijkisarelativetensorofweight + 1.Answer:Westartfromthedefinitionofthedeterminantofarank-2tensorin3Dspacewhichwegaveinthischapterofthebook,thatis:det(A) = [1 ⁄ (3!)]ϵijkϵlmnAliAmjAnk

Now,ifwereplaceAwithJ,whereJstandsfortheJacobianmatrix,thenthelastequationcanbewrittenas:J = [1 ⁄ (3!)]ϵijkϵlmnJliJmjJnk

whereJistheJacobianwhileJli = ∂ul ⁄ ∂ũiandJmjandJnkaredefinedsimilarly.Now,ifwemultiplybothsidesofthelastequationwithϵijkandnotetheidentity:ϵijkϵijk = 3!whichwegaveinthischapterofthebook,thenthelastequationwillbecome:ϵijkJ = ϵlmnJliJmjJnk

ϵijk = J − 1ϵlmnJliJmjJnk

ϵijk = J − 1ϵlmn(∂ul ⁄ ∂ũi)(∂um ⁄ ∂ũj)(∂un ⁄ ∂ũk)

ϵ̃ijk = J − 1ϵlmn(∂ul ⁄ ∂ũi)(∂um ⁄ ∂ũj)(∂un ⁄ ∂ũk)

wherethelaststepisjustifiedbythefactthatthepermutationtensorisconservedacrossallcoordinatesystemsandhenceϵijk = ϵ̃ijk(seeFootnote14in§8↓).Aswesee,thelastequationisthetransformationequationofacovariantrank-3relativetensorofweightw =  − 1fromunbarredtobarredsystemsandhenceϵijkisarelativetensorofweight − 1,asrequired.Ifwerepeattheaboveargumentonthecontravariantpermutationtensor,thenfromthedefinitionofthedeterminantwehave:det(A) = [1 ⁄ (3!)]ϵijkϵlmnAliAmjAnk

Now,ifwereplaceAwithJ − 1,whereJ − 1istheinverseJacobianmatrix,thenthelastequationwillbecome:J − 1 = [1 ⁄ (3!)]ϵijkϵlmn(J − 1)li(J − 1)mj(J − 1)nk

whereJ − 1istheinverseJacobianwhile(J − 1)li = ∂ũl ⁄ ∂uiand(J − 1)mjand(J − 1)nkaredefinedsimilarly.Now,ifwemultiplybothsidesofthelastequationwithϵlmnandnotetheidentity:ϵlmnϵlmn = 3!thenthelastequationwillbecome:ϵlmnJ − 1 = ϵijk(J − 1)li(J − 1)mj(J − 1)nk

ϵlmn = J + 1ϵijk(J − 1)li(J − 1)mj(J − 1)nk

ϵlmn = J + 1ϵijk(∂ũl ⁄ ∂ui)(∂ũm ⁄ ∂uj)(∂ũn ⁄ ∂uk)

ϵ̃lmn = J + 1ϵijk(∂ũl ⁄ ∂ui)(∂ũm ⁄ ∂uj)(∂ũn ⁄ ∂uk)

wherethelaststepisjustifiedbythefactthatthepermutationtensorisconservedacrossallcoordinatesystemsandhenceϵlmn = ϵ̃lmn.Aswesee,thelastequationisthetransformationequationofacontravariantrank-3relativetensorofweightw =  + 1fromunbarredtobarredsystemsandhenceϵijkisarelativetensorofweight + 1,asrequired.

18. Showthatεi1…in = √(g)ϵi1…inandεi1…in = [1 ⁄ √(g)]ϵi1…inareabsolutetensors.Answer:Ifwefollowtheargumentsofthelastquestion,thenweshouldgetthefollowingtransformationequations:ϵ̃i1…in = J − 1ϵj1…jn(∂uj1 ⁄ ∂ũi1)⋯(∂ujn ⁄ ∂ũin)

ϵ̃i1…in = J + 1ϵj1…jn(∂ũi1 ⁄ ∂uj1)⋯(∂ũin ⁄ ∂ujn)

Now,itwasshownearlier(seeExercise183↑of§3↑)thatthedeterminantofarank-2absolutecovarianttensorisarelativescalarofweight2.Onapplyingthisruleonthemetrictensorgij,whichisarank-2absolutecovarianttensor,weget:|g̃ij| = J2|gpq|

g̃ = J2g

√(g̃) = J√(g)

Onmultiplyingthetwosidesofthefirsttransformationequationbythetwosidesofthelastequation,anddividingthetwosidesofthesecondtransformationequationbythetwosidesofthelastequation,weget:√(g̃)ϵ̃i1…in = √(g)ϵj1…jn(∂uj1 ⁄ ∂ũi1)⋯(∂ujn ⁄ ∂ũin)

ϵ̃i1…in ⁄ √(g̃) = [ϵj1…jn ⁄ √(g)](∂ũi1 ⁄ ∂uj1)⋯(∂ũin ⁄ ∂ujn)

Thelasttwoequationsmeanthat√(g)ϵj1…jn ≡ εj1…jnandϵj1…jn ⁄ √(g) ≡ εj1…jn

transformasabsolutetensorsandhencetheyareabsolutetensors,asrequired(notingthetrivialdifferenceinthelabelingofindiceswiththelabelinginthequestion).

19. Writeϵi1⋯inϵj1⋯jninitsdeterminantalformintermsoftheordinaryKroneckerdelta.Answer:

20. Provethefollowingidentity:ϵi1⋯inϵi1⋯in = n!.

Answer:Accordingtothesummationconvention,ϵi1⋯inϵi1⋯inisthesumofproductsofϵi1⋯inandϵi1⋯inwhereboththeseshouldbe0or + 1or − 1andhencethetermsofthissumiseither0or + 1.Now,thenumberoftermsofthissumisequaltothenumberofallpermutationsofi1, i2, ⋯in.However,allthepermutationswithrepetitiveindicesarezeroandhenceonlythetermsofnon-repetitiveindicesareequalto + 1.Becausetherearen!non-repetitivepermutationsofi1, i2, ⋯inthenwehaven!termsof + 1andhencetheirsumisequalton!,asrequired.

21. StateamathematicalrelationrepresentingtheuseoftheordinaryKroneckerdeltatensorasanindexreplacementoperator.Answer:Forexample,intherelationδjiAklj = AklitheordinaryKroneckerdeltatensorreplacedthecovariantindexjofthetensorAkljwiththeindexiandtheresultischangingAkljtoAkli.Similarly,intherelationδijBjk = BiktheordinaryKroneckerdeltatensorreplacedthecontravariantindexjofthetensorBjkwiththeindexiandtheresultischangingBjktoBik.

22. Provethefollowingrelationinductivelybystartingfromwritingitinanexpandedformina2Dspace:δii = n.Answer:Ina2Dspacewehave:δii = 

δ11 + δ22 = 

1 + 1 = 

2 = 

n

Moreover,ifinannDspacethisrelationistrue,thatis:δii = 

δ11 + δ22 + ⋯ + δnn = 

1 + 1 + ⋯ + 1 = 

n

theninan(n + 1)Dspaceitshouldalsobetruebecausewethenhave:δii = 

δ11 + δ22 + ⋯ + δnn + δn + 1n + 1 = 

1 + 1 + ⋯ + 1 + 1 = 

n + 1

Hence,bymathematicalinductiontherelationδii = nistrueinanyspaceofanydimension.

23. RepeatExercise224↑withtherelation:ui, j = δijusingamatrixform.Answer:Ina2Dspacewehave:

Moreover,ifinannDspacethisrelationistrue,thatis:

theninan(n + 1)Dspaceitshouldalsobetruebecausewethenhave:

Hence,bymathematicalinductiontherelationui, j = δijistrueinanyspaceofanydimension.

24. JustifythefollowingrelationassuminganorthonormalCartesiansystem:∂ixj = ∂jxi.Answer:Duetothefactthatthecoordinatesareindependentofeachother,plusthefactthatthecovariantandcontravarianttypesarethesameinorthonormalCartesiancoordinatesystems,wehave:∂ixj ≡ 

∂xj ⁄ ∂xi = 

δji = 

δij = 

∂xi ⁄ ∂xj ≡ 

∂jxi

wherethemiddleequality(i.e.δji = δij)isjustifiedbythesymmetryoftheKroneckerdeltatensor.

25. Justifythefollowingrelationswheretheindexedeareorthonormalvectors:ei⋅ej = δij

eiej:ekel = δikδjl

Answer:Regardingtherelationei⋅ej = δij,becauseeiandejareorthogonalwheni ≠ jthenei⋅ej = 0wheni ≠ j,andbecausetheyarenormalizedthenei⋅ej = 1wheni = j.ThesetwoconditionsareequivalenttotheconditionsoftheKroneckerdeltatensor(i.e.δij = 0wheni ≠ jandδij = 1wheni = j)andhenceei⋅ej = δij,asrequired.Regardingtherelationeiej:ekel = δikδjl,wehave:eiej:ekel = 

(ei⋅ek)(ej⋅el) = 

δikδjl

wherethefirstequalityisjustifiedbythedefinitionofthedoubleinnerproduct(i.e.:)whichisgiveninthebook,whilethesecondequalityisjustifiedbytheanswerofthefirstpartofthisquestion.

26. Showthatδjiδkjδik = n.Answer:Wehave:δjiδkjδik = 

δkiδik = 

δii = 

n

whereinthefirstandsecondequalitiestheKroneckerdeltaisactingasanindexreplacementoperatorwhilethethirdequalityisbasedontheidentityδii = nwhichwasprovedearlier(seeExercise224↑).

27. Writethedeterminantalarrayformofϵijϵkloutliningthepatternofthetensorindicesintheirrelationtotheindicesoftherowsandcolumnsofthedeterminantarray.Answer:ϵijϵkl = 

δikδjl − δilδjk

Thepatternoftheindicesinthedeterminantarrayofthisidentityisthattheindicesofthefirstϵprovidetheindicesfortherowsastheupperindicesofthedeltaswhiletheindicesofthesecondϵprovidetheindicesforthecolumnsasthelowerindicesofthedeltas.

28. Provethefollowingidentityusingatruthtable:ϵijϵkl = δikδjl − δilδjk.Answer:Thetruthtableisgiveninthetextwhereoncomparingthecolumnofϵijϵklwiththecolumnofδikδjl − δilδjkweseethatthecorrespondingentrieshaveidenticalvaluesandhenceϵijϵkl = δikδjl − δilδjk.

29. Provethefollowingidentitiesjustifyingeachstepinyourproofs:ϵilϵkl = δik

ϵijkϵlmk = δilδjm − δimδjl

Answer:Regardingtheidentityϵilϵkl = δik,wehave:ϵilϵkl = 

δikδll − δilδlk = 

2δik − δilδlk = 

2δik − δik = 

δik

wherethefirstlineisbasedontheidentityofthepreviousquestionwiththereplacementofjwithl,thesecondlineisjustifiedbythepreviouslyprovedidentityδii = n(seeExercise224↑)wheren = 2sincethedimensionandrankofϵareidentical,andthethirdlineisbasedontheactionoftheKroneckerdeltaasanindexreplacementoperator.Regardingtheidentityϵijkϵlmk = δilδjm − δimδjl,wehave:ϵijkϵlmk = 

δilδjmδkk + δimδjkδkl + δikδjlδkm − δilδjkδkm − δimδjlδkk − δikδjmδkl = 

3δilδjm + δimδjkδkl + δikδjlδkm − δilδjkδkm − 3δimδjl − δikδjmδkl = 

3δilδjm + δimδjl + δimδjl − δilδjm − 3δimδjl − δilδjm = 

δilδjm − δimδjl

wherethefirstlineisjustifiedbytheidentity:

withthereplacementofnwithk,thesecondlineisjustifiedbythepreviouslyprovedidentityδii = n(seeExercise224↑)wheren = 3sincethedimensionandrankofϵareidentical,andthethirdlineisbasedontheactionoftheKroneckerdeltaasanindexreplacementoperator.

30. Provethefollowingidentitiesusingothermoregeneralidentities:ϵijkϵljk = 2δil

ϵijkϵijk = 6

Answer:Regardingthefirstidentity,wehave:ϵijkϵljk = 

δilδjj − δijδjl = 

3δil − δijδjl = 

3δil − δil = 

2δil

whereinthefirstlineweareusingtheidentityϵijkϵlmk = δilδjm − δimδjlofthepreviousquestionwiththecontractionofjandm,thesecondlineisjustifiedbythepreviouslyprovedidentityδii = n(seeExercise224↑)wheren = 3sincethedimensionandrankofϵareidentical,andthethirdlineisbasedontheactionoftheKroneckerdeltaasanindexreplacementoperator.

Regardingthesecondidentity,wehave:ϵijkϵijk = 

2δii = 

2 × 3 = 

6

whereinthefirstequalityweareusingthefirstidentityofthisquestion(i.e.ϵijkϵljk = 2δil)withthecontractionofiandl,whileinthesecondequalityweareusingthepreviouslyprovedidentityδii = nwithn = 3.

31. OutlinethesimilaritiesanddifferencesbetweentheordinaryKroneckerdeltatensorandthegeneralizedKroneckerdeltatensor.Answer:Therearemanyvalidpointsthatcanbegivenintheanswerofthisquestion;someoftheseare:●BothKroneckerdeltasareconstant,numeric,absolute,isotropictensorsinalldimensions.●BothKroneckerdeltashavecloserelationwiththepermutationtensor.Theycanbothbedefinedintermsofthepermutationtensorandtheysharemanyidentitieswiththistensor.●WhileordinaryKroneckerdeltaisarank-2tensorinanyspace,thegeneralizedKroneckerdeltaisarank-2ntensorinnDspace.●WhileordinaryKroneckerdeltacanbecovariant,contravariantormixed,thegeneralizedKroneckerdeltaisalwaysmixedwithequalnumbersofcovariantandcontravariantindices.

32. GivetheinductivemathematicaldefinitionofthegeneralizedKroneckerdeltatensorδi1…in

j1…jn.Answer:δi1…in

j1…jn =  + 1(j1…jnisevenpermutationofi1…in)

δi1…inj1…jn =  − 1(j1…jnisoddpermutationofi1…in)

δi1…inj1…jn = 0( repeatedi'sorj's)

33. WritethedeterminantalarrayformofthegeneralizedKroneckerdeltatensorδi1…in

j1…jnintermsoftheordinaryKroneckerdeltatensor.

Answer:

34. Defineϵi1…inandϵi1…inintermsofthegeneralizedKroneckerdeltatensor.

Answer:ϵi1…in = δ1…n

i1…in

ϵi1…in = δi1…in1…n

35. Provetherelation:ϵijkϵlmn = δijklmnusingananalyticoraninductiveoratruthtablemethod.Answer:Fromtheinductivedefinitionoftheentriesofthepermutationtensorwehave:

and

So,onmultiplyingϵijkwithϵlmnwehave9cases:ϵijkϵlmn = 1whenijkand

lmnareofthesameparity(i.e.bothevenpermutationsorbothoddpermutations)andϵijkϵlmn =  − 1whenijkandlmnareofdifferentparity(i.e.oneisevenpermutationandtheotherisoddpermutation)whileϵijkϵlmn = 0inalltheremaining5cases(i.e.whenoneorbothofϵijkandϵlmniszeroduetorepetitionofindex).Now,ifwelooktotheinductivedefinitionofthegeneralizedKroneckerdeltatensorina3Dspace:

weseeitisessentiallythesame,i.e.δijklmn = 1whenijkandlmnareofthesameparity(inreferenceto123),δijklmn =  − 1whenijkandlmnareofoppositeparity(seeFootnote15in§8↓),andδijklmn = 0whentheparityofoneorbothisambiguousduetorepetitionofindex.Hence,weconcludethatϵijkϵlmnandδijklmnareinductivelyequalandhenceϵijkϵlmn = δijklmn,asrequired.

36. DemonstratethatthegeneralizedKroneckerdeltaisanabsolutetensor.Answer:Fromtherelation:δi1…in

j1…jn = ϵi1…inϵj1…jnweseethatthegeneralizedKroneckerdeltaisequaltotheproductofthecovariantpermutationtensor(whichisarelativetensorofweightw =  − 1asprovedearlierinExercise174↑)timesthecontravariantpermutationtensor(whichisarelativetensorofweightw =  + 1asprovedearlierinExercise174↑)andhenceitsweightisw = 0,i.e.itisanabsolutetensor,asrequired.Wenotethattheproductoftensorsisatensorwhoseweightisthesumoftheweightsoftheoriginaltensors.

37. Provethefollowingrelationjustifyingeachstepinyourproof:δmnqklq = δmnkl.Answer:δmnqklq = 

δmk(δnlδqq − δnqδql) − δml(δnkδqq − δnqδqk) + δmq(δnkδql − δnlδqk) = 

δmkδnlδqq − δmkδnqδql − δmlδnkδqq + δmlδnqδqk + δmqδnkδql − δmqδnlδqk = 

3δmkδnl − δmkδnqδql − 3δmlδnk + δmlδnqδqk + δmqδnkδql − δmqδnlδqk = 

3δmkδnl − δmkδnl − 3δmlδnk + δmlδnk + δmlδnk − δmkδnl = 

δmkδnl − δmlδnk = 

δmnkl

Justification:Line1isbasedonthedefinitionofthegeneralizedKroneckerdeltain3D.Line2isbasedonthedefinitionofexpansionofdeterminant.Line3isbasedonthedistributivityofmultiplicationoveralgebraicaddition.Line4isbasedontheidentityδii = nwithn = 3(seeExercise224↑).Line5isbasedonusingtheordinaryKroneckerdeltaasanindexreplacementoperator.Line6isbasedontheoperationsofadditionandsubtraction.Line7isbasedonthedefinitionofexpansionofdeterminant(inreverse).Line8isbasedonthedefinitionofthegeneralizedKroneckerdeltain2D.

38. Provethecommonformoftheepsilon-deltaidentity.Answer:Wehave:ϵijkϵlmk = 

δijklmk = 

δijlm = 

δilδjm − δimδjl

Justification:Line1isbasedontheidentityϵijkϵlmn = δijklmn(whichweprovedinExercise354↑)withn = k.Line2isbasedontheidentityδmnqklq = δmnklwhichweprovedinthelastquestion.Line3isbasedonthedefinitionofthegeneralizedKroneckerdeltain2D.Line4isbasedonthedefinitionofexpansionofdeterminant.

39. Provethefollowinggeneralizationoftheepsilon-deltaidentity:gijεiklεjmn = gkmgln − gknglmAnswer:Onmultiplyingthetwosideswithgkmglnweget:gijgkmglnεiklεjmn = gkmglngkmgln − gknglmgkmgln

Onraisingtheindices,weobtain:εjmnεjmn = gkkgll − gmngnm

Now,fromtheidentities:εjmn = ϵjmn ⁄ √(g),εjmn = √(g)ϵjmnandgij = δijweget:

ϵjmnϵjmn = 

δkkδll − δmnδnm = 

δkkδll − δmm

wherethelaststepisbasedonemployingtheKroneckerdeltaasanindexreplacementoperator.Notingthattherankanddimensionofthepermutationtensorareidentical(i.e.3inthiscase),wehave(usingpreviouslyprovedidentities)(seeFootnote16in§8↓):ϵjmnϵjmn = 3! = 6

δkkδll − δmm = (3 × 3) − 3 = 6

andhencetheidentityϵjmnϵjmn = δkkδll − δmm(whichisobtainedfromtheidentitygijεiklεjmn = gkmgln − gknglm)iscorrect.Therefore,theidentitygijεiklεjmn = gkmgln − gknglmisvalid(seeFootnote17in§8↓).

40. Listanddiscussallthemaincharacteristics(e.g.symmetry)ofthemetrictensor.Answer:Themaincharacteristicsare:●Itisarank-2symmetricabsolutenon-singulartensor.●Ithascovariant,contravariantandmixedtypeswiththelatterbeingthesameastheidentitytensor.●Itisusedforraisingandloweringofindicesandhenceitfacilitatestheconversionoftensorsfromonevariancetypetoanother.●Itentersinthedefinitionandformulationofmanystandardconceptsandoperationsoftensorcalculusandhenceitispresenteverywhereintensorcalculusanditsapplications.●Itcontainsvitalinformationaboutthemaincharacteristicfeaturesofthespaceanditsgeometricproperties.

41. Howmanytypesthemetrictensorhas?Answer:Ithasthreetypes:covariantgij,contravariantgijandmixedgji = δji.

42. Investigatetherelationofthemetrictensorofagivenspacetothecoordinatesystemsofthespaceaswellasitsrelationtothespaceitselfbycomparingthecharacteristicsofthemetricindifferentcoordinatesystems

ofthespacesuchasbeingdiagonalornotorhavingconstantorvariablecomponentsandsoon.Hence,assessthestatusofthemetricasapropertyofthespacebutwithaformdeterminedbytheadoptedcoordinatesystemtodescribethespaceandhenceitisalsoapropertyofthecoordinatesysteminthissense.Answer:Asatensor,themetricshouldhavesignificanceregardlessofanycoordinatesystemwherethissignificanceisrepresentedbythefactthatitsummarizesthecharacteristicsofthespaceanditsgeometricnaturesuchasbeingflatorcurved.However,themetrictensoriscloselyrelatedtotheunderlyingcoordinatesystemofthespacethroughtherelations:gij = Ei⋅Ej

gij = Ei⋅Ej

gij = Ei⋅Ej

whichlinktheentriesofthemetrictensortothebasisvectorswhichareintimatelyrelatedtothecoordinatesystemsincethecovariantbasisvectorsarethetangentvectorstothecoordinatecurves,whilethecontravariantbasisvectorsrepresentthegradientofthespacecoordinatesandhencetheyareperpendiculartothecoordinatesurfaces.Thecloserelationbetweenthemetrictensorandthecoordinatesystemisreflectedinmanyaspects.Forexample,themetrictensorisconstantforrectilinearcoordinatesystemsandvariableforcurvilinearcoordinatesystemsbecausethebasisvectorsareconstantfortheformerandvariableforthelatter.Anotherexampleisthatthemetrictensorisdiagonalwhenthecoordinatesystemofthespaceisorthogonalandthisisjustifiedbytheaboverelationsbetweentheentriesofthemetrictensorandthebasisvectorssincethebasisvectorsoforthogonalsystemsaremutuallyperpendicularandhencethedotproductwillvanishwheni ≠ j.Theseexamples,amongothers,representthecloserelationbetweentheadoptedcoordinatesystemandtheformofthemetrictensor.Basedontheabovediscussion,wecanconcludethatalthoughtheessenceofthemetrictensorisrelatedtothenatureofthespaceandhenceitisindependentofthecoordinatesystem,theformofthemetrictensorishighlydependentonthecoordinatesystem.Soinbrief,themetrictensorsummarizesessentialpropertiesofthespaceandhenceitisindependentoftheemployedcoordinatesystemofthespace,buttheformofthemetrictensorishighlydependentontheemployedcoordinatesystemandhenceit

iscloselyrelatedtotheemployedcoordinatesystem.Accordingly,thepremisethatissuggestedinthequestion(i.e.themetricisapropertyofthespacebutwithaformdeterminedbytheadoptedcoordinatesystemtodescribethespaceandhenceitisalsoapropertyofthecoordinatesystem)isfullyjustified.

43. Whatistherelationbetweenthecovariantmetrictensorandthelengthofaninfinitesimalelementofarcdsinageneralcoordinatesystem?Answer:Therelationis:(ds)2 = gijduiduj

wheregijisthecovariantmetrictensor,theindexeduaregeneralcoordinatesandi, j = 1⋯nwithnbeingthedimensionofthespace.

44. HowwilltherelationinQuestion434↑become(a)inanorthogonalcoordinatesystemand(b)inanorthonormalCartesiancoordinatesystem?Answer:Inanorthogonalcoordinatesystemitbecomes:(ds)2 = Σi(hi)2duidui

wherehiisthescalefactoroftheithcoordinateandi = 1⋯nwithnbeingthedimensionofthespace.InanorthonormalCartesiancoordinatesystemitbecomes:(ds)2 = dxidxi

wherexiisaCartesiancoordinate,i = 1⋯n(withnbeingthedimensionofthespace)andsumoveriisimplied.

45. Whatisthecharacteristicfeatureofthemetrictensorinorthogonalcoordinatesystems?Answer:Itisdiagonal,thatis:gij = 0gij = 0(i ≠ j)

gij ≠ 0gij ≠ 0(i = j)46. Writethemathematicalexpressionsforthecomponentsofthecovariant,

contravariantandmixedformsofthemetrictensorintermsofthecovariantandcontravariantbasisvectors,EiandEi.Answer:gij = Ei⋅Ej

gij = Ei⋅Ej

gij = Ei⋅Ej47. Write,infulltensornotation,themathematicalexpressionsforthe

componentsofthecovariantandcontravariantformsofthemetrictensor,gijandgij.Answer:gij = (∂xk ⁄ ∂ui)(∂xk ⁄ ∂uj)

gij = (∂ui ⁄ ∂xk)(∂uj ⁄ ∂xk)48. Whatistherelationbetweenthemixedformofthemetrictensorandthe

ordinaryKroneckerdeltatensor?Answer:Theyareidentical,thatis:gij = δij

49. Explainwhythecovariantandcontravariantmetrictensorisnotnecessarilydiagonalingeneralcoordinatesystemsbutitisnecessarilysymmetric.Answer:Thebasisvectors,whethercovariantorcontravariant,ingeneralcoordinatesystemsarenotnecessarilymutuallyorthogonalandhencethemetrictensorisnotdiagonalingeneralsincethedotproductswhicharegivenintheanswerofapreviousquestion(i.e.gij = Ei⋅Ejandgij = Ei⋅Ej)arenotnecessarilyzerowheni ≠ j.However,sincethedotproductofvectorsisacommutativeoperation(i.e.Ei⋅Ej = Ej⋅EiandEi⋅Ej = Ej⋅Ei),themetrictensorisnecessarilysymmetric(i.e.gij = gjiandgij = gji).

50. Explainwhythediagonalelementsofthecovariantandcontravariantmetrictensoringeneralcoordinatesystemsarenotnecessarilyofunitmagnitudeorpositivebuttheyarenecessarilynon-zero.Answer:Sincethebasisvectors,whethercovariantorcontravariant,ingeneralcoordinatesystemsarenotnecessarilyofunitlength,thenthediagonalelements(whicharegivenbygii = Ei⋅Ei = |Ei|2andgii = Ei⋅Ei = |Ei|2withnosumoni),arenotnecessarilyofunitmagnitude.Also,sincethecoordinatesinsomegeneralcoordinatesystemscanbeimaginary,thenthediagonalelementscanbenegative.However,becausethebasisvectorscannotvanishattheregularpointsofthespace(sincethetangenttothecoordinatecurveandthegradienttothecoordinatesurfacedoexistandtheycannotbezero),thenthedotproducts(i.e.Ei⋅EiandEi⋅Ei),andhencethediagonalelementsofthemetrictensor(i.e.giiandgii),should

necessarilybenon-zero.51. Explainwhythemixedtypemetrictensorinanycoordinatesystemis

diagonalorinfactitistheunitytensor.Answer:Asgivenearlier,themixedtypemetrictensorisgivenbygij = Ei⋅Ej.Now,sincethecovariantandcontravariantbasissetsarereciprocalsystems(i.e.thedotproductbetweentheircorrespondingelementsis1andbetweentheirnon-correspondingelementsis0),thenweshouldhave:gij = Ei⋅Ej = 1(i = j)

gij = Ei⋅Ej = 0(i ≠ j)

Thesecondrelationmeansthatthemixedtypemetrictensorisdiagonalandthefirstrelation(combinedwiththesecondrelation)meansitistheunitytensor.Alternatively,wehave:gikgkj = δji

gji = δji

whereline1representsthefactthatthecovariantandcontravariantmetrictensorsareinversesofeachotherwhilethesecondlineisbasedonusingthemetrictensorasanindexshiftingoperator.Hence,themixedtypemetrictensorisdiagonalanditistheunitytensor.

52. Showthatthecovariantandcontravariantformsofthemetrictensor,gijandgij,areinversesofeachother.Answer:Thecovariantandcontravariantformsofthemetrictensoraregivenby:gij = (∂xk ⁄ ∂ui)(∂xk ⁄ ∂uj)

gij = (∂ui ⁄ ∂xk)(∂uj ⁄ ∂xk)

Multiplyingthemastwomatrices,wehave:gijgjk = 

(∂xm ⁄ ∂ui)(∂xm ⁄ ∂uj)(∂uj ⁄ ∂xn)(∂uk ⁄ ∂xn) = 

(∂xm ⁄ ∂ui)(∂xm ⁄ ∂xn)(∂uk ⁄ ∂xn) = 

(∂xm ⁄ ∂ui)(∂xn ⁄ ∂xm)(∂uk ⁄ ∂xn) = 

(∂xm ⁄ ∂ui)(∂uk ⁄ ∂xm) = 

∂uk ⁄ ∂ui = 

δki

wherelines2,4and5arethechainruleofdifferentiation,line3istheidentity∂ixj = ∂jxiinorthonormalCartesiansystems(seeExercise244↑),andline6istheidentityui, j = δij(seeExercise234↑).Similarly:gijgjk = 

(∂ui ⁄ ∂xm)(∂uj ⁄ ∂xm)(∂xn ⁄ ∂uj)(∂xn ⁄ ∂uk) = 

(∂ui ⁄ ∂xm)(∂xn ⁄ ∂xm)(∂xn ⁄ ∂uk) = 

(∂ui ⁄ ∂xm)(∂xm ⁄ ∂xn)(∂xn ⁄ ∂uk) = 

(∂ui ⁄ ∂xm)(∂xm ⁄ ∂uk) = 

∂ui ⁄ ∂uk = 

δik

wherethelinesaresimilarlyjustifiedasinthefirstpart.Hence,thecovariantandcontravariantformsofthemetrictensorareinversesofeachother,asrequired.

53. Whythedeterminantofthemetrictensorshouldnotvanishatanypointinthespace?Answer:Becausethemetrictensorshouldbeinvertible(i.e.non-singular)ateverypointinthespace(i.e.weshouldhavecovariantandcontravarianttypesgloballyifwearesupposedtohaveanappropriatecoordinatesystem).

54. Ifthedeterminantofthecovariantmetrictensorgijisg,whatisthedeterminantofthecontravariantmetrictensorgij?Answer:Itisthereciprocalofg,i.e.1 ⁄ g.Thiscanbeeasilyinferredfromthefactthatthecovariantandcontravariantformsofthemetrictensorareinversesofeachother.

55. ShowthatthemetrictensorcanberegardedasatransformationoftheordinaryKroneckerdeltatensorinitsdifferentvariancetypesfromanorthonormalCartesiancoordinatesystemtoageneralcoordinatesystem.Answer:Fromthedefinitionofthemetrictensorinitsdifferentvariancetypes,wehave:gij = (∂xk ⁄ ∂ui)(∂xk ⁄ ∂uj)

gij = (∂ui ⁄ ∂xk)(∂uj ⁄ ∂xk)

gij = (∂ui ⁄ ∂xk)(∂xk ⁄ ∂uj)

Theseequationscanbewrittenas:gij = (∂xk ⁄ ∂ui)(∂xl ⁄ ∂uj)δkl

gij = (∂ui ⁄ ∂xk)(∂uj ⁄ ∂xl)δkl

gij = (∂ui ⁄ ∂xk)(∂xl ⁄ ∂uj)δkl

wheretheyarejustifiedbythefactthattheordinaryKroneckerdeltatensorisanindexreplacementoperatorplusthefactthatinorthonormalCartesiansystemsallvariancetypesoftheKroneckerdeltatensorareequal,i.e.δijeiej = δijeiej = δijeiej.Asseen,thelastthreeequationsarethetransformationequationsoftheordinaryKroneckerdeltatensorinitsdifferentvariancetypesfromanorthonormalCartesiancoordinatesystemtoageneralcoordinatesystem,asrequired.

56. Justifytheuseofthemetrictensorasanindexshiftingoperatorusingamathematicalargument.Answer:LethaveavectorAwhichcanbecovariantorcontravariantandhencewehaveA = AjEj = AjEj.So,letinner-multiplythisvectorwithEi,thatis:A⋅Ei = 

AjEj⋅Ei = Ajδji = Ai

A⋅Ei = AjEj⋅Ei = Ajgji

Oncomparingthesetwoequationsandtakingaccountoftheinvarianceofvector(i.e.itsindependenceasatensorfromtheemployedbasisset),weconcludethatAi = Ajgji,i.e.thecovariantmetrictensorisanindexloweringoperator.Similarly,letinner-multiplythisvectorwithEi,thatis:A⋅Ei = AjEj⋅Ei = Ajgji

A⋅Ei = AjEj⋅Ei = Ajδij = Ai

Oncomparingthesetwoequationsandtakingaccountoftheinvarianceofvector,weconcludethatAi = Ajgji,i.e.thecontravariantmetrictensorisanindexraisingoperator.TheaboveargumentwhichemploysavectorAcanbeeasilygeneralizedtonon-scalartensorsofanyrankandhenceweconcludethatthemetrictensorisanindexshiftingoperator,asrequired.

57. Carryoutthefollowingindexshiftingoperationsrecordingtheorderoftheindiceswhennecessary:gijCkljgmnBnstglnDkmn

Answer:

gijCklj = Ckl.i

gmnBn.st = Bmst

glnDkm.n = Dkm.l

58. WhatisthedifferencebetweenthethreeoperationsinQuestion574↑?Answer:Thefirstisanindexraisingoperationusingthecontravariantmetrictensor,thesecondisanindexloweringoperationusingthecovariantmetrictensor,andthethirdisanindexreplacementoperationusingthemixedtypemetrictensor(orKroneckerdelta).

59. Whytheorderoftheraisedandloweredindicesisimportantandhenceitshouldberecorded?Mentiononeformofnotationusedtorecordtheorderoftheindices.Answer:Asindicatedbefore,tensorsarereferredtobasisvectorsetandhencetheorderofindicesoftensorsisimportantbecauseitdeterminestheorderofthebasisvectorstowhichthetensorisreferred.Wemayalsoneedtoreversetheindexshiftingoperationthatweconductedearlierinalaterstage.Therefore,theorderoftheraisedandloweredindicesisimportantandhenceitshouldbekeptandrecorded.Oneformofnotationthatisusedtorecordtheorderoftheindicesistoinsertadottoindicatetheoriginalpositionoftheshiftedindex.

60. Whatistheconditionthatshouldbesatisfiedbythemetrictensorofaflatspace?Givecommonexamplesofflatandcurvedspaces.Answer:Themetrictensorofaflatspacecanberepresentedbyadiagonalformwithallthediagonalelementsbeing + 1or − 1.Examplesofflatspacesareplane,3DEuclideanspaceand4DMinkowskispace.Examplesofcurvedspacesaresphereandellipsoid.

61. Consideringacoordinatetransformation,whatistherelationbetweenthedeterminantsofthecovariantmetrictensorintheoriginalandtransformedcoordinatesystems,gandg̃?Answer:Therelationis:g̃ = J2gwhereJistheJacobianofthetransformation.

62. Bisa“conjugate”or“associate”tensoroftensorA.Whatthismeans?Answer:ItmeansthatBisobtainedfromAbyinnerproductmultiplicationofAbythecovariantorcontravariantmetrictensor.

63. Completeandjustifythefollowingstatement:“Thecomponentsofthemetrictensorareconstantsiff...etc.”.

Answer:“ThecomponentsofthemetrictensorareconstantsifftheChristoffelsymbolsofthespacemetricvanishidentically”.ThiscanbejustifiedbythemathematicaldefinitionsoftheChristoffelsymbolsofthefirstandsecondkind(whicharegiveninthetext)(seeFootnote18in§8↓)sincethederivatives(andhencetheChristoffelsymbols)mustvanishidenticallywhenthecomponentsofthemetrictensorareconstant.Similarly,whentheChristoffelsymbolsvanishidenticallytheninthegeneralcasethecomponentsofthemetrictensormustbeconstant(refertoExercise205↓in§5↓formoredetails).

64. Whatarethecovariantandabsolutederivativesofthemetrictensor?Answer:Themetrictensorislikeaconstantwithrespecttotensordifferentiationandhencethecovariantandabsolutederivativesofthemetrictensorarezeroinallcoordinatesystems.

65. AssuminganorthogonalcoordinatesystemofannDspace,completethefollowingequationswheretheindexedgrepresentsthemetrictensororitscomponents,i ≠ jinthesecondequationandthereisnosuminthethirdequation:det(gij) = ?gij = ?gii = ?Answer:det(gij) = 1 ⁄ (g11g22…gnn)

gij = 0

gii = 1 ⁄ gii66. Write,inmatrixform,thecovariantandcontravariantmetrictensorfor

orthonormalCartesian,cylindricalandsphericalcoordinatesystems.Whatdistinguishesallthesematrices?Explainandjustify.Answer:OrthonormalCartesianin3D:

Cylindrical:

Spherical:

Allthesematricesarediagonalbecauseallthesecoordinatesystemsareorthogonal.

67. ReferringtoQuestion664↑,whatistherelationbetweenthediagonalelementsofthesematricesandthescalefactorshiofthecoordinatesofthesesystems?Answer:Therelationis:gii = (hi)2 = 1 ⁄ gii(nosumoni)

wheregiiandgiiaretheithdiagonalelementsofthecovariantandcontravariantmetrictensorandhiisthescalefactoroftheithcoordinate.

68. ConsideringtheMinkowskimetric,isthespaceofthemechanicsofLorentztransformationsflatorcurved?Isithomogeneousornot?Whateffectthiscanhaveonthelengthofelementofarcds?Answer:ThespaceofthemechanicsofLorentztransformationsisflatbecausethemetrictensorisdiagonalwithallthediagonalelementsbeing + 1or − 1,butitisnothomogeneousbecausethemetrictensorisnottheunitytensor.Theeffectisthatthelengthofelementofarcdscanbeimaginary.

69. Derivethefollowingidentities:gim∂kgmj =  − gmj∂kgim

∂kgij =  − gmjgni∂kgnm

Answer:Wehavegimgmj = gji = δji.Ontakingthepartialderivativeofbothsidesofthisequationweobtain:∂k(gimgmj) = ∂kδji

gim∂kgmj + gmj∂kgim = 0

gim∂kgmj =  − gmj∂kgim

whichisthefirstrelation.WenotethatthesecondlineisjustifiedbytheproductruleofdifferentiationandthefactthatthecomponentsoftheKroneckerdeltatensorareconstants.Regardingthesecondrelation,westartfromthefirstrelationandhencewehave:gim∂kgmj =  − gmj∂kgim

gmj∂kgim =  − gim∂kgmj

gnj∂kgin =  − gin∂kgnj

gmjgnj∂kgin =  − gmjgin∂kgnj

gnm∂kgin =  − gmjgin∂kgnj

δnm∂kgin =  − gmjgin∂kgnj

∂kgim =  − gmjgin∂kgnj

∂kgij =  − gjmgin∂kgnm

∂kgij =  − gmjgni∂kgnm

whereinline2weexchangethetwosidesandmultiplythemby − 1,inline3werelabelmasn,inline4wemultiplybothsidesbygmj,inline5we

usethemetrictensorasanindexshiftingoperator,inline6weusetheidentitygnm = δnm,inline7weusetheKroneckerdeltaasanindexreplacementoperator,inline8weexchangethelabelsoftheindicesmandj,andinline9weusethesymmetryofthemetrictensor.

70. WhatisthedotproductofAandBwhereAisarank-2covarianttensorandBisacontravariantvector?Writethisoperationinstepsprovidingfulljustificationofeachstep.Answer:WehaveA = AijEiEjandB = BkEkandhence:A⋅B = 

(AijEiEj)⋅(BkEk) = 

AijBk(Ei⋅Ek)Ej = 

AijBkδikEj = 

AijBiEj

whereline1isbasedonthedefinitionofdotproductandthedefinitionofAandB,lines2and3arebasedonthedefinitionofdotproductofbasisvectorsandtherelationbetweenthemandtheKroneckerdelta,andline4isbasedonusingtheKroneckerdeltaasanindexreplacementoperator.Wecansimilarlyhave:A⋅B = 

(AijEiEj)⋅(BkEk) = 

AijBk(Ej⋅Ek)Ei = 

AijBkδjkEi = 

AijBjEi

wherethelinesaresimilarlyjustified.71. DeriveanexpressionforthemagnitudeofavectorAwhenAiscovariant

andwhenAiscontravariant.

Answer:IfAiscovariantthenA = AiEiandhence:|A| = 

√(A⋅A) = 

√([AiEi]⋅[AjEj]) = 

√([Ei⋅Ej]AiAj) = 

√(gijAiAj) = 

√(AjAj)

whereline1isadefinitionofthemagnitudeofavector,line2isthedefinitionofcovariantvector,lines3and4arebasedonthedefinitionofdotproductofbasisvectorsandtherelationbetweenthemandthemetrictensor,andline5isbasedonusingthemetrictensorasanindexshiftingoperator.IfAiscontravariantthenA = AiEiandhence:|A| = 

√(A⋅A) = 

√([AiEi]⋅[AjEj]) = 

√([Ei⋅Ej]AiAj) = 

√(gijAiAj) = 

√(AjAj)

wheretheselinesaresimilarlyjustifiedasinthefirstpart.Asseen,theseexpressionsareidentical.

72. Deriveanexpressionforthecosineoftheangleθbetweentwocovariantvectors,AandB,andbetweentwocontravariantvectorsCandD.Answer:Wehave:

cosθ = 

(A⋅B) ⁄ (|A||B|) = 

[AiEi⋅BjEj] ⁄ [√(AkEk⋅AlEl)√(BmEm⋅BnEn)] = 

[gijAiBj] ⁄ [√(gklAkAl)√(gmnBmBn)] = 

[AjBj] ⁄ [√(AlAl)√(BnBn)]

whereline1isadefinitionofthecosineofanglebetweentwovectors,line2isbasedonthedefinitionofcovariantvectorandthedefinitionofmagnitudeofavector,line3isbasedontherelationbetweenthedotproductofbasisvectorsandthemetrictensor,andline4isbasedonusingthemetrictensorasanindexshiftingoperator.

Similarly,wehave:cosθ = 

(C⋅D) ⁄ (|C||D|) = 

[CiEi⋅DjEj] ⁄ [√(CkEk⋅ClEl)√(DmEm⋅DnEn)] = 

[gijCiDj] ⁄ [√(gklCkCl)√(gmnDmDn)] = 

[CjDj] ⁄ [√(ClCl)√(DnDn)]

wheretheselinesaresimilarlyjustified.Asseen,theseexpressionsareidentical(apartfromthedifferentlabelsofvectors).

73. Whatisthemeaningoftheanglebetweentwointersectingsmoothcurves?Answer:Itistheanglebetweenthetangentvectorsofthesecurvesatthepointofintersection.

74. WhatisthecrossproductofAandBwherethesearecovariantvectors?Answer:Itis:A × B = εijkAiBjEk

whereεijkistheabsolutepermutationtensorinitscontravariantform.75. Completethefollowingequationsassumingageneralcoordinatesystemof

a3Dspace:Ei × Ej = ?Ei × Ej = ?Answer:Ei × Ej = εijkEk

Ei × Ej = εijkEk

whereεijkandεijkaretheabsolutepermutationtensorinitscovariantandcontravariantforms.

76. Definetheoperationsofscalartripleproductandvectortripleproductofvectorsusingtensorlanguageandassumingageneralcoordinatesystemofa3Dspace.Answer:ThescalartripleproductofcovariantvectorsA,BandCisgivenby:A⋅(B × C) = εijkAiBjCk

whereεijkisthecontravariantabsolutepermutationtensor.ThescalartripleproductofcontravariantvectorsA,BandCisgivenby:A⋅(B × C) = εijkAiBjCk

whereεijkisthecovariantabsolutepermutationtensor.ThevectortripleproductofacovariantvectorAandtwocontravariantvectorsBandCisgivenby:A × (B × C) = ϵilmϵjklAiBjCkEm

ThevectortripleproductofacontravariantvectorAandtwocovariantvectorsBandCisgivenby:A × (B × C) = ϵilmϵjklAiBjCkEm

77. Whatistherelationbetweentherelativeandabsolutepermutationtensors

intheircovariantandcontravariantforms?Answer:Thecovariantabsolutepermutationtensorεi1…inisequaltothecovariantrelativepermutationtensorϵi1…intimes√(g),thatis:εi1…in = √(g)ϵi1…in

wheregisthedeterminantofthecovariantmetrictensor.Thecontravariantabsolutepermutationtensorεi1…inisequaltothecontravariantrelativepermutationtensorϵi1…individedby√(g),thatis:εi1…in = [1 ⁄ √(g)]ϵi1…in

78. DefinethedeterminantofamatrixBintensornotationassumingageneralcoordinatesystemofa3Dspace.Answer:Itis:det(B) = [1 ⁄ (3!)]δijklmnBliBmjBnkwhereδijklmnisthegeneralizedKroneckerdeltaof3DspaceandBisrepresentedbyitsmixedform.

79. Derivetherelationforthelengthoflineelementingeneralcoordinatesystems:(ds)2 = gijduiduj.Howwillthisrelationbecomewhenthecoordinatesystemisorthogonal?Justifyyouranswer.Answer:Wehave:(ds)2 = 

dr⋅dr = 

Eidui⋅Ejduj = 

(Ei⋅Ej)duiduj = 

gijduiduj

whereristhepositionvectoringeneralcoordinates,EiandEjarecovariantbasisvectors,uiandujaregeneralcoordinatesandgijisthecovariantmetrictensor.Line1isbasedonthedefinitionofdsfromfirstprinciples,line2isbasedonthedefinitionofpositionvectorinitsinfinitesimaldifferentialform,line3isbasedonthedefinitionofdotproduct,andline4isbasedontherelationbetweenthedotproductofbasisvectorsandthemetrictensor.

Whenthecoordinatesystemisorthogonal,thisrelationbecomes:(ds)2 = Σi(hi)2duidui

wherehiisthescalefactoroftheithcoordinate.Thereasonisthatinorthogonalsystemsthemetrictensorisdiagonalwheretheithdiagonal(covariant)elementisgivenbygii = (hi)2,andhence:(ds)2 = 

gijduiduj = 

Σi(hi)2duidui

80. WritetheintegralrepresentingthelengthLofat-parameterizedspacecurveintermsofthemetrictensor.Answer:Theintegralis:L = ⨏t2t1√(gij[dui ⁄ dt][duj ⁄ dt])dt

wheret1andt2arethevaluesoftcorrespondingtothestartandendpointsofthecurverespectively,gijisthecovariantmetrictensorofthespaceanduiandujaregeneralcoordinatesrepresentingthepathofthecurve.Asusual,weareassumingthattheargumentofthesquarerootisnon-negativesincewearedealingwithrealvalues.

81. Usingtheequation(ds)2 = Σi(hi)2duiduiplusthescalefactors(whicharegiveninatableinthebook),developexpressionsfordsinorthonormalCartesian,cylindricalandsphericalcoordinatesystems.Answer:InorthonormalCartesiansystemswehave:h1 = h2 = h3 = 1.Hence:(ds)2 = 

Σi(hi)2duidui = 

(dx1)2 + (dx2)2 + (dx3)2 = 

(dx)2 + (dy)2 + (dz)2

Incylindricalsystemswehave:h1 = h3 = 1andh2 = ρ.Hence:

(ds)2 = 

Σi(hi)2duidui = 

(dρ)2 + ρ2(dφ)2 + (dz)2

Insphericalsystemswehave:h1 = 1,h2 = randh3 = rsinθ.Hence:(ds)2 = 

Σi(hi)2duidui = 

(dr)2 + r2(dθ)2 + r2sin2θ(dφ)282. Derivethefollowingformulafortheareaofadifferentialelementonthe

coordinatesurfaceui = constantina3Dspaceassumingageneralcoordinatesystem:dσ(ui = constant) = √(ggii)dujduk(i ≠ j ≠ k,nosumoni)Howwillthisrelationbecomewhenthecoordinatesystemisorthogonal?Answer:Wehave:dσ(ui = C) = 

|drj × drk| = 

|(∂r ⁄ ∂uj) × (∂r ⁄ ∂uk)|dujduk = 

|Ej × Ek|dujduk = 

|εjkiEi|dujduk = 

|εjki||Ei|dujduk = 

√(g)√(Ei⋅Ei)dujduk = 

√(g)√(gii)dujduk = 

√(ggii)dujduk

whereσrepresentsarea,Cisaconstant,drjanddrkaretheinfinitesimaldisplacementvectorsalongthejthandkthcoordinatecurvesthatdefinetheareaelementwhiletheothersymbolsareasdefinedpreviously.Weshouldalsoimposetheconditioni ≠ j ≠ kwithnosumoveriingii.Line1isbasedonthedefinitionofvectorcrossproductanditsrelationtotheareaoftheparallelogramthatisdefinedbythevectors,line2isthechainruleinmulti-variabledifferentiation,line3isbasedonthedefinitionofcovariantbasisvectorsastangentstothecoordinatecurves,line4isbasedonthedefinitionofcrossproductingeneralcoordinatesystems,line5isbasedontherulesofmodulus,line6isbasedontherelationεjki = √(g)ϵjkiandthefactthat|ϵjki| = 1(sincei ≠ j ≠ k)plusthedefinitionofthemagnitudeofavector,andline7isbasedontherelationbetweenthecontravariantbasisvectorsandtheelementsofthecontravariantmetrictensor.Inorthogonalcoordinatesystemsofa3Dspacewehave:√(ggii) = 

√((hi)2(hj)2(hk)2[1 ⁄ (hi)2]) = 

hjhk

(i ≠ j ≠ k,nosumonanyindex)andhencetheaboverelationbecomes:dσ(ui = C) = hjhkdujduk

wherei ≠ j ≠ kandnosumonjork.83. Usingtheequationdσ(ui = C) = hjhkdujdukplusthescalefactors(whichare

giveninatableinthebook),developexpressionsfordσonthecoordinatesurfacesinorthonormalCartesian,cylindricalandsphericalcoordinatesystems.Answer:InorthonormalCartesiansystemswehave:h1 = h2 = h3 = 1.Hence:dσ(x = C) = dydz

dσ(y = C) = dxdz

dσ(z = C) = dxdy

Incylindricalsystemswehave:h1 = h3 = 1andh2 = ρ.Hence:dσ(ρ = C) = ρdφdz

dσ(φ = C) = dρdz

dσ(z = C) = ρdρdφ

Insphericalsystemswehave:h1 = 1,h2 = randh3 = rsinθ.Hence:dσ(r = C) = r2sinθdθdφ

dσ(θ = C) = rsinθdrdφ

dσ(φ = C) = rdrdθ84. Derivethefollowingformulaforthevolumeofadifferentialelementofa

solidbodyina3Dspaceassumingageneralcoordinatesystem:dτ = √(g)du1du2du3Howwillthisrelationbecomewhenthecoordinatesystemisorthogonal?Answer:Wehave:dτ = 

|dr1⋅(dr2 × dr3)| = 

|(∂r ⁄ ∂u1)⋅([∂ r ⁄ ∂u2] × [∂ r ⁄ ∂u3])|du1du2du3 = 

|E1⋅( E2 ×  E3)|du1du2du3 = 

|E1⋅ε231 E1|du1du2du3 = 

|E1⋅ E1||ε231|du1du2du3 = 

|δ11||ε231|du1du2du3 = 

√(g)du1du2du3

whereτrepresentsvolume,dr1, dr2, dr3arethethreeinfinitesimal

displacementvectorsthatdefinetheelementofvolumealongthecorrespondingcoordinatecurves,gisthedeterminantofthecovariantmetrictensorgijandtheothersymbolsareasdefinedpreviously.Line1isbasedonthedefinitionofscalartripleproductofvectorsanditsrelationtothevolumeoftheparallelepipedthatisdefinedbythevectors,line2isthechainruleinmulti-variabledifferentiation,line3isbasedonthedefinitionofcovariantbasisvectorsastangentstothecoordinatecurves,line4isbasedonthedefinitionofcrossproductingeneralcoordinatesystems,line5isbasedontherulesofmodulus,line6isbasedontherelationbetweenthebasisvectorsandtheKroneckerdelta,andline7isbasedontherelationε231 = √(g)ϵ231andthefactthat|ϵ231| = 1plusthefactthat|δ11| = 1.Inorthogonalcoordinatesystemsofa3Dspacewehave√(g) = h1h2h3andhencetheaboverelationbecomes:dτ = h1h2h3du1du2du3

85. Makeaplotrepresentingthevolumeofaninfinitesimalelementofasolidbodyina3Dspaceasthemagnitudeofascalartripleproductofthreevectors.Answer:TheplotshouldlooksimilartoFigure7↓.

Figure7 Thevolumeofaninfinitesimalelementofasolidbodyina3DspaceintheneighborhoodofagivenpointPasthemagnitudeofthescalartripleproductoftheinfinitesimaldisplacementvectorsinthedirectionsofthethreecoordinatecurvesatP,dr1,dr2anddr3.

86. UsetheexpressionofthevolumeelementingeneralcoordinatesystemsofnDspacestofindtheformulaforthevolumeelementinorthogonalcoordinatesystems.Answer:ThegeneralizedvolumeelementdτingeneralcoordinatesystemsofnDspacesisgivenbytheformula:dτ = √(g)du1…dun

wheregisthedeterminantofthecovariantmetrictensorgijandtheindexeduaregeneralcoordinates.Inorthogonalcoordinatesystemswehave√(g) = h1⋯hnandhencetheformulabecomes:dτ = h1⋯hndu1…dun

withnosummationovern.87. Usingtheequationdτ = h1h2h3du1du2du3plusthescalefactors(whichare

giveninatableinthebook),developexpressionsfordτinorthonormalCartesian,cylindricalandsphericalcoordinatesystems.Answer:InorthonormalCartesiansystemswehave:h1 = h2 = h3 = 1.Hence:dτ = dxdydz

Incylindricalsystemswehave:h1 = h3 = 1andh2 = ρ.Hence:dτ = ρdρdφdz

Insphericalsystemswehave:h1 = 1,h2 = randh3 = rsinθ.Hence:dτ = r2sinθdrdθdφ

Chapter5TensorDifferentiation1. Whytensordifferentiation(representedbycovariantandabsolute

derivatives)isneededingeneralcoordinatesystemstoreplacetheordinarydifferentiation(representedbypartialandtotalderivatives)?Answer:Becausethebasisvectorsinthegeneralcoordinatesystemsarecoordinatedependentandhencethedifferentiationoftensorsshouldextendtothebasisvectorsandnotrestrictedtothecomponentsandthisresultsinthetensordifferentiation,i.e.covariantandabsolutedifferentiation.Inbrief,tensordifferentiationisthedifferentiationofatensorinitstwoparts(i.e.thecomponentsandthebasisvectorstowhichthesecomponentsarereferred)sinceboththesepartsarecoordinatedependentandhencetheyarevariableandshouldbesubjecttotheprocessofdifferentiation.

2. Showthatingeneralcoordinatesystems,theordinarydifferentiationofthecomponentsofnon-scalartensorswithrespecttothecoordinateswillnotproduceatensoringeneral.Answer:LethaveatensorAoftype(m, n)coordinatedbyageneralcoordinatesystemandhenceA = Ai1, ⋯, im

j1, ⋯, jnEi1⋯ EimEj1⋯ EjnwherebothAi1, ⋯, im

j1, ⋯, jnandEi1⋯ EimEj1⋯ Ejnarevariablesthatdependoncoordinates.Now,letdifferentiatethistensorwithrespecttothekthcoordinateusingtheproductruleofdifferentiation,thatis:∂kA = (∂kAi1, ⋯, im

j1, ⋯, jn)Ei1⋯ EimEj1⋯ Ejn + Ai1, ⋯, imj1, ⋯, jn∂k(Ei1⋯ 

EimEj1⋯ Ejn)

Aswesee,theordinarydifferentiationisvalidonlyifthesecondtermontherighthandsidevanishesidenticallywhichisnotthecaseingeneralcoordinatesystemssincethebasisvectorsinthesesystemsarenotconstant.Accordingly,theordinarydifferentiationisjustpartoftensordifferentiationandhenceitwillnotproduceatensoringeneralsinceatensorisproducedonlybythefulldifferentiationprocessofatensor.

3. “TheChristoffelsymbolsareaffinetensorsbutnottensors”.Explainandjustifythisstatement.Answer:Itmeansthatthesesymbolstransformliketensorsinaffine

coordinatesystemsbutnotingeneralcoordinatesystems.Thejustificationisthat:lettransformaChristoffelsymbolofthefirstkindfromonegeneralsystem(unbarred)toanothergeneralsystem(barred)toseeifitwilltransformlikeatensorornot.Toeasethenotation,letuselowercaseindicesfortheoriginalsystem(unbarred)anduppercaseindicesforthetransformedsystem(barred).Accordingly,thecovariantmetrictensoristransformedbythefollowingequation:(1)gIJ = 

grs(∂ur ⁄ ∂uI)(∂us ⁄ ∂uJ) = 

grs∂Iur∂Jus

TheChristoffelsymbolsofthefirstkindinthebarredsystemaredefinedas:(2)[IJ, K] = (1 ⁄ 2)(∂JgKI + ∂IgJK − ∂KgIJ)So,lettakethepartialderivativeofbothsidesofEq.1↑withrespecttoindexKandsubstituteinEq.2↑,thatis:∂KgIJ = 

∂K(grs∂Iur∂Jus) = 

∂Kgrs∂Iur∂Jus + grs∂IKur∂Jus + grs∂Iur∂JKus = 

∂tgrs∂Kut∂Iur∂Jus + grs∂IKur∂Jus + grs∂Iur∂JKus

wherethelaststepisjustifiedbythechainrule.Byusingcyclicreplacementofindicesinthelastequation,weobtain:∂JgKI = ∂sgtr∂Jus∂Kut∂Iur + gtr∂KJut∂Iur + gtr∂Kut∂IJur

∂IgJK = ∂rgst∂Iur∂Jus∂Kut + gst∂JIus∂Kut + gst∂Jus∂KIut

Onrelabelingthedummyindicestounifythenotationandnotingthecommutativityofpartialdifferentialoperators(i.e.∂i∂j = ∂j∂i),weobtainthefollowingrelations:∂KgIJ = ∂tgrs∂Kut∂Iur∂Jus + grs∂IKur∂Jus + grs∂Iur∂JKus

∂JgKI = ∂tgrs∂Jut∂Kur∂Ius + grs∂JKur∂Ius + grs∂Kur∂IJus

∂IgJK = ∂tgrs∂Iut∂Jur∂Kus + grs∂IJur∂Kus + grs∂Jur∂IKus

OnsubstitutingfromtheseexpressionsintoEq.2↑(markingcanceledtermswithsimilartypeofbracketsandnotingthesymmetryofthemetrictensorandtheinsignificanceofthelabelsofdummyindices)weobtain:2[IJ, K] = 

∂JgKI + ∂IgJK − ∂KgIJ = 

∂tgrs∂Jut∂Kur∂Ius + (grs∂JKur∂Ius) + grs∂Kur∂IJus + ∂tgrs∂Iut∂Jur∂Kus + grs∂IJur∂Kus + [grs∂Jur∂IKus] − ∂tgrs∂Kut∂Iur∂Jus − [grs∂IKur∂Jus] − (grs∂Iur∂JKus) = 

∂tgrs∂Jut∂Kur∂Ius + ∂tgrs∂Iut∂Jur∂Kus − ∂tgrs∂Kut∂Iur∂Jus + grs∂Kur∂IJus + grs∂IJur∂Kus

Onrelabelingthedummyindicesinthelastequationnotingthesymmetryofthemetrictensorweget:2[IJ, K] = 

∂tgrs∂Jut∂Kur∂Ius + ∂sgtr∂Ius∂Jut∂Kur − ∂rgst∂Kur∂Ius∂Jut + grs∂Kur∂IJus + grs∂IJus∂Kur = 

(∂tgrs + ∂sgtr − ∂rgst)∂Jut∂Kur∂Ius + 2grs∂Kur∂IJus = 

2[st, r]∂Jut∂Kur∂Ius + 2grs∂Kur∂IJus

Ondividingbothsidesby2weobtain:[IJ, K] = 

[st, r]∂Jut∂Kur∂Ius + grs∂Kur∂IJus = 

[st, r](∂ut ⁄ ∂uJ)(∂ur ⁄ ∂uK)(∂us ⁄ ∂uI) + grs(∂ur ⁄ ∂uK)(∂2us ⁄ ∂uI∂uJ)

i.e.theChristoffelsymbolofthefirstkindtransformslikeatensorbutwithanaddedterm(i.e.thesecondtermontherighthandsideofthelastequation).Hence,theChristoffelsymbolofthefirstkindtransformslikeatensoronlyifthesecondtermvanishesidenticallyandthisistrueonlyinaffinesystemswherethesecondpartialderivativeiszero,i.e.theChristoffelsymbolsofthefirstkindareaffinetensorsbutnotgeneraltensors.TheaboveargumentcanbeeasilyextendedtotheChristoffelsymbolsofthesecondkindsincetheyareobtainedfromtheChristoffelsymbolsofthefirstkindbyraisinganindexandhenceiftheoriginal“tensor”isonlyanaffinetensorandnotageneraltensorthentheraised“tensor”shouldalsobeanaffinetensorandnotageneraltensorbecausetheoperationofraisingindicesdoesnotchangethenatureoftensor.

4. WhatisthedifferencebetweenthefirstandsecondkindsoftheChristoffelsymbols?Answer:Theydifferinanindexwhereitiscovariantinthefirstkindandcontravariantinthesecondkind.

5. ShowthattheChristoffelsymbolsofbothkindsarenotgeneraltensorsbygivingexamplesofthesesymbolsbeingvanishinginsomesystemsbutnotinothersystemsandconsideringtheuniversalityofthezerotensor.Answer:Asweestablishedearlier,thezerotensorisinvariantacrossallcoordinatesystemsandhenceifatensoriszeroinonesystemitmustbezeroinallothersystems.Accordingly,iftheChristoffelsymbolsofbothkindsaretensorsthenwhentheyvanishinonecoordinatesystemtheymustvanishinallothersystems.However,thisisnottruesincetheChristoffelsymbolsofbothkindsvanishinCartesiancoordinatesystemsbutnotincylindricalorsphericalcoordinatesystemsforinstanceandhencetheycannotbetensorsunconditionally,i.e.theyarenotgeneraltensors.However,sincetheyvanishacrossallaffinesystems(andhencetheyareinvariantacrossallthesesystems)thentheyshouldbeaffinetensors.

6. StatethemathematicaldefinitionsoftheChristoffelsymbolsofthefirstandsecondkinds.Howthesetwokindsareobtainedfromeachother?Answer:TheChristoffelsymbolsofthefirstandsecondkindaredefinedrespectivelyby:[ij, k] = (1 ⁄ 2)(∂jgik + ∂igjk − ∂kgij)

Γkij = (gkl ⁄ 2)(∂jgil + ∂igjl − ∂lgij)

wheretheindexedgarethecovariantandcontravariantformsofthemetrictensor.Asseen,thesecondkindisobtainedfromthefirstkindbyraisinganindexwhilethefirstkindisobtainedfromthesecondkindbyloweringanindex.

7. WhatisthesignificanceoftheChristoffelsymbolsbeingsolelydependentonthecoefficientsofthemetrictensorintheirrelationtotheunderlyingspaceandcoordinatesystem?Answer:ThesoledependencyoftheChristoffelsymbolsonthecoefficientsofthemetrictensormeansthattheyarevariablefunctionsofthecoordinatesofthespacelikethecoefficientsofthemetrictensor.Hence,whenthecoefficientsareconstants(i.e.independentofcoordinates)theChristoffelsymbolswillvanishidenticallyoverthewholespace.Inthiscase,theChristoffelsymbolswillreflectapropertyofthespace(i.e.beingflat)andapropertyofthesystem(i.e.beingaffineorrectilinear).

8. DotheChristoffelsymbolsrepresentapropertyofthespace,apropertyofthecoordinatesystem,orapropertyofboth?Answer:Asdiscussedearlier,themetrictensorcontainsessentialinformationaboutthegeometricnatureofthespaceandhenceitisapropertyofthespaceanddependsonthenatureofthespace.However,themetrictensoralsodependsinformontheemployedsystemforcoordinatingthespaceandhenceitcanalsoberegardedasapropertyofthecoordinatesystem.Accordingly,thedependencyoftheChristoffelsymbolsonthecoefficientsofthemetrictensormeansthattheyareacharacteristicpropertyofboththeunderlyingspaceandtheemployedcoordinatesystemandhencetheyreflectcertainfeaturesofthegeometricnatureofthespaceaswellascertainfeaturesoftheemployedcoordinatesystem.

9. IfsomeoftheChristoffelsymbolsvanishinaparticularcurvilinearcoordinatesystem,shouldthesesomenecessarilyvanishinothercurvilinearcoordinatesystems?Justifyyouranswerbygivingsomeexamples.Answer:No.Forexample,incylindricalsystemswehaveonly3non-vanishingChristoffelsymbolsbutinsphericalsystemswehave9non-vanishingsymbolsandhencesomeofthevanishingsymbolsincylindricalsystems,suchas[33, 1],correspondtonon-vanishingsymbolsinsphericalsystems.

10. VerifythattheChristoffelsymbolsofthefirstandsecondkindaresymmetricintheirpairedindicesbyusingtheirmathematicaldefinitions.Answer:TheChristoffelsymbolsofthefirstkindaregivenby:

[ij, l] = (1 ⁄ 2)(∂jgil + ∂igjl − ∂lgij)

Onshiftingthepairedindicesweobtain:[ji, l] = 

(1 ⁄ 2)(∂igjl + ∂jgil − ∂lgji) = 

(1 ⁄ 2)(∂jgil + ∂igjl − ∂lgji) = 

(1 ⁄ 2)(∂jgil + ∂igjl − ∂lgij) = 

[ij, l]

whereinline2wejustexchangedthefirstandsecondterms,whileline3isjustifiedbythesymmetryofthemetrictensor.Similarly,theChristoffelsymbolsofthesecondkindaregivenby:Γkij = (gkl ⁄ 2)(∂jgil + ∂igjl − ∂lgij)

Onshiftingthepairedindicesweobtain:Γkji = 

(gkl ⁄ 2)(∂igjl + ∂jgil − ∂lgji) = 

(gkl ⁄ 2)(∂jgil + ∂igjl − ∂lgji) = 

(gkl ⁄ 2)(∂jgil + ∂igjl − ∂lgij) = 

Γkij

whereinline2wejustexchangedthefirstandsecondterms,whileline3isjustifiedbythesymmetryofthemetrictensor.

11. Correct,ifnecessary,thefollowingequations:∂jEi =  − ΓkijEk

∂jEi =  − ΓimjEm

Answer:Thesecondequationiscorrectwhilethefirstequationshouldbecorrectedbyremovingtheminussign.

12. Whatisthesignificanceofthefollowingequations?Ek⋅∂jEi = Γkij

Ek⋅∂jEi =  − Γikj

Ek⋅∂jEi = [ij, k]

Answer:ThefirstandsecondequationsmeanthattheChristoffelsymbolsofthesecondkindaretheprojectionsofthepartialderivativeofthebasisvectorsinthedirectionofthebasisvectorsoftheoppositevariancetype,whilethethirdequationmeansthattheChristoffelsymbolsofthefirstkindaretheprojectionsofthepartialderivativeofthecovariantbasisvectorsinthedirectionofthebasisvectorsofthesamevariancetype.

13. Derivethefollowingrelationsgivingfullexplanationofeachstep:∂jgil = [ij, l] + [lj, i]

Γjji = ∂i(ln√(g))

Answer:Regardingthefirstrelation,fromthedefinitionoftheChristoffelsymbolsofthefirstkindwehave:[ij, l] = (1 ⁄ 2)(∂jgil + ∂igjl − ∂lgij)

[lj, i] = (1 ⁄ 2)(∂jgli + ∂lgji − ∂iglj)

wherethesecondequationisobtainedfromthefirstbyexchangingiandl.Onaddingthetwosidesoftheseequationsweobtain:[ij, l] + [lj, i] = 

(1 ⁄ 2)(∂jgil + ∂igjl − ∂lgij + ∂jgli + ∂lgji − ∂iglj) = 

(1 ⁄ 2)(∂jgil + ∂igjl − ∂lgij + ∂jgil + ∂lgij − ∂igjl) = 

(1 ⁄ 2)(∂jgil + ∂jgil) = 

∂jgil

whichistherequiredresult.Wenotethatline2isjustifiedbythesymmetryofthemetrictensor.

Regardingthesecondrelation,theChristoffelsymbolsofthesecondkindaregivenby:Γkij = 

Γkji = 

(gkl ⁄ 2)(∂jgil + ∂igjl − ∂lgij)

wherethesymmetryiniandj(accordingtothefirstequality)wasjustifiedinExercise105↑.Oncontractingkwithjweobtain:Γjji = 

(gjl ⁄ 2)(∂jgil + ∂igjl − ∂lgij) = 

(1 ⁄ 2)(gjl∂jgil + gjl∂igjl − gjl∂lgij) = 

(1 ⁄ 2)(glj∂lgij + gjl∂igjl − gjl∂lgij) = 

(1 ⁄ 2)(gjl∂lgij + gjl∂igjl − gjl∂lgij) = 

(1 ⁄ 2)gjl∂igjl = 

[1 ⁄ (2g)]ggjl∂igjl = 

[1 ⁄ (2g)]∂ig = 

(1 ⁄ 2)∂i(lng) = 

∂i[(1 ⁄ 2)lng] = 

∂i(ln√(g))

whichistherequiredresult.Wenotethatinline3werelabeledthedummyindicesjandlinthefirstterm,inline4weusedthesymmetryofthemetrictensor(i.e.glj = gjl),inline7weusedtheexpressionofthederivativeofthedeterminantgofthecovariantmetrictensorgij(seeExercise175↓),inline8weusedtheruleofdifferentiationofnaturallogarithm,andinline10weusedtheruleofpoweroflogarithm.

14. Assuminganorthogonalcoordinatesystem,verifythefollowingrelation:[ij, k] = 0wherei ≠ j ≠ k.Answer:Inorthogonalcoordinatesystemsthemetrictensorisdiagonalandhencegmn = 0whenm ≠ n.FromthedefinitionoftheChristoffelsymbolsofthefirstkindwehave:[ij, k] = (1 ⁄ 2)(∂jgik + ∂igjk − ∂kgij)

Therefore,wheni ≠ j ≠ kthenwehavegik = gjk = gij = 0andhence[ij, k] = 0.

15. Assuminganorthogonalcoordinatesystem,verifythefollowingrelation:Γiji = (1 ⁄ 2)∂jlngiiwithnosumoveri.Answer:AccordingtothedefinitionoftheChristoffelsymbolsofthesecondkindwehave:Γijk = gil[jk, l]

Inorthogonalcoordinatesystemsthemetrictensorisdiagonalandhencegil = 0wheni ≠ l.Therefore,theaboveequationbecomes:Γijk = gii[jk, i]

withnosumoveri.Moreover,inorthogonalsystemswehave:gii = 1 ⁄ gii(nosumoveri)

andhence:Γijk = 

[jk, i] ⁄ gii = 

[1 ⁄ (2gii)](∂kgij + ∂jgki − ∂igjk)

wherethedefinitionoftheChristoffelsymbolsofthefirstkindisusedinthesecondequality.Onunifyingkwithiweobtain:Γiji = 

[1 ⁄ (2gii)](∂igij + ∂jgii − ∂igji) = 

[1 ⁄ (2gii)](∂igij + ∂jgii − ∂igij) = 

[1 ⁄ (2gii)]∂jgii = 

(1 ⁄ 2)∂jlngii

(withnosumoveri)whichistherequiredresult.Wenotethatline2isbasedonthesymmetryofthemetrictensorandline4isbasedontheruleofdifferentiationofnaturallogarithm.

16. ConsideringtheidenticalityanddifferenceoftheindicesoftheChristoffelsymbolsofeitherkind,howmanycaseswehave?Listthesecases.Answer:Wehave4maincases:(a)Alltheindicesareidentical.(b)Onlytwonon-pairedindicesareidentical(seeFootnote19in§8↓).(c)Onlythetwopairedindicesareidentical.(d)Alltheindicesaredifferent.

17. ProvethefollowingrelationwhichisusedinExercise135↑:∂ig = ggjl∂igjl.Answer(seeFootnote20in§8↓):Accordingtothestandarddefinitionofdeterminant(whichisgiveninanylinearalgebratext),thedeterminantgofthecovariantmetrictensorgijisgivenby:g = giaGia

whereGiaisthecofactoroftheentrygiaandsummationconventionappliestoaonlysinceirepresentsagivenrow(orcolumn).Ontakingthepartialderivativeofthetwosidesofthelastequationwithrespecttotheentrygijandusingtheproductruleofdifferentiationweobtain:∂g ⁄ ∂gij = Gia(∂gia ⁄ ∂gij) + gia(∂Gia ⁄ ∂gij)

Now,accordingtothedefinitionofthecofactorGiaoftheentrygia,Gia

containsnogij(i.e.Giaisindependentofgij)andhencethesecondtermiszerobecausethepartialderivativeiszero,thatis:∂g ⁄ ∂gij = Gia(∂gia ⁄ ∂gij)

Moreover,sincetheentriesofarow(oracolumn)areindependentofeachother,weshouldhave(seeFootnote21in§8↓):∂gia ⁄ ∂gij = δjaTherefore,wehave:∂g ⁄ ∂gij = 

Giaδja = 

Gij

Bythechainruleofdifferentiationwealsohave:∂g ⁄ ∂xi = 

(∂g ⁄ ∂gab)(∂gab ⁄ ∂xi) = 

Gab(∂gab ⁄ ∂xi)

Now,thecontravariantmetrictensoristheinverseofthecovariantmetrictensor,andhencefromthedefinitionofmatrixinverseweshouldhave(notingthesymmetryofthemetrictensor):gab = Gab ⁄ gOncomparingthelasttwoequationsweconclude:∂g ⁄ ∂xi = ggab(∂gab ⁄ ∂xi)

Onrelabelingthedummyindicesandusingshorthandnotation,weobtain:∂ig = ggjl∂igjl

whichistherequiredresult.18. Inorthogonalcoordinatesystemsofa3Dspacethenumberofindependent

non-identicallyvanishingChristoffelsymbolsofeitherkindisonly15.Explainwhy.Answer:Ina3Dspacewehave27Christoffelsymbolsofeitherkind

representingallthepossiblepermutationsofthethreeindicesincludingtherepetitiveones.InorthogonalcoordinatesystemstheChristoffelsymbolsofeitherkindvanishwhentheindicesarealldifferent.ThiswasestablishedinExercise145↑forthefirstkind,andwhenthefirstkindvanishesthesecondkindalsovanishesaccordingtothedefinitionofthesecondkindwhichwasgivenearlier.Hence,outofatotalof27symbols,only21non-identicallyvanishingsymbolsareleftsincethe6non-repetitivepermutationsaredropped.Now,sincetheChristoffelsymbolsaresymmetricintheirpairedindices,thenonly15independentnon-identicallyvanishingsymbolswillremainsince6otherpermutationsrepresentingthesesymmetricexchangesarealsodroppedbecausetheyarenotindependent.

19. VerifythefollowingequationsrelatedtotheChristoffelsymbolsinorthogonalcoordinatesystemsina3Dspace:[12, 1] = h1h1, 2

Γ323 = h3, 2 ⁄ h3Answer:Thefirstequationisverifiedinthemaintextandhence,insteadofrepeatingweverifyanotherentry,say[22, 1] =  − h2h2, 1,thatis:[22, 1] = 

 − (1 ⁄ 2)∂1g22 = 

 − (1 ⁄ 2)∂1(h2)2 = 

 − (1 ⁄ 2)2h2∂1h2 = 

 − h2∂1h2 = 

 − h2h2, 1

whereline1isjustifiedbytherelation[ii, j] =  − (1 ⁄ 2)∂jgiiinorthogonalsystems(i ≠ j,nosumoni)whichisgiveninthemaintext,line2isjustifiedbytherelationgii = (hi)2inorthogonalsystems(nosumoni),line3isjustifiedbytherulesofdifferentiation,andlines4and5aresimplealgebraicmanipulationandnotation.Regardingthesecondequation,wehave:

Γ323 = 

[1 ⁄ (2g33)]∂2g33 = 

[1 ⁄ {2(h3)2}]∂2(h3)2 = 

[{2h3} ⁄ {2(h3)2}]∂2h3 = 

(1 ⁄ h3)∂2h3 = 

h3, 2 ⁄ h3

whereline1isjustifiedbytherelationΓiji = [1 ⁄ (2gii)]∂jgiiinorthogonalsystems(nosumoni)whichisgiveninthemaintext,line2isjustifiedbytherelationgii = (hi)2inorthogonalsystems(nosumoni),line3isjustifiedbytherulesofdifferentiation,andlines4and5aresimplealgebraicmanipulationandnotation.

20. Justifythefollowingstatement:“Inanycoordinatesystem,alltheChristoffelsymbolsofeitherkindvanishidenticallyiffallthecomponentsofthemetrictensorinthegivencoordinatesystemareconstants”.Answer:FromthedefinitionoftheChristoffelsymbols,i.e.[ij, k] = (1 ⁄ 2)(∂jgik + ∂igjk − ∂kgij)

Γkij = (gkl ⁄ 2)(∂jgil + ∂igjl − ∂lgij)

wecanseethatbothkindsaresumoftermscontainingpartialderivativesofcomponentsofthemetrictensor.Therefore,ifallthecomponentsofthemetrictensorareconstantsthenallthesepartialderivativeswillvanishidenticallyandhencetheChristoffelsymbolswillalsovanishidentically.Similarly,ifweconsiderthegeneralcaseofcurvilinearsystemsthenwhentheChristoffelsymbolsvanishidenticallythentheindividualpartialderivativesmustvanishidenticallyandhenceallthecomponentsofthemetrictensormustbeconstants.Inotherwords,iftheChristoffelsymbolsofthefirstkindvanishedidenticallybuttheindividualpartialderivativesdidnotvanishidenticallythenweshouldhave:∂jgik + ∂igjk = ∂kgij

whichcannotbetrueingeneralsincethecomponentsofthemetrictensorintheaboveequationareindependentofeachother.ThisargumentsimilarlyappliestotheChristoffelsymbolsofthesecondkind,ascanbeseenfromtheabovedefinitionofΓkij(notingalsothatgklcannotvanishidentically)(seeFootnote22in§8↓).

21. UsingthedefinitionoftheChristoffelsymbolsofthefirstkindwiththemetrictensorofthecylindricalcoordinatesystem,findtheChristoffelsymbolsofthefirstkindcorrespondingtotheEuclideanmetricofcylindricalcoordinatesystems.Answer:Asexplainedearlier(seeExercise185↑),weshouldhave27symbols.However,becausecylindricalcoordinatesystemsareorthogonalthenweshouldhaveonly15independentnon-identicallyvanishingsymbolswhichare(seeFootnote23in§8↓):[11, 1] = (1 ⁄ 2)(∂1g11 + ∂1g11 − ∂1g11) = (1 ⁄ 2)(∂ρ1 + ∂ρ1 − ∂ρ1) = 0

[11, 2] = (1 ⁄ 2)(∂1g12 + ∂1g12 − ∂2g11) = (1 ⁄ 2)(∂ρ0 + ∂ρ0 − ∂φ1) = 0

[11, 3] = (1 ⁄ 2)(∂1g13 + ∂1g13 − ∂3g11) = (1 ⁄ 2)(∂ρ0 + ∂ρ0 − ∂z1) = 0

[12, 1] = [21, 1] = (1 ⁄ 2)(∂2g11 + ∂1g21 − ∂1g12) = (1 ⁄ 2)(∂φ1 + ∂ρ0 − ∂ρ0) = 0

[12, 2] = [21, 2] = 

(1 ⁄ 2)(∂2g12 + ∂1g22 − ∂2g12) = (1 ⁄ 2)(∂φ0 + ∂ρρ2 − ∂φ0) = ρ

[13, 1] = [31, 1] = (1 ⁄ 2)(∂3g11 + ∂1g31 − ∂1g13) = (1 ⁄ 2)(∂z1 + ∂ρ0 − ∂ρ0) = 0

[13, 3] = [31, 3] = (1 ⁄ 2)(∂3g13 + ∂1g33 − ∂3g13) = (1 ⁄ 2)(∂z0 + ∂ρ1 − ∂z0) = 0

[22, 1] = (1 ⁄ 2)(∂2g21 + ∂2g21 − ∂1g22) = (1 ⁄ 2)(∂φ0 + ∂φ0 − ∂ρρ2) =  − ρ

[22, 2] = (1 ⁄ 2)(∂2g22 + ∂2g22 − ∂2g22) = (1 ⁄ 2)(∂φρ2 + ∂φρ2 − ∂φρ2) = 0

[22, 3] = (1 ⁄ 2)(∂2g23 + ∂2g23 − ∂3g22) = (1 ⁄ 2)(∂φ0 + ∂φ0 − ∂zρ2) = 0

[23, 2] = [32, 2] = (1 ⁄ 2)(∂3g22 + ∂2g32 − ∂2g23) = (1 ⁄ 2)(∂zρ2 + ∂φ0 − ∂φ0) = 0

[23, 3] = [32, 3] = (1 ⁄ 2)(∂3g23 + ∂2g33 − ∂3g23) = (1 ⁄ 2)(∂z0 + ∂φ1 − ∂z0) = 0

[33, 1] = (1 ⁄ 2)(∂3g31 + ∂3g31 − ∂1g33) = (1 ⁄ 2)(∂z0 + ∂z0 − ∂ρ1) = 0

[33, 2] = (1 ⁄ 2)(∂3g32 + ∂3g32 − ∂2g33) = (1 ⁄ 2)(∂z0 + ∂z0 − ∂φ1) = 0

[33, 3] = (1 ⁄ 2)(∂3g33 + ∂3g33 − ∂3g33) = (1 ⁄ 2)(∂z1 + ∂z1 − ∂z1) = 0

whiletheremaining6symbolsareidenticallyzero.22. GivealltheChristoffelsymbolsofthefirstandsecondkindofthe

followingcoordinatesystems:orthonormalCartesian,cylindricalandspherical.Answer:OrthonormalCartesian:allsymbolsofbothkindsarezero.

Cylindrical:allsymbolsofthefirstkindarezeroexceptthefollowing:[22, 1] =  − ρ

[12, 2] = [21, 2] = ρ

Allsymbolsofthesecondkindarezeroexceptthefollowing:Γ122 =  − ρ

Γ212 = Γ221 = 1 ⁄ ρ

Spherical:allsymbolsofthefirstkindarezeroexceptthefollowing:[22, 1] =  − r

[33, 1] =  − rsin2θ

[12, 2] = [21, 2] = r

[33, 2] =  − r2sinθcosθ

[13, 3] = [31, 3] = rsin2θ

[23, 3] = [32, 3] = r2sinθcosθ

Allsymbolsofthesecondkindarezeroexceptthefollowing:Γ122 =  − r

Γ133 =  − rsin2θ

Γ212 = Γ221 = 1 ⁄ r

Γ233 =  − sinθcosθ

Γ313 = Γ331 = 1 ⁄ r

Γ323 = Γ332 = cotθ23. MentiontwoimportantpropertiesoftheChristoffelsymbolsofeitherkind

withregardtotheorderandsimilarityoftheirindices.Answer:Onepropertyisthesymmetryofthesesymbolsintheirpairedindices,i.e.[ij, k] = [ji, k]andΓkij = Γkji.Anotherpropertyisthatinorthogonalsystemsthesesymbolsvanishidenticallywhenalltheirindicesaredifferent,i.e.[ij, k] = 0andΓkij = 0wheni ≠ j ≠ k.

24. UsingtheanalyticalexpressionsoftheChristoffelsymbolsofthesecondkindinorthogonalsystemsplustheentriesofthetableofscalefactors(whichisgiveninthebook)andthepropertiesoftheChristoffelsymbolsof

thesecondkind,derivethesesymbolscorrespondingtothemetricsofthecoordinatesystemsofQuestion225↑.Answer:OrthonormalCartesian:allthescalefactorsare1.Moreover,alltheanalyticalexpressionsoftheChristoffelsymbolsofthesecondkindinorthogonalsystemscontainderivativesofthescalefactorsandthesederivativesmustbezero.Hence,allthesesymbolsarezero.

Cylindrical:thescalefactorsareh1 = h3 = 1andh2 = ρ.Hence,allthesesymbolsmustbezeroexceptthosewhoseanalyticalexpressioncontainsderivativeofh2withrespecttoρ,i.e.h2, 1,thatis:Γ212 = h2, 1 ⁄ h2 = (∂ρρ) ⁄ ρ = 1 ⁄ ρ = Γ221

Γ122 =  − [h2h2, 1] ⁄ [(h1)2] =  − (ρ∂ρρ) ⁄ 12 =  − ρ

Spherical:thescalefactorsareh1 = 1,h2 = randh3 = rsinθ.Hence,allthesesymbolsmustbezeroexceptthosewhoseanalyticalexpressioncontainsderivativeofh2withrespecttor(i.e.h2, 1)orderivativeofh3withrespecttor(i.e.h3, 1)orderivativeofh3withrespecttoθ(i.e.h3, 2),thatis:Γ212 = h2, 1 ⁄ h2 = (∂rr) ⁄ r = 

24. 1 ⁄ r = Γ221

Γ122 =  − (h2h2, 1) ⁄ (h1)2 =  − (r∂rr) ⁄ 12 =  − r

Γ313 = h3, 1 ⁄ h3 = [∂r(rsinθ)] ⁄ [rsinθ] = (sinθ) ⁄ (rsinθ) = 1 ⁄ r = Γ331

Γ133 =  − (h3h3, 1) ⁄ (h1)2 =  − [rsinθ∂r(rsinθ)] ⁄ 12 =  − rsin2θ

Γ323 = h3, 2 ⁄ h3 = [∂θ(rsinθ)] ⁄ [rsinθ] = (rcosθ) ⁄ (rsinθ) = cotθ = Γ332

Γ233 =  − (h3h3, 2) ⁄ (h2)2 =  − [rsinθ∂θ(rsinθ)] ⁄ r2 =  − (rsinθrcosθ) ⁄ r2 =  − sinθcosθ

25. WritethefollowingChristoffelsymbolsintermsofthecoordinatesinstead

oftheindicesassumingacylindricalsystem:[12, 1],[23, 1],Γ221andΓ332.Dothesameassumingasphericalsystem.Answer:Cylindrical:[ρφ, ρ],[φz, ρ],ΓφφρandΓzzφ.

Spherical:[rθ, r],[θφ, r],ΓθθrandΓφφθ.26. ShowthatalltheChristoffelsymbolswillvanishwhenthecomponentsof

themetrictensorareconstants.Answer:TheChristoffelsymbolsofthefirstandsecondkindaredefinedas.[ij, k] = (1 ⁄ 2)(∂jgik + ∂igjk − ∂kgij)

Γkij = (gkl ⁄ 2)(∂jgil + ∂igjl − ∂lgij)

Aswesee,bothkindsaresumoftermscontainingpartialderivativesofcomponentsofthemetrictensor.Therefore,whenallthecomponentsofthemetrictensorareconstantsallthesepartialderivativeswillvanishidenticallyandhencetheChristoffelsymbolswillalsovanishidentically.

27. WhytheChristoffelsymbolsofeitherkindmaybesuperscriptedorsubscriptedbythesymboloftheunderlyingmetrictensor?Whenthis(orothermeasuresforindicatingtheunderlyingmetrictensor)becomesnecessary?Mentionsomeoftheothermeasuresusedtoindicatetheunderlyingmetrictensor.Answer:ThepurposeofthesuperscriptsandsubscriptsistoindicatethemetricfromwhichtheChristoffelsymbolsarederived(seethedefinitionoftheChristoffelsymbolswhichisgivenintheanswerofthepreviousquestion).Thisbecomesnecessarywheninagivencontextwehavetwoormoredifferentmetricsrelatedtotwoormoredifferentspacesorsystemsandhencethemetricofthesymbolsrequiresclarification.IndicatingtheunderlyingmetrictensoroftheChristoffelsymbolsmayalsobedonebyusingdifferenttypesofindicesforeachmetric,e.g.byusingLatinoruppercaseindicesfortheChristoffelsymbolsofonemetricandGreekorlowercaseindicesfortheChristoffelsymbolsoftheothermetric.

28. ExplainwhythetotalnumberofindependentChristoffelsymbolsofeachkindisequalto[n2(n + 1)] ⁄ 2wherenisthedimensionofthespace(seeFootnote24in§8↓).

Answer:ConsideringtheidenticalityanddifferenceoftheindicesoftheChristoffelsymbolsofeitherkindingeneralcoordinatesystems,wehave4maincases:(a)Alltheindicesareidentical:thisrepresentsnindependentsymbolssincewehavenvaluesforanyindex.(b)Onlytwonon-pairedindicesareidentical:thisrepresentsn(n − 1)independentsymbols,i.e.nidenticaltimes(n − 1)different(ortheotherwayaround)(seeFootnote25in§8↓).(c)Onlythetwopairedindicesareidentical:thisrepresentsn(n − 1)independentsymbols,i.e.nnon-pairedtimes(n − 1)paired(ortheotherwayaround).(d)Alltheindicesaredifferent:thisrepresentsn(n − 1)(n − 2)symbolswhichisthenumberofnon-repetitivepermutations.However,duetothesymmetryinthepairedindiceswehaveonly[n(n − 1)(n − 2)] ⁄ 2independentsymbols.Accordingly,thetotalnumberofindependentsymbolsis:NCI = 

n + n(n − 1) + n(n − 1) + [{n(n − 1)(n − 2)} ⁄ 2] = 

[2n + 2n(n − 1) + 2n(n − 1) + n(n − 1)(n − 2)] ⁄ 2 = 

[2n + 2n2 − 2n + 2n2 − 2n + n3 − 2n2 − n2 + 2n] ⁄ 2 = 

[n3 + n2] ⁄ 2 = 

[n2(n + 1)] ⁄ 229. Whycovariantdifferentiationoftensorsisregardedasageneralizationof

theordinarypartialdifferentiation?Answer:Becausetheordinarypartialdifferentiationoftensors(i.e.appliedtotheircomponentsonly)isvalidonlyincertaintypesofcoordinatesystems(affineorrectilinear)wherethebasisvectorsarecoordinate-independentandhenceingeneralcurvilinearcoordinatesystemswherethebasisvectorsarecoordinate-dependent,theordinarypartialdifferentiationoftensorsdoesnotproducetensorsingeneral(i.e.itdoesnotsatisfytheinvarianceprincipleoftensors).Incontrast,thecovariantdifferentiationoftensorsnecessarilyproducestensorsandhenceitisageneralizationoftheordinarypartialdifferentiationsinceitisvalidingeneralwhileordinary

partialdifferentiationisvalidonlyinparticularcoordinatesystems.Thisalsoappliestoabsolutedifferentiationasageneralizationoftheordinarytotaldifferentiation.

30. Ingeneralcurvilinearcoordinatesystems,thevariationofthebasisvectorsshouldalsobeconsideredinthedifferentiationprocessofnon-scalartensors.Why?Answer:Becauseingeneralcurvilinearcoordinatesystemsthebasisvectors,aswellasthecomponents,dependonthecoordinatesingeneralandhencetheyarevariablefunctionsofcoordinates.Therefore,bytheproductruleofdifferentiation,appliedtonon-scalartensorsincurvilinearsystems,boththecomponentsandthebasisvectorsshouldbesubjectedtothedifferentiationprocesstotakeaccountofthevariationofbothparts(i.e.componentsandbasisvectors)ofthetensorbecauseanon-scalartensorismadeofcomponentsmultipliedbybasisvectors.Inbrief,ordinarydifferentiationisjustifiedinrectilinearsystemsbecausethebasisvectorsareconstantsandhenceallweneedtodo(accordingtotheproductruleofdifferentiation)istodifferentiatethecomponents,butincurvilinearcoordinatesystemsthisprocess(i.e.differentiationofcomponentsonly)isonlypartofthedifferentiationprocessandhenceitisnotsufficientfordifferentiatinggeneraltensors.Therefore,theotherpartofthedifferentiationprocessaccordingtotheproductruleshouldbeaddedandthiscompletedifferentiationprocess(inwhichboththecomponentsandthebasisvectorsaredifferentiated)iswhatiscalledtensordifferentiation(i.e.covariantandabsolutedifferentiation).

31. StatethemathematicaldefinitionofcontravariantdifferentiationofatensorAi.Answer:Contravariantdifferentiationisachievedbyraisingthedifferentiationindexofthecovariantderivativeusingtheindexraisingoperator.Hence,thecontravariantdifferentiationofacovariantvectorAi,forexample,isgivenby:Ai;j = gjkAi;k

wheregjkisthecontravariantmetrictensor.32. ObtainanalyticalexpressionsforAi;jandBi;jbydifferentiatingthevectors

A = AiEiandB = BiEi.Answer:Wehave:∂j(AiEi) = 

Ei∂jAi + Ai∂jEi = 

Ei∂jAi − AiΓikjEk = 

Ei∂jAi − AkΓkijEi = 

(∂jAi − AkΓkij)Ei = 

Ai;jEi

whereinline1weusetheproductruleofdifferentiationsinceboththecomponentsandbasisvectorsofgeneraltensorsarecoordinate-dependentvariables,inline2weusetheidentity∂jEi =  − ΓikjEkwhichisgiveninthebook,inline3werelabeltheindicesiandk,inline4wetakeoutthecommonfactorEi,andinline5weuseashorthandnotationforthecovariantderivativewhichisbasedonitsdefinition.

Similarly:∂j(BiEi) = 

Ei∂jBi + Bi∂jEi = 

Ei∂jBi + BiΓkijEk = 

Ei∂jBi + BkΓikjEi = 

(∂jBi + BkΓikj)Ei = 

Bi;jEi

whereinline2weusetheidentity∂jEi = ΓkijEkwhichisgiveninthebook,whiletheotherlinesaresimilarlyjustifiedasinthefirstpart.

33. RepeatQuestion325↑withtherank-2tensorsC = CijEiEjandD = DijEiEjtoobtainCij;kandDij;k.

Answer:Wehave:∂k(CijEiEj) = 

EiEj∂kCij + Cij(∂kEi)Ej + CijEi(∂kEj) = 

EiEj∂kCij − Cij(ΓiakEa)Ej − CijEi(ΓjakEa) = 

EiEj∂kCij − CajΓaikEiEj − CiaΓajkEiEj = 

(∂kCij − CajΓaik − CiaΓajk)EiEj = 

Cij;kEiEj

wherethelinesarejustifiedasintheanswerofthepreviousquestion.

Similarly,wehave:∂k(DijEiEj) = 

EiEj∂kDij + Dij(∂kEi)Ej + DijEi(∂kEj) = 

EiEj∂kDij + Dij(ΓaikEa)Ej + DijEi(ΓajkEa) = 

EiEj∂kDij + DajΓiakEiEj + DiaΓjakEiEj = 

(∂kDij + DajΓiak + DiaΓjak)EiEj = 

Dij;k34. ForadifferentiabletensorAoftype(m, n),thecovariantderivativewith

respecttothecoordinateukisgivenby:

Extractfromthepatternofthisexpressionthepracticalrulesthatshouldbe

followedinwritingtheanalyticalexpressionsofcovariantderivativeoftensorsofanyrankandtype.Answer:Thepracticalrulesthatcanbeextractedfromthepatternofthisexpressionaresummarizedinthefollowingwherewecalltheindicesi1i2…imandj1j2…jnthebasisindices(sinceeachoneoftheseindicescorrespondstoonebasisvectorofthebasistensor)(seeFootnote26in§8↓)andcallkthedifferentiationindex.Westartwithanordinarypartialderivativetermofthecomponentofthegiventensorwithrespecttothedifferentiationindex(i.e.thefirsttermontherighthandsideoftheaboveequation).ThenforeachbasisindexofthetensorweaddanextraChristoffelsymboltermwherethistermsatisfiesthefollowingproperties:●Thetermispositiveifthebasisindexiscontravariantandnegativeifthebasisindexiscovariant.●OneofthelowerindicesoftheChristoffelsymbolisthedifferentiationindex.●ThebasisindexofthetensorintheconcernedtermiscontractedwithoneoftheindicesoftheChristoffelsymbolusinganewlabel(i.e.thenewlabelisnotinusealreadyasabasisindex)andhencetheyareoppositeintheirvariancetype.●ThelabelofthebasisindexofthattermistransferredfromthetensortotheChristoffelsymbolkeepingitspositionaslowerorupper.●Alltheotherindicesofthetensorintheconcernedtermkeeptheirlabels,positionandorder.

35. Intheexpressionofcovariantderivative,whatthepartialderivativetermstandsforandwhattheChristoffelsymboltermsrepresent?Answer:Thepartialderivativetermstandsforthederivativeofthecomponentofthetensor,whileeachChristoffelsymboltermstandsforthederivativeofonebasisvectorofthebasistensorofthetensor(e.g.thebasistensorofAijisEiEj).Moreclearly,thepartialderivativetermrepresentstherateofchangeofthetensorcomponentwithchangeofpositionasaresultofmovingalongthecoordinatecurveofthedifferentiationindex,whiletheChristoffelsymboltermsrepresentthechangeexperiencedbythelocalbasisvectorsasaresultofthesamemovement.

36. Forthecovariantderivativeofatype(m, n, w)tensor,obtainthetotalnumberofterms,thenumberofnegativeChristoffelsymboltermsandthenumberofpositiveChristoffelsymbolterms.Answer:Wehave1partialderivativetermandm + nChristoffelsymbol

termsandhencethetotalnumberoftermsism + n + 1.WehavencovariantindicesandhencethenumberofnegativeChristoffelsymboltermsisn.WehavemcontravariantindicesandhencethenumberofpositiveChristoffelsymboltermsism.

37. Whatistherankandtypeofthecovariantderivativeofatensorofrank-nandtype(p, q)?Answer:Thecovariantderivativeofatensorisatensorwhosecovariantrankishigherthanthecovariantrankoftheoriginaltensorbyone.Hence,therankisp + q + 1(orn + 1)andthetypeis(p, q + 1).

38. Thecovariantderivativeofadifferentiablescalarfunctionisthesameastheordinarypartialderivative.Why?Answer:BecausethescalarisnotreferredtoanybasisvectororbasistensorandhenceithasnoChristoffelsymbolterms,sowhatremainsofthecovariantderivativetermsisthepartialderivativetermonlyandhencethecovariantderivativeofascalaristhesameastheordinarypartialderivative.

39. WhatisthesignificanceofthedependenceofthecovariantderivativeontheChristoffelsymbolswithregardtoitsrelationtothespaceandcoordinatesystem?Answer:Asseenearlier,theChristoffelsymbolsaresolelydependentonthemetrictensorandhencetheycanbeseenasacharacteristicpropertyofboththeunderlyingspaceandtheemployedcoordinatesystem.So,thedependencyofthecovariantderivativeontheChristoffelsymbolsimpliesitsdependencyonthespaceandthecoordinatesystemandhenceitischaracterizedbythespaceandthecoordinatesystemanditreflectstheirfeatures.

40. Thecovariantderivativeoftensorsincoordinatesystemswithconstantbasisvectorsisthesameastheordinarypartialderivativeforalltensorranks.Why?Answer:BecausetheChristoffelsymbolterms,whichrepresentthederivativesofthebasisvectors,willvanishsincethebasisvectorsareconstant.So,whatremainsofthecovariantderivativetermsisthepartialderivativetermonlyandhencethecovariantderivativeoftensorsincoordinatesystemswithconstantbasisvectorsisthesameastheordinarypartialderivativeforalltensorranks.

41. Express,mathematically,thefactthatthemetrictensorisinlieuofconstantwithrespecttocovariantdifferentiation.Answer:Thiscanbeexpressedinseveralformssuchas:

∂;kgij = 0

∂;kgij = 0

∂;k(gijAj) = gij∂;kAj

g;k = 0

(g○A);k = g○A;k

wheregisthemetrictensorinsymbolicnotationandthesymbol○denotesaninnerorouterproductoperator.

42. Whichrulesofordinarypartialdifferentiationalsoapplytocovariantdifferentiationandwhichrulesdonot?Statealltheserulessymbolicallyforbothordinaryandcovariantdifferentiation.Answer:Linearity:thisappliestobothordinarypartialdifferentiationandcovariantdifferentiation,thatis:∂(af + bg) ⁄ ∂x = a(∂f ⁄ ∂x) + b(∂g ⁄ ∂x)

(aA±bB);i = aA;i±bB;i

whereaandbarescalarconstants,fandgaredifferentiablescalarfunctionsandAandBaredifferentiabletensors.

Theproductruleofdifferentiation:thisappliestobothordinarypartialdifferentiationandcovariantdifferentiation,thatis:∂(fg) ⁄ ∂x = g(∂f ⁄ ∂x) + f(∂g ⁄ ∂x)

(A○B);i = A;i○B + A○B;i

wherethesymbol○denotesaninnerorouterproductoperator.However,theorderofthetensorsinthecovariantdifferentiationshouldbeobserved.

Commutativityofoperators:thisappliestoordinarypartialdifferentiationbutnottocovariantdifferentiation,thatis:∂i∂j = ∂j∂i

∂;i∂;j ≠ ∂;j∂;i43. Explainwhythecovariantdifferentialoperatorswithrespecttodifferent

indicesdonotcommute,i.e.∂;i∂;j ≠ ∂;j∂;i(i ≠ j).Answer:Thereasonisthatthedifferentiationindices,liketheindicesofthedifferentiatedtensor,arereferredtobasisvectorsandsincethebasisvectorsdonotcommutethenthedifferentiationindices(andhencethecovariantdifferentialoperatorsthatrepresenttheseindices)donotcommute.Thisisunliketheindicesoftheordinarypartialdifferentialoperatorssincetheseindicesarenotreferredtobasisvectors.Inbrief,thecovariantderivativeofatensorisatensorwhosecovariantrankincreasesby1foreachcovariantdifferentiationoperationandhencethedifferentiationindicesarenotdifferentfromtheindicesofthedifferentiatedtensor.Therefore,thedifferentiationindicesshouldfollowthesamerulesthattheindicesofthedifferentiatedtensorfollowandoneoftheserulesistheimportanceoftheorderofindices(i.e.non-commutativityoftensorindices).Thenon-commutativityofcovariantdifferentialoperatorscanalsobeshowndirectlybyperformingtheoperationsofcovariantdifferentiationindifferentorderswhereitcanbeeasilyverifiedthattheresultdependsontheorderoftheoperators(seeExercises525↓and535↓).

44. StatetheRiccitheoremaboutcovariantdifferentiationofthemetrictensorandproveitwithfulljustificationofeachstep.Answer:TheRiccitheoremstatesthatthecovariantderivativeofthemetrictensoriszero.Covariantmetrictensor:∂;kgij = 

∂;k(Ei⋅Ej) = 

(∂;kEi)⋅Ej + Ei⋅(∂;kEj) = 

0⋅Ej + Ei⋅0 = 

0

whereline1istherelationbetweenthemetrictensorandthebasisvectors,line2istheproductruleofdifferentiationwhichappliestocovariantdifferentiationastoordinarydifferentiation,andline3isthefactthatthe

covariantderivativeofthebasisvectorsisidenticallyzero(asdemonstratedinthebookandinExercise575↓).

Contravariantmetrictensor:∂;kgij = 

∂;k(Ei⋅Ej) = 

(∂;kEi)⋅Ej + Ei⋅(∂;kEj) = 

0⋅Ej + Ei⋅0 = 

0

wherethelinesaresimilarlyjustifiedasinthecovariantcase.

Mixedmetrictensor:theproofcanbesimilartotheproofofthecovariantandcontravarianttypes,i.e.∂;kgji = ∂;k(Ei⋅Ej) = ⋯etc.Alternatively:∂;kgji = 

∂;k(giagaj) = 

(∂;kgia)gaj + gia(∂;kgaj) = 

0 + 0 = 

0

whereline1isbasedontheuseofindexshiftingoperator,line2istheproductruleofdifferentiation,andline3isbasedontheresultsthatwealreadyobtainedinthisquestionforthecovariantandcontravarianttypes.

45. State,symbolically,thecommutativepropertyofthecovariantderivativeoperatorwiththeindexshiftingoperator(whichisbasedontheRiccitheorem)usingthesymbolicnotationonetimeandtheindicialnotationanother.Answer:Symbolicnotation:

∂;k(g⋅A) = g⋅(∂;kA)

Indicialnotation:∂;k(gijAj) = gij(∂;kAj)or∂;k(gijAj) = gij(∂;kAj)

46. VerifythattheordinaryKroneckerdeltatensorisconstantwithrespecttocovariantdifferentiation.Answer:Mixedtype:wehave:∂;kδij = 

∂kδij + δajΓiak − δiaΓajk = 

0 + δajΓiak − δiaΓajk = 

0 + Γijk − Γijk = 

0

whereline1isbasedontherulesofcovariantderivative,line2isjustifiedbythefactthatallthecomponentsoftheKroneckerdeltaareconstant(i.e.either0or1),andline3isbasedonusingtheKroneckerdeltaasanindexreplacementoperator.

Covarianttype:wehave:∂;kδij = 

∂;k(giaδaj) = 

(∂;kgia)δaj + gia(∂;kδaj) = 

0 + 0 = 

0

whereline1isbasedonusingindexshiftingoperator,line2isbasedontheproductruleofdifferentiationwhichappliestocovariantdifferentiationastoordinarydifferentiation,andline3isbasedonthefactthatthecovariantderivativeofthemetrictensorandthecovariantderivativeofthemixedKroneckerdeltaarebothzero(seeExercise445↑andthefirstpartofthecurrentquestion).

Contravarianttype:wehave:∂;kδij = 

∂;k(gjaδia) = 

(∂;kgja)δia + gja(∂;kδia) = 

0 + 0 = 

0

wherethelinesarejustifiedasforthecovarianttype.47. State,symbolically,thefactthatcovariantdifferentiationandcontractionof

indexoperationscommutewitheachother.Answer:Wehave:(∂;mAijk)δkj = ∂;m(Aijkδkj)

wherethelefthandsideistheorder“covariantdifferentiationfirstfollowedbycontractionofindex”whiletherighthandsideistheorder“contractionofindexfirstfollowedbycovariantdifferentiation”.

48. Whatistheconditiononthecomponentsofthemetrictensorthatmakesthecovariantderivativebecomeordinarypartialderivativeforalltensorranks?Answer:Theconditionisthatthecomponentsofthemetrictensorareconstants,becauseinthiscasealltheChristoffelsymbolswillvanishidentically(accordingtotheirdefinitionwhichwasgivenearlier)andhencealltheChristoffelsymboltermsofthecovariantderivativewillvanishaswell,sowhatremainsofthecovariantderivativetermsisthepartialderivativetermonlyandhencethecovariantderivativebecomesanordinarypartialderivativeforalltensorranks.

49. Provethatcovariantdifferentiationandcontractionofindicescommute.Answer:Wehavetwocases:

(a)Covariantdifferentiationfirstfollowedbycontractionofindex(i.e.contractingjwithkinthefollowingequation),thatis:(∂;mAijk)δkj = 

(Aijk;m)δkj = 

Aijk;mδkj = 

Aikk;m

(b)Contractionofindexfirstfollowedbycovariantdifferentiation:∂;m(Aijkδkj) = 

∂;m(Aikk) = 

Aikk;m

Oncomparingthetwoequationsweobtain:(∂;mAijk)δkj = ∂;m(Aijkδkj)

i.e.covariantdifferentiationandcontractionofindicescommute,asrequired.

50. Whatisthemathematicalconditionthatisrequiredifthemixedsecondorderpartialderivativesshouldbeequal,i.e.∂i∂j = ∂j∂i(i ≠ j)?Answer:ItistheC2continuitycondition,i.e.thedifferentiatedfunctionandallitsfirstandsecondpartialderivativesdoexistandtheyarecontinuousintheirdomain.

51. Whatisthemathematicalconditionthatisrequiredifthemixedsecondordercovariantderivativesshouldbeequal,i.e.∂;i∂;j = ∂;j∂;i(i ≠ j)?Answer:TheconditionisthevanishingoftheRiemann-Christoffelcurvaturetensor.

52. DeriveanalyticalexpressionsforAi;jkandAi;kjandhenceverifythatAi;jk ≠ Ai;kj.Answer:Wehave:Ai;jk = 

(Ai;j);k = 

∂kAi;j − ΓaikAa;j − ΓajkAi;a = 

∂k(∂jAi − ΓbijAb) − Γaik(∂jAa − ΓbajAb) − Γajk(∂aAi − ΓbiaAb) = 

∂k∂jAi − Γbij∂kAb − Ab∂kΓbij − Γaik∂jAa + ΓaikΓbajAb − Γajk∂aAi + ΓajkΓbiaAb

wherealltheselinesarejustifiedbythedefinitionofcovariantdifferentiationandsomeotherbasicoperations.

Similarly,wehave:Ai;kj = 

(Ai;k);j = 

∂jAi;k − ΓaijAa;k − ΓakjAi;a = 

∂j(∂kAi − ΓbikAb) − Γaij(∂kAa − ΓbakAb) − Γakj(∂aAi − ΓbiaAb) = 

∂j∂kAi − Γbik∂jAb − Ab∂jΓbik − Γaij∂kAa + ΓaijΓbakAb − Γakj∂aAi + ΓakjΓbiaAb

So,lettakethedifferencebetweenthesetwoequationstoseeifAi;jk − Ai;kj = 0(andhenceAi;jk = Ai;kj)orAi;jk − Ai;kj ≠ 0(andhenceAi;jk ≠ Ai;kj),thatis:Ai;jk − Ai;kj = 

∂k∂jAi − Γbij∂kAb − Ab∂kΓbij − Γaik∂jAa + ΓaikΓbajAb − Γajk∂aAi + ΓajkΓbiaAb − (∂j∂kAi − Γbik∂jAb − Ab∂jΓbik − Γaij∂kAa + ΓaijΓbakAb − Γakj∂aAi + ΓakjΓbiaAb) = 

 − Ab∂kΓbij + ΓaikΓbajAb − ( − Ab∂jΓbik + ΓaijΓbakAb) = 

 − Ab∂kΓbij + ΓaikΓbajAb + Ab∂jΓbik − ΓaijΓbakAb = 

AbRbijk

whereRbijkistheRiemann-Christoffelcurvaturetensor(seeFootnote27in§8↓).Now,sincethistensordoesnotvanishidenticallyingeneral,thenAi;jk − Ai;kj ≠ 0ingeneralandhenceAi;jk ≠ Ai;kj,asrequired.

53. FromtheresultofExercise525↑plusthedefinitionoftheRiemann-Christoffelcurvaturetensorofthesecondkind,verifythefollowingrelation:Ai;jk − Ai;kj = AbRbijk.Answer:TheRiemann-Christoffelcurvaturetensorofthesecondkindisgivenby:Rijkl = ∂kΓijl − ∂lΓijk + ΓrjlΓirk − ΓrjkΓirl

OninnermultiplyingwithAbandrelabelingtheindicesweobtain:AbRbijk = Ab∂jΓbik − Ab∂kΓbij + AbΓaikΓbaj − AbΓaijΓbak

whichisthesameastheresultofAi;jk − Ai;kjthatweobtainedinthepreviousexercise(i.e.theequationbeforethelastintheanswerofthepreviousquestion).Hence,weconcludethatAi;jk − Ai;kj = AbRbijk,asrequired.

54. Whatisthecovariantderivativeofarelativescalarfofweightw?Whatisthecovariantderivativeofarank-2relativetensorAijofweightw?Answer:Theyare:∂;kf = ∂kf − wfΓaak

∂;kAij = ∂kAij + ΓiakAaj − ΓajkAia − wAijΓaak55. Whythecovariantderivativeofanon-scalartensorwithconstant

componentsisnotnecessarilyzeroingeneralcoordinatesystems?Whichtermofthecovariantderivativeofsuchatensorwillvanish?Answer:Becausethecovariantderivativeofanon-scalartensorismadeofsumofterms,andonlyoneofthesetermsistheordinarypartialderivativeoftheconstantcomponent.Hence,evenifthistermvanishedtheotherterms(i.e.theChristoffelsymbolterms)donotnecessarilyvanishandhencethecovariantderivativeofanon-scalartensorwithconstantcomponentsisnotnecessarilyzero.Inotherwords,althoughthepartialderivativetermwillvanishbecauseitrepresentstherateofchangeoftheconstantcomponent,theChristoffelsymboltermswhichrepresenttherate

ofchangeofthebasisvectorsdonotnecessarilyvanishbecausetheconstancyofthecomponentdoesnotimplytheconstancyofthebasisvectors.Asindicatedintheanswerofthefirstpartofthequestion,thevanishingtermistheordinarypartialderivativeterm.

56. Showthat:Ai;j = Aj;iwhereAisagradientofascalarfield.Answer:Letfbeadifferentiablescalarfield,Aisitsgradient(i.e.Ai = ∂if)andAi;jisthecovariantderivativeofthisgradient.Hence,wehave:Ai;j = 

∂jAi − ΓkijAk = 

∂j∂if − ΓkijAk = 

∂i∂jf − ΓkijAk = 

∂iAj − ΓkijAk = 

∂iAj − ΓkjiAk = 

Aj;i

whereline3isjustifiedbythecommutativityoftheordinarypartialdifferentialoperators,andline5isjustifiedbythesymmetryoftheChristoffelsymbolsintheirpairedindices.

57. Showthatthecovariantderivativeofthebasisvectorsofthecovariantandcontravarianttypesisidenticallyzero,i.e.Ei;j = 0andEi;j = 0.Answer:Wehave:Ei;j = 

∂jEi − ΓkijEk = 

ΓkijEk − ΓkijEk = 

0

whereline2isbasedontheidentity∂jEi = ΓkijEkwhichisgiveninthebook.

Similarly:Ei;j = 

∂jEi + ΓikjEk = 

 − ΓikjEk + ΓikjEk = 

0

whereline2isbasedontheidentity∂jEi =  − ΓikjEkwhichisgiveninthebook.

58. Provethefollowingidentity:∂k(gijAiBj) = Ai;kBi + AiBi;kAnswer:Wehave:∂k(gijAiBj) = 

(∂kgij)AiBj + gij(∂kAi)Bj + gijAi(∂kBj) = 

([ik, j] + [jk, i])AiBj + gijBj∂kAi + gijAi∂kBj = 

(gajΓaik + gaiΓajk)AiBj + gijBj∂kAi + gijAi∂kBj = 

gajΓaikAiBj + gaiΓajkAiBj + gijBj∂kAi + gijAi∂kBj = 

gijBj∂kAi + gajΓaikAiBj + gijAi∂kBj + gaiΓajkAiBj = 

gijBi∂kAj + gijΓjakAaBi + gijAi∂kBj + gijΓjakAiBa = 

gij(∂kAj + ΓjakAa)Bi + Aigij(∂kBj + ΓjakBa) = 

gij(Aj;k)Bi + Aigij(Bj;k) = 

Ai;kBi + AiBi;k

whereline1istheproductruleofdifferentiation,line2istheidentity∂kgij = [ik, j] + [jk, i],line3istherelationbetweentheChristoffelsymbolsofthefirstandsecondkind,line5isreorderingofterms,line6isrelabelingofdummyindiceswithuseofsymmetryofmetrictensor,line8isthedefinitionofcovariantderivative,andline9isanindexshiftingoperation.

59. Defineabsolutedifferentiationdescriptivelyandmathematically.Whataretheothernamesofabsolutederivative?Answer:Absolutedifferentiationofatensoralongat-parameterizedcurveC(t)inannDspacewithrespecttotheparametertistheinnermultiplicationofthecovariantderivativeofthetensorandthetangentvectortothecurve.TheabsolutedifferentiationofavectorAofcovarianttypeAiandcontravarianttypeAiisdefinedmathematicallyby:δAi ⁄ δt ≡ Ai;a(dua ⁄ dt) = (dAi ⁄ dt) − ΓbiaAb(dua ⁄ dt)

δAi ⁄ δt ≡ Ai;a(dua ⁄ dt) = (dAi ⁄ dt) + ΓibaAb(dua ⁄ dt)

whereδ ⁄ δtistheabsolutedifferentiationoperator,theindexeduaregeneralcoordinatesandtisthecurveparameterwhiletheothersymbolsareasdefinedpreviously.Thisdefinitionistriviallygeneralizedtotensorsofhigherranks.Absolutederivativeisalsoknownasintrinsicderivativeorabsolutecovariantderivative.

60. WritethemathematicalexpressionfortheabsolutederivativeofthetensorfieldAijkwhichisdefinedoveraspacecurveC(t).Answer:δAijk ⁄ δt = (dAijk ⁄ dt) + ΓibaAbjk(dua ⁄ dt) + ΓjbaAibk(dua ⁄ dt) − ΓbkaAijb(dua ⁄ dt)

61. Whytheabsolutederivativeofadifferentiablescalaristhesameasitsordinarytotalderivative,i.e.δf ⁄ δt = df ⁄ dt?Answer:Becausethecovariantderivativeofascalaristhesameasthe

ordinarypartialderivativesinceascalarhasnoassociationwithbasisvectorstodifferentiateandthereforetherewillbenoChristoffelsymbolterms.Accordingly,theabsolutederivative,whichisaninnerproductofthecovariantderivative,willhaveonlyanordinaryderivativetermandhenceitisthesameastheordinarytotalderivativeofthescalar.Insymbolicterms:δf ⁄ δt = 

f;a(dua ⁄ dt) = 

f, a(dua ⁄ dt) = 

(∂f ⁄ ∂ua)(dua ⁄ dt) = 

df ⁄ dt

whereline1isbasedonthedefinitionofabsolutederivative,line2isbasedonthefactthatthecovariantderivativeofadifferentiablescalaristhesameasitsordinarypartialderivative,line3isanotation,andline4isbasedonthechainruleinmulti-variabledifferentiationnotingthatfisparametricallydependentontonly.

62. Whytheabsolutederivativeofadifferentiablenon-scalartensoristhesameasitsordinarytotalderivativeinrectilinearcoordinatesystems?Answer:BecauseinrectilinearsystemsthecovariantderivativeisthesameastheordinarypartialderivativesincetheChristoffelsymbolsarezerointhesesystems.Accordingly,theabsolutederivative,whichisaninnerproductofthecovariantderivative,willhaveonlyanordinaryderivativetermandhenceitisthesameastheordinarytotalderivativeofthetensor.Insymbolicterms(usingAijkasaninstance):δAijk ⁄ δt = 

Aijk;a(dua ⁄ dt) = 

(∂aAijk + 0 − 0 − 0)(dua ⁄ dt) = 

∂aAijk(dua ⁄ dt) = 

(∂Aijk ⁄ ∂ua)(dua ⁄ dt) = 

dAijk ⁄ dt

wherethelinesaresimilarlyjustifiedasintheanswerofthepreviousquestion.

63. Fromthepatternofcovariantderivativeofageneraltensor,obtainthepatternofitsabsolutederivative.Answer:Thepatternofthecovariantderivativewasexplainedindetailearlier(seeExercise345↑),soallweneedtoobtainthepatternofabsolutederivativeistoaddthefollowingrule:thecovariantderivativeisinnermultipliedwithdua ⁄ dtwheretheindexaiscontractedwiththedifferentiationindexofthecovariantderivative.

64. WehaveA = AijkEiEjEk.Applytheordinarytotaldifferentiationprocess(i.e.dA ⁄ dt)ontothistensor(includingitsbasisvectors)toobtainitsabsolutederivative.Answer:Wehave:dA ⁄ dt = 

d(AijkEiEjEk) ⁄ dt = 

(dAijk ⁄ dt)EiEjEk + Aijk(dEi ⁄ dt)EjEk + AijkEi(dEj ⁄ dt)Ek + AijkEiEj(dEk ⁄ dt) = 

([∂Aijk ⁄ ∂ua][dua ⁄ dt])EiEjEk + Aijk([∂Ei ⁄ ∂ua][dua ⁄ dt])EjEk + AijkEi([∂Ej ⁄ ∂ua][dua ⁄ dt])Ek + AijkEiEj([∂Ek ⁄ ∂ua][dua ⁄ dt]) = 

([∂Aijk ⁄ ∂ua][dua ⁄ dt])EiEjEk + Aijk(ΓbiaEb[dua ⁄ dt])EjEk + AijkEi(ΓbjaEb[dua ⁄ dt])Ek + AijkEiEj( − ΓkbaEb[dua ⁄ dt]) = 

([∂Aijk ⁄ ∂ua][dua ⁄ dt])EiEjEk + Acjk(ΓicaEi[dua ⁄ dt])EjEk + AickEi(ΓjcaEj[dua ⁄ dt])Ek + AijcEiEj( − ΓckaEk[dua ⁄ dt]) = 

([∂Aijk ⁄ ∂ua][dua ⁄ dt] + AcjkΓica[dua ⁄ dt] + AickΓjca[dua ⁄ dt] − AijcΓcka[dua ⁄ dt])EiEjEk = 

([∂Aijk ⁄ ∂ua] + AcjkΓica + AickΓjca − AijcΓcka)(dua ⁄ dt)EiEjEk = 

Aijk;a(dua ⁄ dt)EiEjEk = 

(δAijk ⁄ δt)EiEjEk

whereequality2istheproductruleofdifferentiation,equality3isthechainruleofdifferentiation,equality4istheidentities∂jEi = ΓkijEkand∂jEi =  − ΓikjEk,equality5isrelabelingofdummyindices,equality6istidyingup,equality7istakingcommonfactor,equality8isthedefinitionofcovariantderivative,andequality9isthedefinitionofabsolutederivative.

65. Whichrulesofordinarytotaldifferentiationalsoapplytointrinsicdifferentiationandwhichrulesdonot?Statealltheserulessymbolicallyforbothordinaryandintrinsicdifferentiation.Answer:Linearity:thisappliestobothordinarytotaldifferentiationandintrinsicdifferentiation,thatis:d(af + bg) ⁄ dt = a(df ⁄ dt) + b(dg ⁄ dt)

δ(aA±bB) ⁄ δt = a(δA ⁄ δt)±b(δB ⁄ δt)

whereaandbarescalarconstants,fandgaredifferentiablescalarfunctionsandAandBaredifferentiabletensors.

Theproductruleofdifferentiation:thisappliestobothordinarytotaldifferentiationandintrinsicdifferentiation,thatis:d(fg) ⁄ dt = g(df ⁄ dt) + f(dg ⁄ dt)

δ(A○B) ⁄ δt = (δA ⁄ δt)○B + A○(δB ⁄ δt)

wherethesymbol○denotesaninnerorouterproductoperator.However,theorderofthetensorsintheintrinsicdifferentiationshouldbeobserved.

Commutativityofoperators:thisappliestoordinarydifferentiationbutnottointrinsicdifferentiation.Infact,thisisnomorethanthecommutativityofordinarypartialoperatorsandnon-commutativityofcovariantoperators.To

bemoreclear,lethaveasecondordercovariantdifferentiationofavectorAkwithrespecttotheindicesiandj.Now,sinceAk;ji ≠ Ak;ijthenweshouldhave:Ak;ji(dui ⁄ dt) ≠ Ak;ij(duj ⁄ dt)So,wehave:∂i∂j = ∂j∂i

∂;i∂;j(dui ⁄ dt) ≠ ∂;j∂;i(duj ⁄ dt)

wherethefirstequationrepresentsordinarydifferentiation(whichispartialdifferentiationbecauseweareassumingthedependencyonmorethanonevariable)whilethesecondequationrepresentstheintrinsicdifferentiation(seeFootnote28in§8↓).

66. Usingyourknowledgeaboutcovariantdifferentiationandthefactthatabsolutedifferentiationfollowsthestyleofcovariantdifferentiation,obtainalltherulesofabsolutedifferentiationofthemetrictensor,theKroneckerdeltatensorandtheindexshiftingandindexreplacementoperators.Expressalltheserulesinwordsandinsymbols.Answer:Alltheserulescanbeeasilyobtainedfromthefactthatabsolutedifferentiationisnomorethanacovariantdifferentiationfollowedbyaninnerproductoperation.Now,becausethecovariantdifferentiationofthemetrictensor,theKroneckerdeltatensorandtheindexshiftingandindexreplacementoperatorsiszero,thentheabsolutedifferentiationoftheseshouldalsobezero.Theserulescanbeexpressedsymbolicallyas:δg ⁄ δt = 0

δδ ⁄ δt = 0

δ(gijAj) ⁄ δt = gij(δAj ⁄ δt)

δ(δjiAj) ⁄ δt = δji(δAj ⁄ δt)67. Justifythefollowingstatement:“Forcoordinatesystemsinwhichallthe

componentsofthemetrictensorareconstants,theabsolutederivativeisthesameastheordinarytotalderivative”.Answer:BecauseinsuchcoordinatesystemstheChristoffelsymbolsareidenticallyzero(asestablishedearlierinExercise205↑)andhenceallthetermsoftheabsolutederivativewillvanishexceptthefirstwhichisthe

ordinarytotalderivativeterm.Therefore,theabsolutederivativebecomesanordinarytotalderivative.Insymbolicterms(usingAijkasaninstance):δAijk ⁄ δt = 

(dAijk ⁄ dt) + 0 + 0 − 0 = 

dAijk ⁄ dt68. Theabsolutederivativeofatensoralongagivencurveisunique.Whatthis

means?Answer:Itmeansthatwewillobtainthesameabsolutederivativealongthegivencurveregardlessofthecoordinatesystemthatweareusing.Thisisbasedontheobjectivityoftheabsolutederivativesinceifitiswelldefinedandshouldhaveanyrealisticandusefulmeaningitshouldbeuniqueandindependentoftheemployedcoordinatesystem.Thisisalsobasedontheinvarianceoftensors(inthegeneralsenseofthisinvariance)acrossallcoordinatesystems.

69. Summarizeallthemainpropertiesandrulesthatgoverntensordifferentiation(i.e.covariantandabsolutedifferentiation).Answer:Themainpropertiesandrulesoftensordifferentiationare:●Tensordifferentiationisthesameasordinarydifferentiation(i.e.partialandtotaldifferentiation)butwiththeapplicationofthedifferentiationprocessonboththetensorcomponentsanditsbasistensorusingtheproductruleofdifferentiation.●Thecovariantandabsolutederivativesoftensorsaretensors.●Therankofthecovariantderivativeis1covariantrankhigherthantherankofthedifferentiatedtensorwhiletherankoftheabsolutederivativeisthesameastherankofthedifferentiatedtensor.Henceatensoroftype(m, n)willhaveacovariantderivativeoftype(m, n + 1)andanabsolutederivativeoftype(m, n)(seeFootnote29in§8↓).●Thesumandproductrulesofdifferentiationapplytotensordifferentiationasforordinarydifferentiation.However,theorderoftensorsintensordifferentiationshouldberespectedinthetensorproduct.●Thecovariantandabsolutederivativesofscalarsandaffinetensorsofhigherranksarethesameastheordinaryderivatives(i.e.partialandtotal).●Thecovariantandabsolutederivativesofthemetric,Kroneckerandpermutationtensorsaswellasthebasisvectorsvanishidenticallyinanycoordinatesystem.●Unlikeordinarydifferentialoperators,tensordifferentialoperatorsdonot

commutewitheachother.●Tensordifferentialoperatorscommutewiththecontractionofindices.●Tensordifferentialoperatorscommutewiththeindexreplacementoperatorandindexshiftingoperators.

Chapter6DifferentialOperations1. Describebrieflythenablabaseddifferentialoperatorsandoperations

consideringtheinteractionofthenablaoperatorwiththetensorswhichareacteduponbythisoperator.Answer:Inbrief:●Thenabladifferentialoperatormayactdirectlyonatensor(initsgeneralsensethatincludesscalarandvector)resultinginthegradientofthetensor.●Thenabladifferentialoperatormayactonanon-scalartensorthroughdotproductmultiplicationresultinginthedivergenceofthetensor.●Thenabladifferentialoperatormayactonanon-scalartensorthroughcrossproductmultiplicationresultinginthecurlofthetensor.●ThenabladifferentialoperatormayactonanothernablaoperatorthroughdotproductmultiplicationresultingintheLaplacianoperator.Thereareotherlesscommonoperationsandoperators,buttheyarenotinvestigatedinthebook.

2. Whataretheadvantagesanddisadvantagesofusingthecoordinatesassuffixesforlabelingtheoperators,basisvectorsandtensorcomponentsincylindricalandsphericalsystemsinsteadofindexedgeneralcoordinates?Whataretheadvantagesanddisadvantagesoftheopposite?Answer:Themainadvantageofusingthecoordinatesassuffixesisthatthisnotationisintuitive,unambiguousandwidelyused.Themaindisadvantageisthatitcannotbeputincompacttensorformusingtensornotation(orindicialnotation)whichisbasedonindices.Theadvantageoftheoppositeisthatwecanputitinacompacttensorformusingtensornotation.Themaindisadvantageisthatitisnotasintuitiveandcommonlyusedastheuseofcoordinates;moreover,someambiguityandconfusionmayarisewithregardtothecorrespondencebetweentheindicesandthecoordinates(e.g.iftheindex2insphericalsystemsreferstoθorφ).

3. “Thedifferentiationofatensorincreasesitsrankbyone,byintroducinganextracovariantindex,unlessitimpliesacontractioninwhichcaseitreducestherankbyone”.Justifythisstatementgivingcommonexamplesfromvectorandtensorcalculus.Answer:Asseeninthepreviouschapter,covariantdifferentiation

introducesanewcovariantindex(i.e.thedifferentiationindex)tothetensorandhenceitincreasesitscovariantrankby1.Forexample,Aiisarank-1tensorbutitscovariantderivativewithrespecttothejindexisAi;jwhichisarank-2tensor.However,somedifferentialoperationsincludeacontractionoperationandhencealthoughtheyintroduceanewindextheyconsumetwoindicesbycontractionandhencetheresultisatensorthatis1ranklowerthantherankoftheoriginaltensor(seeFootnote30in§8↓).Themostobviousexamplesarethegradient(whichincreasestherankofthedifferentiatedtensorby1sinceitintroduces1differentiationindexwithnocontraction),andthedivergence(whichreducestherankofthedifferentiatedtensorby1sinceitintroduces1differentiationindexbutconsumes2indicesbycontraction).

4. Writethefollowingsubsidiarynablabasedoperatorsintensornotation:A⋅∇andA × ∇.Isthisnotationconsistentwiththenotationofdotandcrossproductofvectors?Answer:Usingtensornotation,theseoperatorsaredefinedinCartesiancoordinatesas:A⋅∇ = Ai∂i

[A × ∇]i = ϵijkAj∂k

whereAisavector.Yes,thisnotationisconsistentwiththenotationofdotandcrossproductofvectors.

5. Whyingeneralwehave:A⋅∇ ≠ ∇⋅AandA × ∇ ≠ ∇ × A?Answer:Theorderisimportantbecauseitdeterminesthemeaningoftheoperatorandthenatureoftheactionthatissupposedtobeconductedbyit.Forexample,A⋅∇meansthat∇isnotactingonAbutitisactingonsomethingelse,while∇⋅Ameansthat∇isactingonA(i.e.takingthedivergenceofA)althoughinbothcasesAand∇areinvolvedinadotproductoperation.Similarly,A × ∇meansthat∇isnotactingonAbutitisactingonsomethingelse,while∇ × Ameansthat∇isactingonA(i.e.takingthecurlofA)althoughinbothcasesAand∇areinvolvedinacrossproductoperation.

6. DefinethenablavectoroperatorandtheLaplacianscalaroperatorinCartesiancoordinatesystemsusingtensornotation.Answer:∇i = 

∂ ⁄ ∂xi = ∂i

∇2 = ∂2 ⁄ ∂xi∂xi = δij(∂2 ⁄ ∂xi∂xj) = ∂ii

7. FindthegradientofthefollowingvectorfieldinaCartesiancoordinatesystem:A = (x, 2x2, π).Answer:Thecomponentsofthegradient(whichisarank-2tensor)are:∂xx = 1∂yx = 0∂zx = 0

∂x(2x2) = 4x∂y(2x2) = 0∂z(2x2) = 0

∂xπ = 0∂yπ = 0∂zπ = 0

Hence,itcanberepresentedbythefollowingmatrix:

wherei, j = 1, 2, 3andireferstothecomponentsofAwhilejreferstothecoordinatesx, y, z.

8. DefinethedivergenceofadifferentiablevectordescriptivelyandmathematicallyassumingaCartesiancoordinatesystem.

Answer:ThedivergenceofadifferentiablevectorfieldAisascalardefinedasthedotproductofthenablaoperatorandthevectorA(inthisorder),thatis:∇⋅A = δij(∂Ai ⁄ ∂xj) = ∂iAi

whereweareassumingaCartesiancoordinatesystem.9. WhatisthedivergenceofthefollowingvectorfieldinCartesian

coordinates:A = (2z, y3, ex)?Answer:∇⋅A = 

[∂(2z) ⁄ ∂x] + [∂y3 ⁄ ∂y] + [∂ex ⁄ ∂z] = 

0 + 3y2 + 0 = 

3y210. Writesymbolically,usingtensornotation,thefollowingtwoformsofthe

divergenceofarank-2tensorfieldAinCartesiancoordinates:∇⋅Aand∇⋅AT.Answer:[∇⋅A]i = ∂jAji

[∇⋅A]j = ∂iAji11. Definethecurl∇ × AinCartesiancoordinatesusingtensornotationwhere

(a)Aisarank-1tensorand(b)Aisarank-2tensor(notethetwopossibilitiesinthelastcase).Answer:(a)Aisarank-1tensor:[∇ × A]i = ϵijk∂jAk

(b)Aisarank-2tensor:[∇ × A]ij = ϵimn∂mAnj

[∇ × A]ik = ϵimn∂mAkn12. WhatisthecurlofthefollowingvectorfieldassumingaCartesian

coordinatesystem:A = (5e2x, πxy, z2)?Answer:∇ × A = 

i[(∂z2 ⁄ ∂y) − (∂πxy ⁄ ∂z)] − j[(∂z2 ⁄ ∂x) − (∂5e2x ⁄ ∂z)] + k[(∂πxy ⁄ ∂x) − (∂5e2x ⁄ ∂y)] = 

i(0 − 0) −  j(0 − 0) +  k(πy − 0) = 

πyk13. FindtheLaplacianofthefollowingvectorfieldinCartesiancoordinates:

A = (x2y, 2ysinz, πzecoshx)Answer:∇2 A = 

∇2( ix2y + j2ysinz + kπzecoshx) = 

i∇2(x2y) + j∇2(2ysinz) + k∇2(πzecoshx) = 

i[(∂2x2y ⁄ ∂x2) + (∂2x2y ⁄ ∂y2) + (∂2x2y ⁄ ∂z2)] +  j[(∂22ysinz ⁄ ∂x2) + (∂22ysinz ⁄ ∂y2) + (∂22ysinz ⁄ ∂z2)] +  k[(∂2πzecoshx ⁄ ∂x2) + (∂2πzecoshx ⁄ ∂y2) + (∂2πzecoshx ⁄ ∂z2)] = 

i(2y + 0 + 0) +  j(0 + 0 − 2ysinz) + k(πzecoshxsinh2x + πzecoshxcoshx + 0 + 0) = 

(2y)i − (2ysinz)j + πzecoshx(sinh2x + coshx)k14. DefinethenablaoperatorandtheLaplacianoperatoringeneralcoordinate

systemsusingtensornotation.

Answer:∇ = Ei∂i

∇2 = divgrad = ∇⋅∇ = [1 ⁄ √(g)]∂i[√(g)gij∂j]

whereEiisacontravariantbasisvector,gisthedeterminantofthecovariantmetrictensorandgijisthecontravariantmetrictensor.

15. ObtainanexpressionforthegradientofacovariantvectorA = AiEiingeneralcoordinatesjustifyingeachstepinyourderivation.RepeatthequestionwithacontravariantvectorA = AiEi.Answer:Wehave(seeFootnote31in§8↓):∇A = 

Ej∂j(AiEi) = 

EjEi∂jAi + EjAi∂jEi = 

EjEi∂jAi + EjAi( − ΓibjEb) = 

EjEi∂jAi − EjEiΓbijAb = 

EjEi(∂jAi − ΓbijAb) = 

EjEiAi;j

whereline2istheproductruleofdifferentiation,line3istheidentity∂jEi =  − ΓikjEk,line4isrelabelingofdummyindices,andline6isthedefinitionofcovariantderivative.Similarly,wehave:∇A = 

Ej∂j(AiEi) = 

EjEi∂jAi + EjAi∂jEi = 

EjEi∂jAi + EjAi(ΓbijEb) = 

EjEi∂jAi + EjEiΓibjAb = 

EjEi(∂jAi + ΓibjAb) = 

EjEiAi;j

whereline3istheidentity∂jEi = ΓkijEkwhiletheotherlinesarejustifiedasinthepreviouspart.

16. RepeatQuestion156↑witharank-2mixedtensorA = AijEiEj.Answer:Wehave:∇A = 

Ek∂k(AijEiEj) = 

Ek(∂kAij)EiEj + EkAij(∂kEi)Ej + EkAijEi(∂kEj) = 

EkEiEj∂kAij + EkAij(ΓbikEb)Ej + EkAijEi( − ΓjbkEb) = 

EkEiEj∂kAij + EkEiEjAbjΓibk − EkEiEjAibΓbjk = 

EkEiEj(∂kAij + AbjΓibk − AibΓbjk) = 

EkEiEjAij;k

wherethelinesarejustifiedasintheanswerofthepreviousquestion.17. Define,intensorlanguage,thecontravariantformofthegradientofascalar

fieldf.Answer:Thecontravariantformofthegradientofascalarfieldfisgivenby:[∇f]i = 

∂if = 

gij∂jf = 

gijf, j = 

f, i

wheregijisthecontravariantmetrictensor.18. Definethedivergenceofadifferentiablevectordescriptivelyand

mathematicallyassumingageneralcoordinatesystem.Answer:ThedivergenceofadifferentiablecontravariantvectorfieldAjisascalarobtainedbycontractingthedifferentiationindexofthecovariantderivativeofthevectorwiththecontravariantindexofthevector,thatis:∇⋅A = 

δijAj;i = 

Ai;i = 

[1 ⁄ √(g)]∂i[√(g)Ai]

whereδijistheKroneckerdeltaandgisthedeterminantofthecovariantmetrictensor.

19. DerivethefollowingexpressionforthedivergenceofacontravariantvectorAingeneralcoordinates:∇⋅A = Ai;i.Answer:Wehave:∇⋅A = 

Ei∂i⋅(AjEj) = 

Ei⋅∂i(AjEj) = 

Ei⋅(Aj;iEj) = 

(Ei⋅Ej)Aj;i = 

δijAj;i = 

Ai;i

whereline3isthedefinitionofcovariantderivative,line5istherelationbetweenthebasisvectorsandthemixedtypemetrictensor,andline6isanindexreplacementoperation.

20. VerifythefollowingformulaforthedivergenceofacontravariantvectorAingeneralcoordinates:∇⋅A = [1 ⁄ √(g)]∂i[√(g)Ai].Repeatthequestionwiththeformula:∇⋅A = gjiAj;iwhereAisacovariantvector.Answer:Wehave:∇⋅A = 

Ai;i = 

∂iAi + ΓijiAj = 

∂iAi + Aj[1 ⁄ √(g)]∂j[√(g)] = 

∂iAi + Ai[1 ⁄ √(g)]∂i[√(g)] = 

[1 ⁄ √(g)]∂i[√(g)Ai]

whereline1isthedefinitionofdivergence,line2isthedefinitionofcovariantderivative,line3istheidentityΓiji = [1 ⁄ √(g)]∂j[√(g)]whichwasestablishedinthebook,line4isrelabelingthedummyindexj,andline5istheproductruleofdifferentiation.Similarly,wehave:gjiAj;i = 

(gjiAj);i = 

(Ai);i = 

Ai;i = 

∇⋅A

whereline1istheconstancyofthemetrictensorwithrespecttocovariantdifferentiation(Riccitheorem),line2isanindexraisingoperation,andline4isthedefinitionofthedivergenceofacontravariantvector.

21. RepeatQuestion206↑withtheformula:∇⋅A = EkAik;iwhereAisarank-2contravarianttensor.Answer:Wehave:∇⋅A = 

Ei∂i⋅(AjkEjEk) = 

Ei⋅∂i(AjkEjEk) = 

Ei⋅(Ajk;iEjEk) = 

(Ei⋅Ej)EkAjk;i = 

δijEkAjk;i = 

EkAik;i

whereline3isthedefinitionofcovariantderivative,line4istheintendeddotproduct(sincethedifferentiationindexistobecontractedwiththefirstindexofthetensor),line5istherelationbetweenthebasisvectorsandthemixedtypemetrictensor,andline6isanindexreplacementoperation.

22. Provethatthedivergenceofacontravariantvectorisascalar(i.e.rank-0tensor)byshowingthatitisinvariantundercoordinatetransformations.Answer:Asexplainedearlier,thedivergenceofadifferentiablecontravariantvectoristheresultofcontractingthedifferentiationindexofthecovariantderivativeofthevectorwiththecontravariantindexofthevector.Moreover,thecovariantderivativeofatensor(inthiscasearank-1contravariantvector)isatensorwhichis1covariantrankhigherthantherankoftheoriginaltensorandhencethecovariantderivativeofa

contravarianttensorisarank-2mixedtypetensor.Now,sincethecontractionofindexoftensorsproducesatensor(i.e.invariantundercoordinatetransformations)whichis2ranklowerthantherankoftheoriginaltensor,thenthedivergenceofacontravariantvectorisarank-0tensororscalar(i.e.invariantundercoordinatetransformations),asrequired.Therearemoreformalapproachestothisquestionbuttheaboveargumentshouldbesufficient.

23. Derive,fromthefirstprinciples,thefollowingformulaforthecurlofacovariantvectorfieldAingeneralcoordinates:[∇ × A]k = [ϵijk ⁄ √(g)](∂iAj − ΓljiAl).Answer:Wehave:∇ × A = 

Ei∂i × AjEj = 

Ei × ∂i(AjEj) = 

Ei × (Aj;iEj) = 

Aj;i(Ei × Ej) = 

Aj;iεijkEk = 

εijkAj;iEk = 

[ϵijk ⁄ √(g)](∂iAj − ΓljiAl)Ek

whereline3isthedefinitionofcovariantderivative,line5isanidentityaboutthecrossproductofbasisvectorswhichwasestablishedinthebook,andline7istheexpressionofthecovariantderivativeofacovariantvectorplusthedefinitionoftheabsolutecontravariantpermutationtensor.Hence,thekthcontravariantcomponentofcurlAis:[∇ × A]k = [ϵijk ⁄ √(g)](∂iAj − ΓljiAl)

24. ShowthattheformulainExercise236↑willreduceto[∇ × A]k = [ϵijk ⁄ 

√(g)]∂iAjduetothesymmetryoftheChristoffelsymbolsintheirlowerindices.Answer:Thepermutationtensorisnon-zeroonlywheni ≠ j ≠ k.Hence,foreachkthcomponentwehaveonlytwonon-vanishingtermsintheformulaofExercise236↑.Thesetermscorrespondtothetwopermutationsofij(withi ≠ j ≠ k)whereinoneofthesepermutationsϵijkis + 1andintheotherpermutationϵijkis − 1.Now,ifweuseuppercaseindices(i.e.I, J, K)toindicatethattheseindiceshavefixedvalueswithI ≠ J ≠ K(e.g.I = 1,J = 2andK = 3),thentheformulaofExercise236↑canbewrittenas:[∇ × A]K = 

[ϵIJK ⁄ √(g)](∂IAJ − ΓbJIAb) + [ϵJIK ⁄ √(g)](∂JAI − ΓbIJAb) = 

[ϵIJK ⁄ √(g)](∂IAJ − ΓbJIAb) − [ϵIJK ⁄ √(g)](∂JAI − ΓbIJAb) = 

[ϵIJK ⁄ √(g)](∂IAJ − ΓbJIAb − ∂JAI + ΓbIJAb) = 

[ϵIJK ⁄ √(g)](∂IAJ − ΓbJIAb − ∂JAI + ΓbJIAb) = 

[ϵIJK ⁄ √(g)](∂IAJ − ∂JAI) = 

[ϵIJK ⁄ √(g)]∂IAJ + [ϵJIK ⁄ √(g)]∂JAI

whereline2isbasedonthefactthatthepermutationtensoristotallyanti-symmetric,andline4isbasedonthesymmetryoftheChristoffelsymbolsofthesecondkindintheirlowerindices.Now,ifwereturntoourordinarylowercaseindexnotationthenthelastlineisnomorethanasumoftermsrepresentingallthepermutationsofijk(includingthezerotermswhichcorrespondtotherepetitivepermutations),andhencethelastequationcanbewrittencompactlyas:[∇ × A]k = [ϵijk ⁄ √(g)]∂iAj

25. Derive,fromthefirstprinciples,thefollowingexpressionfortheLaplacianofascalarfieldfingeneralcoordinates:∇2f = [1 ⁄ √(g)]∂i[√(g)gij∂jf].Answer:∇2f = 

∇⋅(∇f) = 

Ei∂i⋅(Ej∂jf) = 

Ei⋅∂i(Ej∂jf) = 

Ei⋅∂i(Ejf, j) = 

Ei⋅(Ejf, j;i) = 

(Ei⋅Ej)f, j;i = 

gijf, j;i = 

(gijf, j);i = 

(gij∂jf);i = 

∂i(gij∂jf) + (gkj∂jf)Γiki = 

∂i(gij∂jf) + (gij∂jf)Γkik = 

∂i(gij∂jf) + (gij∂jf)[1 ⁄ √(g)][∂i√(g)] = 

[1 ⁄ √(g)][√(g)∂i(gij∂jf) + (gij∂jf)∂i√(g)] = 

[1 ⁄ √(g)]∂i[√(g)gij∂jf]

whereline1isthedefinitionofLaplacianasdivergenceofgradient,line5isthedefinitionofcovariantderivative,line8istheconstancyofthemetrictensorwithrespecttotensordifferentiation(Riccitheorem),line10istheexpressionofthecovariantderivativeofacontravariantvector(i.e.gij∂jf = ∂if),line12istheidentityΓiji = [1 ⁄ √(g)]∂j[√(g)]whichwasestablishedinthebook,andline14istheproductruleofdifferentiationwhiletheotherlinesareobviousorjustifiedearlier.

26. WhythebasicdefinitionoftheLaplacianofascalarfieldfingeneralcoordinatesas∇2f = div(gradf)cannotbeusedasitistodevelopaformulabeforeraisingtheindexofthegradient?Answer:BecausethedivergenceintheabovedefinitionofLaplacianimpliesacontractionoperationbetweenthegradientindexandthedivergenceindex,andsincethecontractionoperationingeneralcoordinatesystemsshouldbebetweenacovariantindexandacontravariantindexthentheindexofthegradient(whichisacovariantindex)shouldberaisedbeforethecontractionoperationcantakeplace.

27. Define,intensorlanguage,thenablaoperatorandtheLaplacianoperatorassuminganorthogonalcoordinatesystemofa3Dspace.Answer:Theyare(seeFootnote32in§8↓):∇ = Σi(qi ⁄ hi)(∂ ⁄ ∂qi)

∇2 = [1 ⁄ (h1h2h3)]Σ3i = 1(∂ ⁄ ∂qi)([{h1h2h3} ⁄ (hi)2][∂ ⁄ ∂qi])

whereqiarebasisunitvectorsoforthogonalsystems,qiaregeneralorthogonalcoordinatesandtheindexedharescalefactors.

28. Usingtheexpressionofthedivergenceofavectorfieldingeneralcoordinates,obtainanexpressionforthedivergenceinorthogonalcoordinatesofa3Dspace.Answer:Wehave:∇⋅A = 

[1 ⁄ √(g)]∂i[√(g)Ai] = 

[1 ⁄ √(g)]∂[√(g)Ai] ⁄ ∂qi = 

[1 ⁄ √(g)]Σ3i = 1∂[√(g)Ai] ⁄ ∂qi = 

[1 ⁄ (h1h2h3)]Σ3i = 1∂(h1h2h3Ai) ⁄ ∂qi = 

[1 ⁄ (h1h2h3)]Σ3i = 1∂([{h1h2h3} ⁄ hi]Âi) ⁄ ∂qi = 

[1 ⁄ (h1h2h3)][{∂(h2h3Â1) ⁄ ∂q1} + {∂(h1h3Â2) ⁄ ∂q2} + {∂(h1h2Â3) ⁄ ∂q3}]

whereline1istheexpressionofthedivergenceingeneralcoordinateswhichweobtainedearlier(seeExercise206↑),line2isbasedonthefactthatthesystemisorthogonalandhenceweuseorthogonalcoordinates,line3isbasedonthesummationconventionandassuminga3Dspace,line4isbasedonthefactthatinorthogonalsystemsthemetrictensorisdiagonalandhenceg(whichisthedeterminantofthecovariantmetrictensor)isgivenby:g = g11g22g33 = (h1h2h3)2

andline5isbasedonusingphysicalcomponentsÂi( = hiAiwithnosumoni)whileline6isjustanexpansionofline5forthesakeofclarity.

29. DefinethecurlofavectorfieldAinorthogonalcoordinatesofa3Dspaceusingdeterminantalformandtensornotationform.Answer:Determinantalform:

wherethesymbolsareasexplainedinthelasttwoquestions.Tensornotationform:[∇ × A]i = 

Σ3k = 1[(ϵijkhi) ⁄ (h1h2h3)][∂(hkÂk) ⁄ ∂qj]

withnosumoveri.30. UsingtheexpressionoftheLaplacianofascalarfieldingeneral

coordinates,deriveanexpressionfortheLaplacianinorthogonalcoordinatesofa3Dspace.

Answer:Wehave:∇2f = 

[1 ⁄ √(g)]∂i[√(g)gij∂jf] = 

[1 ⁄ √(g)]∂[√(g)gij(∂f ⁄ ∂qj)] ⁄ ∂qi = 

[1 ⁄ √(g)]Σ3i = 1∂[√(g)Σ3j = 1gij(∂f ⁄ ∂qj)] ⁄ ∂qi = 

[1 ⁄ √(g)]Σ3i = 1∂[√(g)gii(∂f ⁄ ∂qi)] ⁄ ∂qi = 

[1 ⁄ (h1h2h3)]Σ3i = 1∂[{(h1h2h3) ⁄ (hi)2}{∂f ⁄ ∂qi}] ⁄ ∂qi

whereline1istheexpressionoftheLaplacianingeneralcoordinateswhichweobtainedearlier(seeExercise256↑),line2isbasedonthefactthatthesystemisorthogonalandhenceweuseorthogonalcoordinates,line3isbasedonthesummationconventionandassuminga3Dspace,line4isbasedonthefactthatinorthogonalsystemswehavegij = 0wheni ≠ jandhencethesumoverjreducestogii(∂f ⁄ ∂qi)wheregiirepresentstheithdiagonalcomponent,andline5isbasedonthefactthatinorthogonalsystemswehave:√(g) = h1h2h3andgii = 1 ⁄ (hi)2

31. Whythecomponentsoftensorsincylindricalandsphericalcoordinatesarephysical?Answer:Becauseallthebasisvectorsarenormalizedandhencetheyaredimensionlesswithunitmagnitude(seeFootnote33in§8↓).Consequently,allthecomponentsinthesesystemsarephysicalwithunifiedphysicaldimensions.

32. DefinethenablaandLaplacianoperatorsincylindricalcoordinates.Answer:Theyare:∇ = eρ∂ρ + eφ(1 ⁄ ρ)∂φ + ez∂z

∇2 = ∂ρρ + (1 ⁄ ρ)∂ρ + (1 ⁄ ρ2)∂φφ + ∂zz

whereeρ, eφ, ezareunitbasisvectorsand∂ρρ = ∂ρ∂ρ,∂φφ = ∂φ∂φand∂zz = ∂z∂z.

33. Usethedefinitionofthegradientofascalarfieldfinorthogonalcoordinatesandthetableofscalefactors(whichisgiveninthebook)toobtainanexpressionforthegradientincylindricalcoordinates.Answer:Thegradientofadifferentiablescalarfieldfinorthogonalcoordinatesystemsofa3Dspaceisgivenby:∇f = (q1 ⁄ h1)(∂f ⁄ ∂q1) + ( q2 ⁄ h2)(∂f ⁄ ∂q2) + ( q3 ⁄ h3)(∂f ⁄ ∂q3)

Now,incylindricalsystemswehave:q1 = ρq2 = φq3 = z

q1 =  eρq2 =  eφq3 =  ez

h1 = 1h2 = ρh3 = 1

Hence,theaboveequationbecomes:∇f = eρ(∂f ⁄ ∂ρ) + eφ(1 ⁄ ρ)(∂f ⁄ ∂φ) + ez(∂f ⁄ ∂z)

34. UsethedefinitionofthedivergenceofavectorfieldAinorthogonalcoordinatesandthetableofscalefactors(whichisgiveninthebook)toobtainanexpressionforthedivergenceincylindricalcoordinates.Answer:ThedivergenceofadifferentiablevectorfieldAinorthogonalcoordinatesystemsofa3Dspaceisgivenby:∇⋅A = [1 ⁄ (h1h2h3)][{∂(h2h3Â1) ⁄ ∂q1} + {∂(h1h3Â2) ⁄ ∂q2} + {∂(h1h2Â3) ⁄ ∂q3}]

Now,incylindricalsystemswehave:q1 = ρq2 = φq3 = z

h1 = 1h2 = ρh3 = 1

Â1 = AρÂ2 = AφÂ3 = Az

Hence,theaboveequationbecomes:∇⋅A = 

[1 ⁄ ρ][{∂(ρAρ) ⁄ ∂ρ} + {∂(Aφ) ⁄ ∂φ} + {∂(ρAz) ⁄ ∂z}] = 

[1 ⁄ ρ][{∂(ρAρ) ⁄ ∂ρ} + {∂Aφ ⁄ ∂φ} + ρ{∂Az ⁄ ∂z}]35. WritethedeterminantalformofthecurlofavectorfieldAincylindrical

coordinates.Answer:

wherethesymbolsareasexplainedearlier.

36. UsethedefinitionoftheLaplacianofascalarfieldfinorthogonalcoordinatesandthetableofscalefactors(whichisgiveninthebook)toobtainanexpressionfortheLaplacianincylindricalcoordinates.Answer:TheLaplacianofadifferentiablescalarfieldfinorthogonalcoordinatesystemsofa3Dspaceisgivenby:∇2f = [1 ⁄ (h1h2h3)]Σ3i = 1∂({(h1h2h3) ⁄ (hi)2}{∂f ⁄ ∂qi}) ⁄ ∂qi

Now,incylindricalsystemswehave:q1 = ρq2 = φq3 = z

h1 = 1h2 = ρh3 = 1

Hence,theaboveequationbecomes:∇2f = 

[(1 ⁄ ρ)∂(ρ{∂f ⁄ ∂ρ}) ⁄ ∂ρ] + [(1 ⁄ ρ)∂({1 ⁄ ρ}{∂f ⁄ ∂φ}) ⁄ ∂φ] + [(1 ⁄ ρ)∂(ρ{∂f ⁄ ∂z}) ⁄ ∂z] = 

[(ρ ⁄ ρ)∂(∂f ⁄ ∂ρ) ⁄ ∂ρ] + [(1 ⁄ ρ)(∂ρ ⁄ ∂ρ)(∂f ⁄ ∂ρ)] + [(1 ⁄ ρ)∂({1 ⁄ ρ}{∂f ⁄ ∂φ}) ⁄ ∂φ] + [(1 ⁄ ρ)∂(ρ{∂f ⁄ ∂z}) ⁄ ∂z] = 

[∂2f ⁄ ∂ρ2] + [(1 ⁄ ρ)(∂f ⁄ ∂ρ)] + [(1 ⁄ ρ)∂({1 ⁄ ρ}{∂f ⁄ ∂φ}) ⁄ ∂φ] + [(1 ⁄ ρ)∂(ρ{∂f ⁄ ∂z}) ⁄ ∂z] = 

[∂2f ⁄ ∂ρ2] + [(1 ⁄ ρ)(∂f ⁄ ∂ρ)] + [(1 ⁄ ρ)(1 ⁄ ρ)∂(∂f ⁄ ∂φ) ⁄ ∂φ] + [(ρ ⁄ ρ)∂(∂f ⁄ ∂z) ⁄ ∂z] = 

[∂2f ⁄ ∂ρ2] + [(1 ⁄ ρ)(∂f ⁄ ∂ρ)] + [(1 ⁄ ρ2)(∂2f ⁄ ∂φ2)] + [∂2f ⁄ ∂z2] = 

∂ρρf + (1 ⁄ ρ)∂ρf + (1 ⁄ ρ2)∂φφf + ∂zzf

whereinline2weusedtheproductruleofdifferentiation,andinline4weusedthefactthatthecoordinatesareindependentofeachother.

37. Ascalarfieldincylindricalcoordinatesisgivenby:f(ρ, φ, z) = ρ.WhatarethegradientandLaplacianofthisfield?Answer:Wehave:∇f = 

eρ∂ρf + eφ(1 ⁄ ρ)∂φf + ez∂zf = 

eρ∂ρρ + eφ(1 ⁄ ρ)∂φρ + ez∂zρ = 

eρ + 0 + 0 = 

eρ

∇2f = 

∂ρρf + (1 ⁄ ρ)∂ρf + (1 ⁄ ρ2)∂φφf + ∂zzf = 

∂ρρρ + (1 ⁄ ρ)∂ρρ + (1 ⁄ ρ2)∂φφρ + ∂zzρ = 

0 + (1 ⁄ ρ) + 0 + 0 = 

1 ⁄ ρ38. Avectorfieldincylindricalcoordinatesisgivenby:A(ρ, φ, z) = (3z, 

πφ2, z2cosρ).Whatarethedivergenceandcurlofthisfield?Answer:Wehave:∇⋅A = 

[1 ⁄ ρ][∂ρ(ρAρ) + ∂φAφ + ρ∂zAz] = 

[1 ⁄ ρ][∂ρ(ρ3z) + ∂φ(πφ2) + ρ∂z(z2cosρ)] = 

[1 ⁄ ρ][3z + 2πφ + 2ρzcosρ]

∇ × A = 

[1 ⁄ ρ]eρ[∂φ(z2cosρ) − ∂z(ρπφ2)] − [1 ⁄ ρ]ρeφ[∂ρ(z2cosρ) − ∂z(3z)] + [1 ⁄ ρ]ez[∂ρ(ρπφ2) − ∂φ(3z)] = 

[1 ⁄ ρ]eρ[0 − 0] −  eφ[ − z2sinρ − 3] + [1 ⁄ ρ]ez[πφ2 − 0] = 

eφ(z2sinρ + 3) +  ez([π ⁄ ρ]φ2)39. RepeatExercise336↑withsphericalcoordinates.

Answer:Thegradientofadifferentiablescalarfieldfinorthogonalcoordinatesystemsofa3Dspaceisgivenby:∇f = (q1 ⁄ h1)(∂f ⁄ ∂q1) + ( q2 ⁄ h2)(∂f ⁄ ∂q2) + ( q3 ⁄ h3)(∂f ⁄ ∂q3)

Now,insphericalsystemswehave:q1 = rq2 = θq3 = φ

q1 =  erq2 =  eθq3 =  eφ

h1 = 1h2 = rh3 = rsinθ

Hence,theaboveequationbecomes:∇f = er(∂f ⁄ ∂r) + (eθ ⁄ r)(∂f ⁄ ∂θ) + (eφ ⁄ {rsinθ})(∂f ⁄ ∂φ)

40. RepeatExercise346↑withsphericalcoordinates.Answer:ThedivergenceofadifferentiablevectorfieldAinorthogonalcoordinatesystemsofa3Dspaceisgivenby:∇⋅A = [1 ⁄ (h1h2h3)][{∂(h2h3Â1) ⁄ ∂q1} + {∂(h1h3Â2) ⁄ ∂q2} + {∂(h1h2Â3) ⁄ ∂q3}]

Now,insphericalsystemswehave:q1 = rq2 = θq3 = φ

h1 = 1h2 = rh3 = rsinθ

Â1 = ArÂ2 = AθÂ3 = Aφ

Hence,theaboveequationbecomes:∇⋅A = 

[1 ⁄ (r2sinθ)][{∂(r2sinθAr) ⁄ ∂r} + {∂(rsinθAθ) ⁄ ∂θ} + {∂(rAφ) ⁄ ∂φ}] = 

[1 ⁄ (r2sinθ)][sinθ{∂(r2Ar) ⁄ ∂r} + r{∂(sinθAθ) ⁄ ∂θ} + r{∂Aφ ⁄ ∂φ}]41. RepeatExercise356↑withsphericalcoordinates.

Answer:

wherethesymbolsareasexplainedearlier.

42. RepeatExercise366↑withsphericalcoordinates.Answer:TheLaplacianofadifferentiablescalarfieldfinorthogonalcoordinatesystemsofa3Dspaceisgivenby:∇2f = [1 ⁄ (h1h2h3)]Σ3i = 1∂([(h1h2h3) ⁄ (hi)2][∂f ⁄ ∂qi]) ⁄ ∂qi

Now,insphericalsystemswehave:q1 = rq2 = θq3 = φ

h1 = 1h2 = rh3 = rsinθ

Hence,theaboveequationbecomes:∇2f = 

{[1 ⁄ (r2sinθ)]∂[r2sinθ(∂f ⁄ ∂r)] ⁄ ∂r} + {[1 ⁄ (r2sinθ)]∂[(r2sinθ ⁄ r2)(∂f ⁄ ∂θ)] ⁄ ∂θ} + {[1 ⁄ (r2sinθ)]∂[(r2sinθ ⁄ r2sin2θ)(∂f ⁄ ∂φ)] ⁄ ∂φ} = 

[(1 ⁄ r2)∂(r2{∂f ⁄ ∂r}) ⁄ ∂r] + [(1 ⁄ {r2sinθ})∂(sinθ{∂f ⁄ ∂θ}) ⁄ ∂θ] + [(1 ⁄ {r2sin2θ})(∂2f ⁄ ∂φ2)] = 

[(r2 ⁄ r2)(∂2f ⁄ ∂r2)] + [(1 ⁄ r2)(∂r2 ⁄ ∂r)(∂f ⁄ ∂r)] + [(sinθ ⁄ {r2sinθ})(∂2f ⁄ ∂θ2)] + [(cosθ ⁄ {r2sinθ})(∂f ⁄ ∂θ)] + [(1 ⁄ {r2sin2θ})(∂2f ⁄ ∂φ2)] = 

[∂2f ⁄ ∂r2] + [(2 ⁄ r)(∂f ⁄ ∂r)] + [(1 ⁄ r2)(∂2f ⁄ ∂θ2)] + [(cosθ ⁄ {r2sinθ})(∂f ⁄ ∂θ)] + [(1 ⁄ {r2sin2θ})(∂2f ⁄ ∂φ2)]

whereinstep3weusedtheproductruleofdifferentiation.43. Ascalarfieldinsphericalcoordinatesisgivenby:f(r, θ, φ) = r2 + θ.What

arethegradientandLaplacianofthisfield?Answer:Wehave:∇f = 

er∂rf + eθ(1 ⁄ r)∂θf + eφ[1 ⁄ (rsinθ)]∂φf = 

er2r + eθ(1 ⁄ r) + 0 = 

er2r + eθ(1 ⁄ r)

∇2f = 

∂rrf + (2 ⁄ r)∂rf + (1 ⁄ r2)∂θθf + [cosθ ⁄ (r2sinθ)]∂θf + [1 ⁄ (r2sin2θ)]∂φφf = 

2 + (2 ⁄ r)(2r) + 0 + [cosθ ⁄ (r2sinθ)] + 0 = 

6 + [cosθ ⁄ (r2sinθ)]44. Avectorfieldinsphericalcoordinatesisgivenby:A(r, θ, φ) = (er, 5sinφ, 

lnθ).Whatarethedivergenceandcurlofthisfield?Answer:Wehave:∇⋅A = 

[1 ⁄ (r2sinθ)][sinθ{∂(r2Ar) ⁄ ∂r} + r{∂(sinθAθ) ⁄ ∂θ} + r{∂Aφ ⁄ ∂φ}] = 

[1 ⁄ (r2sinθ)][sinθ{∂(r2er) ⁄ ∂r} + r{∂(sinθ5sinφ) ⁄ ∂θ} + r{∂lnθ ⁄ ∂φ}] = 

[1 ⁄ (r2sinθ)][sinθ(2rer) + sinθ(r2er) + 5rcosθsinφ + 0] = 

[1 ⁄ (r2sinθ)][2rersinθ + r2ersinθ + 5rcosθsinφ] = 

[1 ⁄ (rsinθ)][2ersinθ + rersinθ + 5cosθsinφ]

whereinline3weusedtheproductruleofdifferentiation.∇ × A = 

[er ⁄ (r2sinθ)][∂θ(rsinθlnθ) − ∂φ(r5sinφ)] − [(reθ) ⁄ (r2sinθ)][∂r(rsinθlnθ) − ∂φ(er)] + [(rsinθeφ) ⁄ (r2sinθ)][∂r(r5sinφ) − ∂θ(er)] = 

[er ⁄ (r2sinθ)][rcosθlnθ + rsinθ(1 ⁄ θ) − r5cosφ] − [eθ ⁄ (rsinθ)][sinθlnθ − 0] + [ eφ ⁄ r][5sinφ − 0] = 

[er ⁄ (rsinθ)][cosθlnθ + (1 ⁄ θ)sinθ − 5cosφ] − eθ[(lnθ) ⁄ r] + eφ[(5sinφ) ⁄ r]

Chapter7TensorsinApplication1. Summarizethereasonsforthepopularityoftensorcalculustechniquesin

mathematical,scientificandengineeringapplications.Answer:Thesetechniquesarebeautiful,powerfulandsuccinct.

2. State,intensorlanguage,thedefinitionofthefollowingmathematicalconceptsassumingCartesiancoordinatesofa3Dspace:traceofmatrix,determinantofmatrix,inverseofmatrix,multiplicationoftwocompatiblesquarematrices,dotproductoftwovectors,crossproductoftwovectors,scalartripleproductofthreevectorsandvectortripleproductofthreevectors.Answer:tr(A) = Aii

det(A) = [1 ⁄ (3!)]ϵijkϵlmnAilAjmAkn

[A − 1]ij = [1 ⁄ {2 det(A)}]ϵipqϵjmnAmpAnq

[AB]ik = AijBjk

a⋅b = δijaibj = aibi

[a × b]i = ϵijkajbk

a⋅(b × c) = ϵijkaibjck

[a × (b × c)]i = ϵijkϵklmajblcm

whereAandBaresquarematricesanda,bandcarevectors.3. FromthetensordefinitionofA × (B × C),obtainthetensordefinitionof

(A × B) × C.Answer:Wehave:

[A × (B × C)]i = ϵijkϵklmAjBlCm

[(B × C) × A]i =  − ϵijkϵklmAjBlCm

[(A × B) × C]i =  − ϵijkϵklmCjAlBm

[(A × B) × C]i = ϵikjϵklmCjAlBm

[(A × B) × C]i = ϵikmϵkjlAjBlCm

[(A × B) × C]i = ϵikmϵjlkAjBlCm

whereline2isjustifiedbytheanti-commutativepropertyofcrossproductofvectors[i.e.vectorAandvector(B × C)],inline3werelabelthethreevectors,inline4weusetheanti-symmetricpropertyofthepermutationtensor,inline5werelabelthedummyindices,andinline6weusethecyclicpropertyoftheindicesofthepermutationtensor.

4. WehavethefollowingtensorsinorthonormalCartesiancoordinatesofa3Dspace:A = (22, 3π, 6.3)

B = (3e, 1.8, 4.9)

C = (47, 5e, 3.5)

Usethetensorexpressionsfortherelevantmathematicalconceptswith

systematicsubstitutionoftheindicesvaluestofindthefollowing:tr(D)det(E)D − 1

E⋅DA⋅CC × BC⋅(A × B)B × (C × A).Answer:●tr(D):tr(D) = 

Dii = 

D11 + D22 = 

π + e

●det(E):det(E) = 

ϵijEi1Ej2 = 

ϵ11E11E12 + ϵ22E21E22 + ϵ12E11E22 + ϵ21E21E12 = 

0 + 0 + E11E22 − E21E12 = 

(3 × 7) − (π3e2) = 

21 − π3e2

●D − 1:weusetheformula:[D − 1]ij = 

[1 ⁄  det(D)]δimjnDnm = 

[1 ⁄  det(D)](δi1j1D11 + δi1j2D21 + δi2j1D12 + δi2j2D22)

Therefore,wehave:[D − 1]11 = 

[1 ⁄  det(D)](δ1111D11 + δ1112D21 + δ1211D12 + δ1212D22) = 

[1 ⁄  det(D)](0 + 0 + 0 + D22) = 

D22 ⁄  det(D)

[D − 1]12 = 

[1 ⁄  det(D)](δ1121D11 + δ1122D21 + δ1221D12 + δ1222D22) = 

[1 ⁄  det(D)](0 + 0 − D12 + 0) = 

 − D12 ⁄  det(D)

[D − 1]21 = 

[1 ⁄  det(D)](δ2111D11 + δ2112D21 + δ2211D12 + δ2212D22) = 

[1 ⁄  det(D)](0 − D21 + 0 + 0) = 

 − D21 ⁄  det(D)

[D − 1]22 = 

[1 ⁄  det(D)](δ2121D11 + δ2122D21 + δ2221D12 + δ2222D22) = 

[1 ⁄  det(D)](D11 + 0 + 0 + 0) = 

D11 ⁄  det(D)

Now,det(D) = πe − 12andhence:

●E⋅D:weusetheformula:[E⋅D]ij = 

EikDkj = 

Ei1D1j + Ei2D2j

thatis:[E⋅D]11 = 

E11D11 + E12D21 = 

3π + e24 = 

3π + 4e2

[E⋅D]12 = 

E11D12 + E12D22 = 

3 × 3 + e2e = 

9 + e3

[E⋅D]21 = 

E21D11 + E22D21 = 

π3π + 7 × 4 = 

π4 + 28

[E⋅D]22 = 

E21D12 + E22D22 = 

π33 + 7e = 

3π3 + 7e

Hence:

●A⋅C:A⋅C = 

AiCi = 

A1C1 + A2C2 + A3C3 = 

22 × 47 + 3π × 5e + 6.3 × 3.5 = 

1034 + 15πe + 22.05≃

1184.15

●C × B:weusetheformula:[C × B]i = ϵijkCjBk

Now,sinceϵijkiszerowhenwehaverepetitiveindicesthen:

wheni = 1wehaveϵ111 = ϵ112 = ϵ113 = ϵ121 = ϵ122 = ϵ131 = ϵ133 = 0.

wheni = 2wehaveϵ211 = ϵ212 = ϵ221 = ϵ222 = ϵ223 = ϵ232 = ϵ233 = 0.

wheni = 3wehaveϵ311 = ϵ313 = ϵ322 = ϵ323 = ϵ331 = ϵ332 = ϵ333 = 0.

Hence,weshouldhave:[C × B]1 = 

ϵ123C2B3 + ϵ132C3B2 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 

C2B3 − C3B2 = 

5e × 4.9 − 3.5 × 1.8 = 

24.5e − 6.3

[C × B]2 = 

ϵ213C1B3 + ϵ231C3B1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 

 − C1B3 + C3B1 = 

 − 47 × 4.9 + 3.5 × 3e = 

 − 230.3 + 10.5e

[C × B]3 = 

ϵ312C1B2 + ϵ321C2B1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 

C1B2 − C2B1 = 

47 × 1.8 − 5e × 3e = 

84.6 − 15e2

Hence:C × B = 

(24.5e − 6.3, 10.5e − 230.3, 84.6 − 15e2)≃

(60.30,  − 201.76,  − 26.24)

●C⋅(A × B):weusetheformula:C⋅(A × B) = ϵijkCiAjBk

Now,sinceϵijkiszerowhenwehaverepetitiveindicesthenweshouldhaveonly6non-vanishingtermswhichrepresentthenon-repetitivepermutationsof123,thatis:C⋅(A × B) = 

ϵ123C1A2B3 + ϵ312C3A1B2 + ϵ231C2A3B1 + ϵ132C1A3B2 + ϵ213C2A1B3 + ϵ321C3A2B1 = 

 + C1A2B3 + C3A1B2 + C2A3B1 − C1A3B2 − C2A1B3 − C3A2B1 = 

 + 47 × 3π × 4.9 + 3.5 × 22 × 1.8 + 5e × 6.3 × 3e − 47 × 6.3 × 1.8 − 5e × 22 × 4.9 − 3.5 × 3π × 3e = 

690.9π − 394.38 + 94.5e2 − 539e − 31.5πe≃

740.26

●B × (C × A):weusetheformula:[B × (C × A)]i = ϵijkϵklmBjClAm

Now,sinceϵijkiszerowhenwehaverepetitiveindicesthenweshouldhaveonly6non-vanishingtermswhichrepresentthenon-repetitivepermutationsof123.Thissimilarlyappliestoϵklm;howevertheindexkofϵklm,isfixedbytheindexkofϵijkandhencewehaveonly2non-vanishingpermutationsforeachoneofthesixpermutationsofϵijk(i.e.thenon-repetitivepermutationsoflmwithl ≠ kandm ≠ k).Accordingly,weshouldhaveatotalof12non-vanishingterms,i.e.4non-vanishingtermsforeachvalueofi,thatis:[B × (C × A)]1 = 

ϵ123ϵ312B2C1A2 + ϵ132ϵ213B3C1A3 + ϵ123ϵ321B2C2A1 + ϵ132ϵ231B3C3A1 = 

B2C1A2 + B3C1A3 − B2C2A1 − B3C3A1 = 

1.8 × 47 × 3π + 4.9 × 47 × 6.3 − 1.8 × 5e × 22 − 4.9 × 3.5 × 22≃

1332.71

[B × (C × A)]2 = 

ϵ213ϵ312B1C1A2 + ϵ231ϵ123B3C2A3 + ϵ213ϵ321B1C2A1 + ϵ231ϵ132B3C3A2 = 

 − B1C1A2 + B3C2A3 + B1C2A1 − B3C3A2 = 

 − 3e × 47 × 3π + 4.9 × 5e × 6.3 + 3e × 5e × 22 − 4.9 × 3.5 × 3π≃

 − 915.99

[B × (C × A)]3 = 

ϵ312ϵ213B1C1A3 + ϵ321ϵ123B2C2A3 + ϵ312ϵ231B1C3A1 + ϵ321ϵ132B2C3A2 = 

 − B1C1A3 − B2C2A3 + B1C3A1 + B2C3A2 = 

 − 3e × 47 × 6.3 − 1.8 × 5e × 6.3 + 3e × 3.5 × 22 + 1.8 × 3.5 × 3π≃

 − 1881.48

Hence:B × (C × A)≃(1332.71,  − 915.99,  − 1881.48)

5. Statethematrixandtensordefinitionsofthemainthreeindependentscalarinvariants(I, IIandIII)ofrank-2tensors.Answer:Theyare:I = tr(A) = Aii

II = tr(A2) = AijAji

III = tr(A3) = AijAjkAki

whereAisarank-2tensor.6. Expressthemainthreeindependentscalarinvariants(I, IIandIII)ofrank-

2tensorsintermsofthethreesubsidiaryscalarinvariants(I1, I2andI3).Answer:Theyare(seeFootnote34in§8↓):I = I1

II = (I1)2 − 2I2

III = (I1)3 − 3I1I2 + 3I37. ReferringtoQuestion47↑,findthethreescalarinvariants(I, IIandIII)of

Dandthethreescalarinvariants(I1, I2andI3)ofEusingthetensordefinitionsoftheseinvariantswithsystematicindexsubstitution.Answer:●Wehave:I(D) = 

Dii = 

D11 + D22 = 

π + e≃

5.86

II(D) = 

DijDji = 

D11D11 + D12D21 + D21D12 + D22D22 = 

π × π + 3 × 4 + 4 × 3 + e × e = 

π2 + 24 + e2≃

41.26

III(D) = 

DijDjkDki = 

D11D11D11 + D12D21D11 + D11D12D21 + D12D22D21 + D21D11D12 + D22D21D12 + D21D12D22 + D22D22D22 = 

π3 + 12π + 12π + 12e + 12π + 12e + 12e + e3 = 

π3 + 36π + 36e + e3≃

262.05

●Wehave:I1( E) = 

Eii = 

E11 + E22 = 

3 + 7 = 

10

I2( E) = 

(1 ⁄ 2)(EiiEjj − EijEji) = 

(1 ⁄ 2)(E11E11 + E11E22 + E22E11 + E22E22) − (1 ⁄ 2)(E11E11 + E12E21 + E21E12 + E22E22) = 

(1 ⁄ 2)(9 + 21 + 21 + 49) − (1 ⁄ 2)(9 + e2π3 + π3e2 + 49) = 

21 − e2π3≃

 − 208.11

I3( E) = 

det(E) = 

ϵijEi1Ej2 = 

ϵ11E11E12 + ϵ22E21E22 + ϵ12E11E22 + ϵ21E21E12 = 

0 + 0 + E11E22 − E21E12 = 

(3 × 7) − (π3e2) = 

21 − π3e2≃

 − 208.1

8. Statethefollowingvectoridentitiesintensornotation:∇ × r = 0

∇⋅(fA) = f∇⋅A + A⋅∇f

A × (∇ × B) = (∇B)⋅A − A⋅∇B

∇ × (A × B) = (B⋅∇)A + (∇⋅B)A − (∇⋅A)B − (A⋅∇)B

Answer:AssumingCartesiancoordinates,wehave:ϵijk∂jxk = 0

∂i(fAi) = f∂iAi + Ai∂if

ϵijkϵklmAj∂lBm = (∂iBm)Am − Al(∂lBi)

ϵijkϵklm∂j(AlBm) = (Bm∂m)Ai + (∂mBm)Ai − (∂jAj)Bi − (Aj∂j)Bi9. StatethedivergenceandStokestheoremsforavectorfieldinCartesian

coordinatesusingvectorandtensornotations.Also,defineallthesymbolsinvolved.Answer:AssumingCartesiancoordinates,wehave:●Divergencetheorem:∭Ω∇⋅Adτ = ∬SA⋅ndσ

 ⨏Ω∂iAidτ =  ⨏SAinidσ

whereAisadifferentiablevectorfield,ΩisaboundedregioninannDspaceenclosedbyageneralizedsurfaceS,dτanddσaregeneralizedvolumeandareadifferentials,nandniaretheunitvectornormaltothesurfaceanditsithcomponent,andtheindexirangesover1, …, n.●Stokestheorem:∬S(∇ × A)⋅ndσ =  ⨏CA⋅dr

 ⨏Sϵijk∂jAknidσ =  ⨏CAidxi

whereCstandsfortheperimeterofthesurfaceS,anddrisadifferentialofthepositionvectorwhichistangenttotheperimeter,xiisaCartesian

coordinatewhiletheothersymbolsareasdefinedinthefirstpart.10. Provethefollowingvectoridentitiesusingtensornotationandtechniques

withfulljustificationofeachstep:∇⋅r = n

∇⋅(∇ × A) = 0

A⋅(B × C) = C⋅(A × B)

∇ × (∇ × A) = ∇(∇⋅A) − ∇2 A

Answer:AssumingCartesiancoordinates,wehave:●∇⋅r = n:∇⋅r = 

∂ixi = 

δii = 

n

whereline1isthedefinitionofdivergence,line2istheidentity∂jxi = δijwithj = i,andline3istheidentityδii = nwhichisgiveninthebookandprovedinExercise224↑of§4↑.●∇⋅(∇ × A) = 0:∇⋅(∇ × A) = 

∂i[∇ × A]i = 

∂i(ϵijk∂jAk) = 

ϵijk∂i∂jAk = 

ϵijk∂j∂iAk = 

 − ϵjik∂j∂iAk = 

 − ϵijk∂i∂jAk = 

0

whereline1isthedefinitionofdivergence,line2isthedefinitionofcurl,line3istheconstancyofcomponentsofthepermutationtensor,line4isthecommutativityofpartialdifferentialoperators,line5istheanti-symmetryofthepermutationtensor,line6isrelabelingofdummyindices,andline7isbasedoncomparingline6withline3plusthefactthatonly0isequaltoitsnegative.●A⋅(B × C) = C⋅(A × B):A⋅(B × C) = 

ϵijkAiBjCk = 

ϵkijAiBjCk = 

ϵkijCkAiBj = 

C⋅(A × B)

whereline1isthedefinitionofscalartripleproduct,line2isthecyclicpropertyofϵijk,line3isthecommutativityofordinarymultiplication,andline4isthedefinitionofscalartripleproduct.●∇ × (∇ × A) = ∇(∇⋅A) − ∇2 A:[∇ × (∇ × A)]i = 

ϵijk∂j[∇ × A]k = 

ϵijk∂j(ϵklm∂lAm) = 

ϵijkϵklm∂j(∂lAm) = 

ϵijkϵlmk∂j∂lAm = 

(δilδjm − δimδjl)∂j∂lAm = 

δilδjm∂j∂lAm − δimδjl∂j∂lAm = 

∂m∂iAm − ∂l∂lAi = 

∂i(∂mAm) − ∂llAi = 

[∇(∇⋅A)]i − [∇2 A]i = 

[∇(∇⋅A) − ∇2 A]i

whereline1isthedefinitionofcurl(firstcurl),line2isthedefinitionofcurl(secondcurl),line3istheconstancyofcomponentsofthepermutationtensor,line4isthecyclicpropertyofϵklm,line5istheepsilon-deltaidentity,line6isthedistributivityofproductoveralgebraicsum,line7isindexreplacementoperation,line8isthecommutativityofpartialdifferentialoperatorsandthedefinitionofsecondderivative,line9isbasedonthedefinitionsofgradient,divergenceandLaplacian,line10isthedistributivityofindexingoveralgebraicsumoftensorterms.Sinceiisafreeindex,thentheidentityequallyappliestoallcomponentsandhence:∇ × (∇ × A) = ∇(∇⋅A) − ∇2 Aasrequired.

11. Whatisthetype,intheformof(m, n, w),oftheRiemann-Christoffelcurvaturetensorofthefirstandsecondkinds?Answer:Thetypeofthefirstkindis(0, 4, 0)whilethetypeofthesecondkindis(1, 3, 0).

12. WhataretheothernamesusedtolabeltheRiemann-Christoffelcurvaturetensorofthefirstandsecondkinds?Answer:ThefirstkindmaybecalledthecovariantortotallycovariantRiemann-Christoffelcurvaturetensor,whilethesecondkindmaybecalledthemixedRiemann-Christoffelcurvaturetensor.

13. WhatistheimportanceoftheRiemann-Christoffelcurvaturetensorwithregardtocharacterizingthespaceasflatorcurved?Answer:TheRiemann-Christoffelcurvaturetensorvanishesidenticallyiffthespaceisgloballyflat.Hence,bytestingtheRiemann-Christoffelcurvaturetensorwecandetermineifthespaceisflat(ifthetensorvanishesidentically)orcurved(ifnot).

14. StatethemathematicaldefinitionoftheRiemann-Christoffelcurvature

tensorofeitherkindsindeterminantalform.Answer:

15. HowcanweobtaintheRiemann-Christoffelcurvaturetensorofthefirstkindfromthesecondkindandviceversa?Answer:Thefirstkindcanbeobtainedfromthesecondkindbyloweringthefirstindexofthesecondkind,whilethesecondkindcanbeobtainedfromthefirstkindbyraisingthefirstindexofthefirstkind.

16. UsingthedefinitionofthesecondordermixedcovariantderivativeofavectorfieldandthedefinitionofthemixedRiemann-Christoffelcurvaturetensor,verifythefollowingequation:Aj;kl − Aj;lk = RijklAi.Repeatthequestionwiththeequation:Aj;kl − Aj;lk = RjilkAi.Answer:●InQuestion525↑of§5↑weobtained:Ai;jk − Ai;kj = 

 − Ab∂kΓbij + ΓaikΓbajAb + Ab∂jΓbik − ΓaijΓbakAb = 

(∂jΓbik − ∂kΓbij + ΓaikΓbaj − ΓaijΓbak)Ab = 

RbijkAb

Onrelabelingtheindices,weobtain:Aj;kl − Aj;lk = RijklAi

whichistherequiredresult.●Inthebookweobtainedthefollowingequations:Ai;jk = ∂k∂jAi + Γiaj∂kAa − Γajk∂aAi + Γiak∂jAa + Aa(∂kΓiaj − ΓbjkΓiba + 

ΓibkΓbaj)

Ai;kj = ∂j∂kAi + Γiak∂jAa − Γakj∂aAi + Γiaj∂kAa + Aa(∂jΓiak − ΓbkjΓiba + ΓibjΓbak)

Ontakingthedifferenceweobtain:Ai;jk − Ai;kj = 

∂k∂jAi + Γiaj∂kAa − Γajk∂aAi + Γiak∂jAa + Aa(∂kΓiaj − ΓbjkΓiba + ΓibkΓbaj) − [∂j∂kAi + Γiak∂jAa − Γakj∂aAi + Γiaj∂kAa + Aa(∂jΓiak − ΓbkjΓiba + ΓibjΓbak)] = 

Aa(∂kΓiaj + ΓibkΓbaj) − Aa(∂jΓiak + ΓibjΓbak) = 

(∂kΓiaj − ∂jΓiak + ΓibkΓbaj − ΓibjΓbak)Aa = 

RiakjAa

Onrelabelingtheindices,weobtain:Aj;kl − Aj;lk = RjilkAi

whichistherequiredresult.17. BasedontheequationsinQuestion167↑,whatisthenecessaryand

sufficientconditionforthecovariantdifferentialoperatorstobecomecommutative?Answer:ThecovariantdifferentialoperatorsbecomecommutativeifftheRiemann-Christoffelcurvaturetensorvanishesidentically.

18. State,mathematically,theanti-symmetricandblocksymmetricpropertiesoftheRiemann-Christoffelcurvaturetensorofthefirstkindinitsfourindices.Answer:Rjikl =  − Rijkl

Rijlk =  − Rijkl

Rklij =  + Rijkl

whereline1istheanti-symmetricpropertyinthefirsttwoindices,line2istheanti-symmetricpropertyinthelasttwoindices,andline3istheblocksymmetricproperty.

19. Basedonthetwoanti-symmetricpropertiesofthecovariantRiemann-Christoffelcurvaturetensor,listalltheformsofthecomponentsofthetensorthatareidenticallyzero(e.g.Riijk).Answer:Anycomponentwithatleastapairofidenticalanti-symmetricindicesshouldvanishidentically.Hence,theformsofthecomponentsthatareidenticallyzeroare:

Twoindicesidentical:Riijk,Rijkk,Riikk.

Threeindicesidentical:Riiik,Riiji,Rkjkk,Rikkk.

Fourindicesidentical:Riiii.20. Verifytheblocksymmetricpropertyandthetwoanti-symmetricproperties

ofthecovariantRiemann-Christoffelcurvaturetensorusingitsdefinition.Answer:●Blocksymmetricproperty:Rklij = 

(1 ⁄ 2)(∂i∂lgjk + ∂j∂kgli − ∂i∂kglj − ∂j∂lgik) + grs([kj, r][li, s] − [ki, r][lj, s]) = 

(1 ⁄ 2)(∂l∂igjk + ∂k∂jgli − ∂k∂iglj − ∂l∂jgik) + grs([kj, r][li, s] − [ki, r][lj, s]) = 

(1 ⁄ 2)(∂k∂jgli + ∂l∂igjk − ∂k∂iglj − ∂l∂jgik) + grs([li, s][kj, r] − [ki, r][lj, s]) = 

(1 ⁄ 2)(∂k∂jgli + ∂l∂igjk − ∂k∂igjl − ∂l∂jgki) + grs([il, s][jk, r] − [ik, r][jl, s]) = 

(1 ⁄ 2)(∂k∂jgli + ∂l∂igjk − ∂k∂igjl − ∂l∂jgki) + grs([il, r][jk, s] − [ik, r][jl, s]) = 

Rijkl

whereline1isobtainedfromthegivendefinitionofthecovariantRiemann-Christoffelcurvaturetensor(whichisgiveninthebookasRijkl)

withrelabelingtheindices(ijkl → klij),line2isthecommutativityofpartialdifferentialoperators,line3isreorderingoftermsandfactors,line4isthesymmetryofthemetrictensorandthesymmetryoftheChristoffelsymbolsintheirpairedindices,line5isrelabelingofdummyindicesplusthesymmetryofthemetrictensor,andline6isthedefinitionofRijklwhichisgiveninthebook.●Anti-symmetricpropertyinthefirsttwoindices:Rjikl = 

(1 ⁄ 2)(∂k∂iglj + ∂l∂jgik − ∂k∂jgil − ∂l∂igkj) + grs([jl, r][ik, s] − [jk, r][il, s]) = 

 − [(1 ⁄ 2)(∂k∂jgil + ∂l∂igkj − ∂k∂iglj − ∂l∂jgik) + grs([jk, r][il, s] − [jl, r][ik, s])] = 

 − [(1 ⁄ 2)(∂k∂jgli + ∂l∂igjk − ∂k∂igjl − ∂l∂jgki) + gsr([il, s][jk, r] − [ik, s][jl, r])] = 

 − [(1 ⁄ 2)(∂k∂jgli + ∂l∂igjk − ∂k∂igjl − ∂l∂jgki) + grs([il, r][jk, s] − [ik, r][jl, s])] = 

 − Rijkl

whereline1isobtainedfromthegivendefinitionofthecovariantRiemann-Christoffelcurvaturetensorwithrelabelingtheindices(ijkl → jikl),line2istakingacommonfactorof − 1,line3isthesymmetryofthemetrictensorplusreordering,line4isrelabelingofdummyindices,andline5isthedefinitionofRijklwhichisgiveninthebook.●Anti-symmetricpropertyinthelasttwoindices:Rijlk = 

(1 ⁄ 2)(∂l∂jgki + ∂k∂igjl − ∂l∂igjk − ∂k∂jgli) + grs([ik, r][jl, s] − [il, r][jk, s]) = 

 − [(1 ⁄ 2)(∂l∂igjk + ∂k∂jgli − ∂l∂jgki − ∂k∂igjl) + grs([il, r][jk, s] − [ik, r][jl, s])] = 

 − [(1 ⁄ 2)(∂k∂jgli + ∂l∂igjk − ∂k∂igjl − ∂l∂jgki) + grs([il, r][jk, s] − [ik, r]

[jl, s])] = 

 − Rijkl

whereline1isobtainedfromthegivendefinitionofthecovariantRiemann-Christoffelcurvaturetensorwithrelabelingtheindices(ijkl → ijlk),line2istakingacommonfactorof − 1,line3isreorderingofterms,andline4isthedefinitionofRijklwhichisgiveninthebook.

21. RepeatQuestion207↑fortheanti-symmetricpropertyofthemixedRiemann-Christoffelcurvaturetensorinitslasttwoindices.Answer:Rijlk = 

∂lΓijk − ∂kΓijl + ΓrjkΓirl − ΓrjlΓirk = 

 − (∂kΓijl − ∂lΓijk + ΓrjlΓirk − ΓrjkΓirl) = 

 − Rijkl

whereline1isobtainedfromthegivendefinitionofthemixedRiemann-Christoffelcurvaturetensor(whichisgiveninthebookasRijkl)withrelabelingtheindices(ijkl → ijlk),line2istakingacommonfactorof − 1,andline3isthedefinitionofRijklwhichisgiveninthebook.

22. Basedontheblocksymmetricandanti-symmetricpropertiesofthecovariantRiemann-Christoffelcurvaturetensor,find(withfulljustification)thenumberofdistinctnon-vanishingentriesofthethreemaintypesofthistensor(seetheequationsforN2,N3andN4whicharegiveninthebook).Hence,findthetotalnumberoftheindependentnon-zerocomponentsofthistensor.Answer:Wehavethreemaincases:(a)EntrieswithonlytwodistinctindicesoftypeRijij:duetotheanti-symmetricproperties,theindices1and2shouldbedifferentandtheindices3and4shouldbedifferentsothatthecomponentdoesnotvanishidentically.Moreover,sincewehaveonlytwodistinctindicesthentheformofthistypeshouldbeeitherRijijorRijji.However,sincethesetwoformsdifferonlybysignduetotheanti-symmetricpropertyinthelasttwoindices

(orthefirsttwoindices),thenalltheindependentnon-zerocomponentsofthistypecanberepresentedbyjustoneoftheseforms,sayRijij.Now,thenumberofcomponentsofthisformisequaltothenumberofpermutationsofijandbecauseiandjaredistinctthenthenumberofpermutationsisn(n − 1).However,duetotheanti-symmetricpropertiesthepermutationscorrespondingtoIJ(say23)andthepermutationscorrespondingtoJI(say32)areidenticalandhencetheyarenotdistinct.Therefore,onlyhalfofthesepermutationswillcontributetothenumberofindependentnon-zerocomponentsofthistensor,thatis(seeFootnote35in§8↓):N2 = [n(n − 1)] ⁄ 2

(b)EntrieswithonlythreedistinctindicesoftypeRijki:duetotheanti-symmetricproperties,theindices1and2shouldbedifferentandtheindices3and4shouldbedifferentsothatthecomponentdoesnotvanishidentically.Therefore,theidenticalindicesshouldbeeither1and3(i.e.Rijik),or1and4(i.e.Rijki),or2and3(i.e.Rjiik),or2and4(i.e.Rjiki).Now,duetoanti-symmetrytheformsRijikandRijkiarenotindependentandtheformsRjiikandRjikiarenotindependentandhenceweareleftwithRijki(representingthefirsttwoforms)andRjiik(representingthelasttwoforms).However,duetoanti-symmetrywehaveRijki =  − Rjiki = Rjiikandhenceeventhesetwoformsarenotindependent.Therefore,alltheindependentnon-zerocomponentswithonlythreedistinctindicesarerepresentedbyasingleform,sayRijki.Now,sincethelastiisnotindependent(andhenceallthecomponentsrepresentedbyRijkicanbesimilarlyrepresentedbyRijk)thenthenumberofcomponentsrepresentedbythisformisequaltothenumberofpermutationsofijkandsincethesethreeindicesaredistinctthenweshouldhaven(n − 1)(n − 2)permutations.However,duetotheanti-symmetricpropertyinthefirsttwoindicesthenRijk =  − Rjik(e.g.R123 =  − R213)andhenceonlyhalfofthesepermutationswillcontributetothenumberofindependentnon-zerocomponentsofthistensor,thatis(seeFootnote36in§8↓):N3 = [n(n − 1)(n − 2)] ⁄ 2

(c)EntrieswithfourdistinctindicesoftypeRijkl:thenumberofcomponentsrepresentedbyRijklisequaltothenumberofpermutationsofijklwhichisgivenbyn(n − 1)(n − 2)(n − 3)sincealltheseindicesaredistinct.However,

duetotheanti-symmetricpropertyinthefirsttwoindicesweshouldhalvethisnumbersincetheshiftinindicesproducesonlydifferenceinsignandhencethecomponentsthatareoppositeinsignarenotindependent.Thissimilarlyappliestotheanti-symmetricpropertyinthelasttwoindicesandhenceweshouldhalveagain.So,weareleftwith[n(n − 1)(n − 2)(n − 3)] ⁄ 4non-vanishingandpotentiallyindependentcomponents.Now,theblocksymmetricpropertywillreducethisnumberbyafactorof2becausealltheremainingcomponentshavenoanti-symmetriccorrespondencesinceitisalreadyremovedandhencethetwoblocksareliketwoindicesandhencebytheblocksymmetryweshouldalsohalve.Thismeansthatweareleftwith[n(n − 1)(n − 2)(n − 3)] ⁄ 8non-vanishingandpotentiallyindependentcomponents.Finally,thefirstBianchiidentitylinkseachsetofthreeoftheremainingcomponentsandhenceoneofthethreecanbeexpressedintermsoftheothertwoandthereforeitisnotindependent.Thismeansthatthenumberwillbereducedbyafactorof2 ⁄ 3(sinceonethirdofthethreeisdependentonthetwothirds).Hence,thenumberoftheindependentnon-zerocomponentsthatarecontributedbythistypeis(seeFootnote37in§8↓):N4 = 

[{n(n − 1)(n − 2)(n − 3)} ⁄ 8] × [2 ⁄ 3] = 

[n(n − 1)(n − 2)(n − 3)] ⁄ 12

Toclarifycase(c),wepresentinthefollowingtablehowN4isobtainedforthecaseof4Dspace(i.e.n = 4)whereincolumn1weincludeallthepermutations[i.e.n(n − 1)(n − 2)(n − 3) = 24],incolumn2weapplyanti-symmetryinthefirsttwoindices[i.e.{n(n − 1)(n − 2)(n − 3)} ⁄ 2 = 12],incolumn3weapplyanti-symmetryinthelasttwoindices[i.e.{n(n − 1)(n − 2)(n − 3)} ⁄ 4 = 6],incolumn4weapplyblocksymmetry[i.e.{n(n − 1)(n − 2)(n − 3)} ⁄ 8 = 3],andincolumn5weapplythefirstBianchiidentity[i.e.{n(n − 1)(n − 2)(n − 3)} ⁄ 12 = 2]sinceR1324 = R1234 + R1423(seeFootnote38in§8↓).

1234 1234 1234 1234 12341243 12431324 1324 1324 1324

1342 13421423 1423 1423 1423 14231432 1432213421432314 2314 23142341 23412413 2413 24132431 243131243142321432413412 3412 34123421 3421412341324213423143124321

Accordingly,thetotalnumberoftheindependentnon-zerocomponentsofthistensoris:NRI = 

N2 + N3 + N4 = 

[{n(n − 1)} ⁄ 2] + [{n(n − 1)(n − 2)} ⁄ 2] + [{n(n − 1)(n − 2)(n − 3)} ⁄ 12] = 

[(n2 − n) ⁄ 2] + [(n3 − n2 − 2n2 + 2n) ⁄ 2] + [{(n3 − n2 − 2n2 + 2n)(n − 3)} ⁄ 12] = 

[(n3 − 2n2 + n) ⁄ 2] + [{(n3 − 3n2 + 2n)(n − 3)} ⁄ 12] = 

[(n3 − 2n2 + n) ⁄ 2] + [(n4 − 3n3 + 2n2 − 3n3 + 9n2 − 6n) ⁄ 12] = 

(6n3 − 12n2 + 6n + n4 − 3n3 + 2n2 − 3n3 + 9n2 − 6n) ⁄ 12 = 

(n4 − n2) ⁄ 12 = 

[n2(n2 − 1)] ⁄ 1223. UsetheformulaefoundinQuestion227↑andotherformulaegiveninthe

texttofindthenumberofallcomponents,thenumberofnon-zerocomponents,thenumberofzerocomponentsandthenumberofindependentnon-zerocomponentsofthecovariantRiemann-Christoffelcurvaturetensorin2D,3Dand4Dspaces.Answer:WesymbolizethenumberofallcomponentswithNa,thenumberofnon-zerocomponentswithNnz,thenumberofzerocomponentswithNzandthenumberofindependentnon-zerocomponentswithNRI,whileweusentosymbolizethedimensionalityofthespaceandhencewehave(seeFootnote39in§8↓):

24. Provethefollowingidentitywithfulljustificationofeachstepofyourproof:Raakl = 0.Answer:Wehave:Raakl = 

∂kΓaal − ∂lΓaak + ΓralΓark − ΓrakΓarl = 

∂kΓaal − ∂lΓaak + ΓralΓark − ΓarkΓral = 

∂kΓaal − ∂lΓaak = 

∂k[∂l{ln√(g)}] − ∂l[∂k{ln√(g)}] = 

∂k∂l[ln√(g)] − ∂l∂k[ln√(g)] = 

∂k∂l[ln√(g)] − ∂k∂l[ln√(g)] = 

0

whereline1isthedefinitionofRijkl(whichisgiveninthebook)withi = j = a,line2isrelabelingofdummyindicesinthelastterm,line4istheidentityΓjji = ∂i[ln√(g)]whichisgiveninthebook,andline6isthecommutativityofpartialdifferentialoperators.

25. MakealistofallthemainpropertiesoftheRiemann-Christoffelcurvaturetensor(i.e.rank,type,symmetry,etc.).Answer:Someofthesepropertiesare:●Itisabsolutetensor.●Itisrank-4tensor.●Itcanbecovariantoftype(0, 4)ormixedoftype(1, 3).●Thecovarianttypeisanti-symmetricinitsfirsttwoindicesandinitslasttwoindicesandblocksymmetricinthefirstandsecondpairsofindices,whilethemixedtypeisanti-symmetricinitslasttwoindices.●Itdependsonlyonthemetrictensor.●Itcharacterizesthespaceandhenceitisusedforexampleasatestforthecurvatureofspacesinceitvanishesidenticallyiffthespaceisflat.●Whenitvanishesthecovariantdifferentialoperatorsbecomecommutative.

26. ProvethefollowingidentityusingtheBianchiidentities:Rijkl;s + Riljk;s = Riksl;j + Rikjs;l.Answer:Wehave:Rijkl;m + Rijlm;k + Rijmk;l = 0

Rikjs;l + Riksl;j + Riklj;s = 0

 − Riklj;s = Riksl;j + Rikjs;l

( − Riklj);s = Riksl;j + Rikjs;l

(Rijkl + Riljk);s = Riksl;j + Rikjs;l

Rijkl;s + Riljk;s = Riksl;j + Rikjs;l

whereline1isthesecondoftheBianchiidentitiesasgiveninthebook,line2isrelabelingtheindices(i.e.ijklm → ikjsl),line5isthefirstBianchiidentity(i.e.Rijkl + Riljk + Riklj = 0),andline6isthedistributivityofcovariantderivative.

27. WritethefirstBianchiidentityinitsfirstandsecondkinds.Answer:ThefirstandsecondkindsofthefirstBianchiidentityaregivenrespectivelyby:Rijkl + Riljk + Riklj = 0

Rijkl + Riljk + Riklj = 0

wheretheindexedRarethecovariantandmixedtypeRiemann-Christoffelcurvaturetensor.

28. VerifythefollowingformofthefirstBianchiidentityusingthemathematicaldefinitionoftheRiemann-Christoffelcurvaturetensor:Rijkl + Rkijl + Rjkil = 0.Answer:Wehave:Rijkl + Rkijl + Rjkil = 

Rijkl + Rkijl + Riljk = 

Rijkl − Rikjl + Riljk = 

Rijkl + Riklj + Riljk = 

∂k[jl, i] − ∂l[jk, i] + [il, r]Γrjk − [ik, r]Γrjl + ∂l[kj, i] − ∂j[kl, i] + [ij, r]Γrkl − [il, r]Γrkj + ∂j[lk, i] − ∂k[lj, i] + [ik, r]Γrlj − [ij, r]Γrlk = 

∂k[jl, i] − ∂l[jk, i] + [il, r]Γrjk − [ik, r]Γrjl + ∂l[jk, i] − ∂j[kl, i] + [ij, r]Γrkl − [il, r]Γrjk + ∂j[kl, i] − ∂k[jl, i] + [ik, r]Γrjl − [ij, r]Γrkl = 

0

whereequality1istheblocksymmetryinthelastterm,equalities2and3areanti-symmetryinthesecondterm,equality4isthedefinitionoftheRiemann-Christoffelcurvaturetensorwithrequiredrelabelingofindices,andequality5isthesymmetryoftheChristoffelsymbolsintheirpairedindices.

29. WhatisthepatternoftheindicesinthesecondBianchiidentity?Answer:Thepatternisthatthefirsttwoindicesarefixedwhilethelastthreeindicesarecyclicallypermutedinthethreeterms.

30. WritethedeterminantalformoftheRiccicurvaturetensorofthefirstkind.Answer:

31. StartingfromthedeterminantalformoftheRiccicurvaturetensorofthefirstkind,obtainthefollowingformoftheRiccicurvaturetensorwithjustificationofeachstepinyourderivation:Rij = ∂j∂i[ln√(g)] + ΓabjΓbia − [1 ⁄ √(g)]∂a[√(g)Γaij]

Answer:UsingthedeterminantalformoftheRiccicurvaturetensorofthefirstkind(whichisgivenintheanswerofthepreviousquestion)wehave:Rij = 

∂jΓaia − ∂aΓaij + ΓabjΓbia − ΓabaΓbij = 

∂j∂i[ln√(g)] − ∂aΓaij + ΓabjΓbia − [∂b{ln√(g)}]Γbij = 

∂j∂i[ln√(g)] + ΓabjΓbia − ∂aΓaij − Γbij∂b[ln√(g)] = 

∂j∂i[ln√(g)] + ΓabjΓbia − ∂aΓaij − Γaij∂a[ln√(g)] = 

∂j∂i[ln√(g)] + ΓabjΓbia − ∂aΓaij − Γaij[1 ⁄ √(g)]∂a√(g) = 

∂j∂i[ln√(g)] + ΓabjΓbia − [√(g) ⁄ √(g)]∂aΓaij − Γaij[1 ⁄ √(g)]∂a√(g) = 

∂j∂i[ln√(g)] + ΓabjΓbia − [1 ⁄ √(g)][√(g)∂aΓaij + Γaij∂a√(g)] = 

∂j∂i[ln√(g)] + ΓabjΓbia − [1 ⁄ √(g)]∂a[√(g)Γaij]

whereline2istheexpansionofthedeterminantalform,line3istheidentityΓjij = ∂i[ln√(g)],line4isreorderingofterms,line5isrelabelingofdummyindexinthelastterm,line6istheruleofdifferentiationofnaturallogarithm,line7ismultiplyingthethirdtermwith1,line8istakingcommonfactorfromthelasttwoterms,andline9istheproductruleofdifferentiation.

32. VerifythesymmetryoftheRiccitensorofthefirstkindinitstwoindices.Answer:Wehave:Rij = 

∂j∂i[ln√(g)] + ΓabjΓbia − [1 ⁄ √(g)]∂a[√(g)Γaij] = 

∂i∂j[ln√(g)] + ΓabjΓbia − [1 ⁄ √(g)]∂a[√(g)Γaij] = 

∂i∂j[ln√(g)] + ΓaibΓbaj − [1 ⁄ √(g)]∂a[√(g)Γaij] = 

∂i∂j[ln√(g)] + ΓabiΓbja − [1 ⁄ √(g)]∂a[√(g)Γaji] = 

Rji

whereline1istheresultthatweobtainedinthepreviousquestion,line2isthecommutativityofthepartialdifferentialoperatorsinthefirstterm,line3isreorderinginthesecondtermwithrelabelingofthedummyindices,line4isthesymmetryoftheChristoffelsymbolsintheirlowerindices,andline5istheresultthatweobtainedinthepreviousquestionwithrelevantindexlabeling(i.e.line4isidenticaltoline1withanexchangeofiandj).

33. WhatisthenumberofdistinctentriesoftheRiccicurvaturetensorofthe

firstkind?Answer:BecausetheRiccicurvaturetensorofthefirstkindissymmetric,thenthenumberofitsdistinct(seeFootnote40in§8↓)entriesis:NRD = [n(n + 1)] ⁄ 2

34. HowcanweobtaintheRiccicurvaturescalarfromthecovariantRiemann-Christoffelcurvaturetensor?Writeanorderlylistofalltherequiredstepstodothisconversion.Answer:Wedothefollowing:(a)WeobtainthemixedRiemann-ChristoffelcurvaturetensorbyraisingthefirstindexofthecovariantRiemann-Christoffelcurvaturetensor,thatis:Raijk = gabRbijk

(b)WeobtainthecovariantRiccitensorbycontractingthecontravariantindexwiththelastcovariantindexofthemixedRiemann-Christoffelcurvaturetensor,thatis:Rij = Raija

(c)WeobtainthemixedRiccitensorbyraisingthefirstindexofthecovariantRiccitensor,thatis:Rkj = gkiRij

(d)WeobtaintheRiccicurvaturescalarℛbycontractingtheindicesofthemixedRiccitensor,thatis:ℛ = δjkRkj = Rjj

35. MakealistofallthemainpropertiesoftheRiccicurvaturetensor(rank,type,symmetry,etc.)andtheRiccicurvaturescalar.Answer:SomeofthemainpropertiesoftheRiccicurvaturetensorare:(a)ItisderivedfromtheRiemann-Christoffelcurvaturetensor.(b)Itisabsoluterank-2tensor.(c)Itdependsonlyonthemetrictensor.(d)Itisusedtocharacterizethespaceandexpressitscurvature.(e)Itcanbecovariant(firstkind)oftype(0, 2)ormixed(secondkind)oftype(1, 1).(f)Thecovarianttypeissymmetric.SomeofthemainpropertiesoftheRiccicurvaturescalarare:(A)ItisderivedfromtheRiccicurvaturetensor(andultimatelyfromthe

Riemann-Christoffelcurvaturetensor).(B)Itisabsoluterank-0tensor(i.e.scalar).(C)Itdependsonlyonthemetrictensor.(D)Itisusedtocharacterizethespaceandexpressitscurvature.

36. OutlinetheimportanceoftheRiccicurvaturetensorandtheRiccicurvaturescalarincharacterizingthespace.Answer:BecausetheRiccicurvaturetensorandtheRiccicurvaturescalararederivedfromtheRiemann-Christoffelcurvaturetensor,theyplaysimilarrolestothoseoftheRiemann-Christoffelcurvaturetensorincharacterizingthespaceanddepictingitscurvaturequantitativelyandqualitatively.Hence,theyhaveimportantusesandapplicationsinseveralmathematicalbranchesandphysicaltheorieslikedifferentialgeometryandgeneralrelativity.

37. Write,intensornotation,themathematicalexpressionsofthefollowingtensorsinCartesiancoordinatesdefiningallthesymbolsinvolved:infinitesimalstraintensor,stresstensor,firstandseconddisplacementgradienttensors,Fingerstraintensor,Cauchystraintensor,velocitygradienttensor,rateofstraintensorandvorticitytensor.Answer:Infinitesimalstraintensor:γij = (∂idj + ∂jdi) ⁄ 2whereγijistheinfinitesimalstraintensorandtheindexeddisthedisplacementvector.

Stresstensor:Ti = σijnjwhereTiisthetractionvector,σijisthestresstensorandnjisthenormalvector.

Firstandseconddisplacementgradienttensors:Eij = ∂xi ⁄ ∂x’jΔij = ∂x’i ⁄ ∂xjwhereEijisthefirstdisplacementgradienttensor,Δijistheseconddisplacementgradienttensorandtheindexedxandx’aretheCartesiancoordinatesofanobservedcontinuumparticleatthepresentandpasttimesrespectively.

Fingerstraintensor:Bij = (∂xi ⁄ ∂x’k)(∂xj ⁄ ∂x’k)whereBijistheFingerstraintensorandtheothersymbolsareasdefinedbefore.

Cauchystraintensor:Bij − 1 = (∂x’k ⁄ ∂xi)(∂x’k ⁄ ∂xj)whereBij − 1istheCauchystraintensorandtheothersymbolsareasdefinedbefore.

Velocitygradienttensor:[∇v]ij = ∂ivjwhere∇visthevelocitygradienttensorwhilevandvjrepresentthevelocityvector.

Rateofstraintensor:Sij = (∂ivj + ∂jvi) ⁄ 2whereSijistherateofstraintensorandtheindexedvisasdefinedbefore.

Vorticitytensor:S̃ij = (∂ivj − ∂jvi) ⁄ 2whereS̃ijisthevorticitytensorandtheindexedvisasdefinedbefore.

38. WhichofthetensorsinQuestion377↑aresymmetric,anti-symmetricorneither?Answer:Infinitesimalstraintensor:symmetric.

Stresstensor:notnecessarilysymmetricalthoughitcanbe.

Firstandseconddisplacementgradienttensors:neither.

Fingerstraintensor:symmetric.

Cauchystraintensor:symmetric.

Velocitygradienttensor:neither.

Rateofstraintensor:symmetric.

Vorticitytensor:anti-symmetric.39. WhichofthetensorsinQuestion377↑areinversesofeachother?

Answer:Thefirstandseconddisplacementgradienttensorsareinversesofeachother.

FingerstraintensorandCauchystraintensorareinversesofeachother.40. WhichofthetensorsinQuestion377↑arederivedfromothertensorsin

thatlist?Answer:Fingerstraintensorisderivedfromthefirstdisplacementgradienttensor.

Cauchystraintensorisderivedfromtheseconddisplacementgradienttensor.

Therateofstraintensorisderivedfromtheinfinitesimalstraintensor.

SincetheFirstandseconddisplacementgradienttensorsareinversesofeachother,theymaybeconsideredasderivedfromeachother.

SincetheFingerstraintensorandtheCauchystraintensorareinversesofeachother,theymaybeconsideredasderivedfromeachother.

Sincethevelocitygradienttensoristhesumoftherateofstraintensorandthevorticitytensor,itmaybeconsideredasderivedfromthesetensors.

Sincetherateofstraintensoristhesymmetricpartofthevelocitygradienttensor,itmaybeconsideredasderivedfromthevelocitygradienttensor.

Sincethevorticitytensoristheanti-symmetricpartofthevelocitygradienttensor,itmaybeconsideredasderivedfromthevelocitygradienttensor.

41. Whatistherelationbetweenthefirstandseconddisplacementgradienttensors?Answer:Asindicatedearlier,theyareinversesofeachother,thatis:EikΔkj = δij

whereδijistheKroneckerdeltaandtheothersymbolsareasdefinedearlier.42. Whatistherelationbetweenthevelocitygradienttensorandtherateof

straintensor?Answer:Asindicatedearlier,therateofstraintensoristhesymmetricpartofthevelocitygradienttensor,thatis:S = [∇v + (∇v)T] ⁄ 2

whereSistherateofstraintensorandtheothersymbolsareasdefinedearlier.

43. Whatistherelationbetweenthevelocitygradienttensorandthevorticitytensor?Answer:Asindicatedearlier,thevorticitytensoristheanti-symmetricpartofthevelocitygradienttensor,thatis:S̃ = [∇v − (∇v)T] ⁄ 2

whereS̃isthevorticitytensorandtheothersymbolsareasdefinedearlier.44. Whatistherelationbetweentherateofstraintensorandtheinfinitesimal

straintensor?Answer:Therateofstraintensoristhetimederivativeoftheinfinitesimalstraintensor,thatis:S = ∂γ ⁄ ∂t

whereSistherateofstraintensor,γistheinfinitesimalstraintensorandtistime.

45. Whataretheothernamesgiventothefollowingtensors:stresstensor,deformationgradienttensors,leftCauchy-Greendeformationtensor,Cauchystraintensorandrateofdeformationtensor?Answer:Stresstensor:alsoknownasCauchystresstensor.

Deformationgradienttensors:alsoknownasdisplacementgradienttensors.

LeftCauchy-Greendeformationtensor:alsoknownasFingerstraintensor.

Cauchystraintensor:alsoknownasrightCauchy-Greendeformationtensor.

Rateofdeformationtensor:alsoknownasrateofstraintensor.

46. Whatisthephysicalsignificanceofthefollowingtensors:infinitesimalstraintensor,stresstensor,firstandseconddisplacementgradienttensors,Fingerstraintensor,velocitygradienttensor,rateofstraintensorandvorticitytensor?Answer:Infinitesimalstraintensordescribesandquantifiesthestateofstrainusuallyincontinuummediasuchasviscousfluids.

Stresstensordescribesandquantifiesthestateofstressinphysicalobjects.Itisalsousedforexampleintransforminganormalvectortoasurfacetoatractionvectoractingonthatsurface.

Firstandseconddisplacementgradienttensorsdescribeandquantifythedisplacementofaphysicalobject(e.g.particleofcontinuum)initshistoricaldevelopment(i.e.relationshipbetweenitspresentandpastposition).

Fingerstraintensordescribesandquantifiesthestateofstraininphysicalobjects(e.g.continuummedia)initshistoricaldevelopment.Thiscanbeinferredfromthefactthatitisderivedfromthefirstdisplacementgradienttensor.

Velocitygradienttensordescribesandquantifiesthegradientofvelocity(i.e.itsrateofvariationoverspace)inphysicalobjectssuchasliquidsandgases.

Rateofstraintensordescribesandquantifiesthelocalrateatwhichneighboringmaterialelementsofadeformingcontinuummovewithrespecttoeachother.

Vorticitytensordescribesandquantifiesthelocalrateofrotationofadeformingcontinuummedium.

AuthorNotes●Allcopyrightsofthisbookareheldbytheauthor.●Thisbook,likeanyotheracademicdocument,isprotectedbythetermsandconditionsoftheuniversallyrecognizedintellectualpropertyrights.Hence,anyquotationoruseofanypartofthebookshouldbeacknowledgedandcitedaccordingtothescholarlyapprovedtraditions.

FootnotesFootnote1.Thisanswerisaboutthesymbolismofthisbook(whichisgenerallyofcommonuse),andhencesomeconditions(e.g.beingofloweroruppercase)arenotuniversal.Thereadersshouldthereforeconsulteachauthorabouthisownconventionabouttheseconditions.Footnote2.Thisalsoimpliestheinvarianceofwhatthetensorrepresentsofabstractmathematicalentityorphysicalentityandhencethe“reality”oftherepresentedentityisindependentoftheformandtypeofrepresentation.Forexample,avectorofmagnitude1meterpointingtothenorthwillbesoinanycoordinatesystemandinanyformofrepresentationandhenceitwillnotbeofmagnitude1inonesystemandofmagnitude2inanothersystemorpointingnorthinoneformofrepresentationandpointingeastinanotherformofrepresentation.Footnote3.Wenotethattheperspectiveofcomponentsandbasisvectorsmynotbeentirelyconsistentinthisanswer.However,themainfocusistheconditionforsymmetry.Footnote4.Whenwesay“transformationfromCartesiantocylindrical”wemeanusingtheCartesiantocylindricaltransformationequations.Thissimilarlyappliestothefollowingtransformations.Footnote5.Tobeconcise,wereplacex2 + y2 + z2withr2.Footnote6.Moreaccurately,“commutative”isanattributetothecompositionoftransformations.Footnote7.Thesubscriptindicesaremeanttolabelthevectorswithnosignificanceabouttheirvariancetype.Footnote8.Thesymbol||standsforthedeterminantoftheenclosedmatrix.Footnote9.WearereferringheretotheequationofExercise163↑whereascalarfshouldtransform(accordingtothatequation)as:f̃ = Jwfandhencew = 0forabsolutescalarandw ≠ 0forrelativescalar.Footnote10.Forimproperrotation,thisismoregeneralthanbeingisotropic.Footnote11.Inthiscontext,“distinct”means“independent”.Footnote12.Infact,thisisnotarigorousproofbutratherademonstrationsincecertainrestrictionsandextensions(e.g.withregardtopartialsymmetryoranti-symmetry)areneededtoobtaintherequiredresult.Footnote13.Wenotethatsomeoftheseconditionsareoverlapping;however

weprefertoputitinthiswayforclarity.Footnote14.Thereadershouldnotethattheover-barreferstothebarredsystemandhenceitshouldnotbeconfusedwiththeunder-barwhichweusetolabelabsolutepermutationtensors.Footnote15.Theseimplicationsareultimatelybasedontherulesofparity,thatis:even × even = even

odd × odd = even

even × odd = odd

odd × even = odd

wherethefirstimplicationisrepresentedbythefirsttwoequalitieswhilethesecondimplicationisrepresentedbythelasttwoequalities.Footnote16.Therelationδkkδll − δmm = (3 × 3) − 3 = 6maynotbeobvioustosome.Hence,wecanrelabelthedummyindexmtoobtain:δkkδll − δmm = 

δkkδll − δkk = 

δkk(δll − 1) = 

3 × 2 = 

3! = 

6Footnote17.Alternatively,wemayreversetheprooftoobtaintherequiredresult.Footnote18.Thesedefinitionsare:[ij, k] = (1 ⁄ 2)(∂jgik + ∂igjk − ∂kgij)

Γkij = (gkl ⁄ 2)(∂jgil + ∂igjl − ∂lgij)Footnote19.Infact,wehavetwopossibilitiesforthiscasebutbecauseofthesymmetryoftheChristoffelsymbolsintheirpairedindicestheyarecombinedin

asinglecase.Footnote20.Thisanswerisalmostanexactreplica(withsomeexplanatoryremarks)oftheproofgivenintheSokolnikoffbookwhichIcitedintheReferencesofmybook.Footnote21.Anupperindexinthedenominatorofpartialderivativeislikealowerindexinthenumerator,andhencealowerindexinthedenominator(i.e.jingij)shouldbelikeanupperindexinthenumerator(i.e.jinδja).Thissimilarlyappliestothenextequation.Footnote22.Themoreformalwayofprovingthe“onlyif”part(i.e.iftheChristoffelsymbolsvanishidenticallythenallthecomponentsofthemetrictensorinthegivencoordinatesystemareconstants)istousethefactthatwhentheChristoffelsymbolsvanishidenticallythecovariantderivativebecomespartialderivative.Now,bytheRiccitheorem,thecovariantderivativeofthemetrictensoriszeroandhencethepartialderivativeiszerointhiscase.Accordingly,themetrictensormustbeconstant,asrequired.However,wethinkourargumentissimple,sufficientandmoreclear.Footnote23.Wenotethat∂1,∂2,and∂3mean∂ρ,∂φ,and∂z.Wealsonotethatincylindricalsystemswehaveg11 = 1,g22 = ρ2andg33 = 1whilealltheotherentriesarezero.Also,“non-identicallyvanishing”heremeansfromtheperspectiveoforthogonalsystemsalthoughsomeareidenticallyvanishingfromtheperspectiveofcylindricalsystems.Footnote24.Thereadershouldnotethatthisquestionisaboutgeneralcoordinatesystemsandhenceitshouldnotbeconfusedwithcertainspecialcoordinatesystemslikeorthogonalsystems.Footnote25.Asindicatedbefore(seethefootnoteofExercise165↑),wehavetwopossibilitiesforthiscasewhereforeachpossibilitywehave[n(n − 1)] ⁄ 2independentsymbolsandhencethetotalnumberofindependentsymbolsinthiscaseisn(n − 1).Footnote26.Forexample,thebasistensorofAijisEiEj.Footnote27.WenotethatinthelastsetofequationswerelabeledsomedummyindicesandusedthesymmetryoftheChristoffelsymbolsintheirpairedindicestoreachourfinalresult.Alltheseoperationsshouldbeobvioustothereader.Also,inthelastlineweusedthedefinitionoftheRiemann-Christoffelcurvaturetensorwhichisgiveninthebook.Footnote28.Theaboveexplanationisbasedonacertaininterpretationofthepropertyofcommutativitywithregardtotheintrinsicdifferentiationandthecorrespondingordinarydifferentiationoperation.However,wemayassumeotherinterpretationsandhencetherulesandexplanationcouldchange.The

detailsareirrelevanttoourobjective.Moreover,mostofthesedetailscanbeeasilyobtainedfromfirstprincipleswherethebasicrulethatshouldbefollowedisthatordinarydifferentialoperatorsarecommutativewhiletensordifferentialoperatorsarenot.Weshouldalsonotethatweareassumingthatintrinsicdifferentiationworkswiththesecondordercovariantdifferentiationaswiththefirstordercovariantdifferentiation.Footnote29.Weareconsideringherefirstordercovariantderivative.Inbrief,eachcovariantdifferentiationoperationincreasesthecovariantrankby1.Footnote30.Wenotethatthisisdifferentfromtheabsolutedifferentiationwhichkeepsthetypeandrankoftheoriginaltensorbecauseinabsolutedifferentiationthedifferentiationindexiscontractedwiththeindexofthetangentvectorandnotwithanindexoftheoriginaltensor,i.e.inthisprocessweintroducetwoindices(onefromcovariantdifferentiationandonefromthetangentvector)andconsumethesetwoindicesbycontractionwithouttouchingtheindicesoftheoriginaltensorandthereforetheoriginaltensorkeepsitstypeandrank.Thiscanbeeasilyseenfromthedefinitionofabsolutedifferentiation,i.e.δA ⁄ δt = A;k(duk ⁄ dt)Footnote31.Aswenotedinthebook,thebasisvectorthatassociatesthederivativeoperatorinthefollowingequations(aswellasinsimilarequationsandexpressions)shouldbethelastoneinthebasistensor.Footnote32.Infact,thesearevectorcalculusdefinitionsmorethantensorcalculusdefinitions.However,weprefertheseforclarityandtoavoidlengthyclarifications.Footnote33.Inthisquestionandanswerwearereferringtothecommonlyusedcylindricalandsphericalcoordinatesystemsnotingthatthesesystemscanalsoberepresentedbycovariantandcontravariantforms.Footnote34.Wenotethatsomeofthesedefinitionsmaybelongto3Dspecifically.Footnote35.Wenotethatthegivenformulaapplieseventon = 1sinceitgivesN2 = 0.Footnote36.Wenotethatthegivenformulaapplieseventon = 1andn = 2sinceitgivesN3 = 0.Footnote37.Wenotethatthegivenformulaapplieseventon = 1,n = 2andn = 3sinceitgivesN4 = 0.Footnote38.Thiscanbeshownasfollows:R1234 + R1423 + R1342 = 0

R1234 + R1423 − R1324 = 0

R1324 = R1234 + R1423

whereline1isthefirstBianchiidentityandline2isanti-symmetryinthelasttwoindices.Footnote39.Wenotethatsomeofthefollowingformulaearenotgiveninthetextorinthepreviousquestion.Footnote40.Infact,theyarenotnecessarilydistinctbuttheyareindependent,hence“distinct”insuchcontextsmeans“independent”.Thissimilarlyappliestotheentriesofanti-symmetrictensorswheredistinctinsuchcontextsmeansindependentandhenceanentryanditsnegativearenotdistinct(i.e.notindependent)althoughtheyaredistinctinagenericsense;moreoversomeindependententriesmaynotbedistinctinagenericsense.