Orthogonal curvilinear coordinates - PDF document

Chapter5OrthogonalCurvilinearCoordinatesLastupdate:22Nov2010Syllabussection:4.Orthogonalcurvilinearcoordinates;lengthoflineelement;grad,divandcurlincurvilinearcoordinates;sphericalandcylindricalpolarcoordinatesasexamples.SofarwehaveonlyusedCartesianx;y;zcoordinates.Sometimes,becauseofthegeometryofagivenproblem,itiseasiertoworkinsomeothercoordinatesystem.Hereweshowhowtodothis,restrictingthegeneralityonlybyanorthogonalitycondition.5.1PlanePolarCoordinatesInCalculusIIandChapter2,wemetthesimplecurvilinearcoordinatesintwodimensions,planepolars,denedbyx=rcosq;y=rsinq:Wecaneasilyinverttheserelationstogetr=p x2+y2;q=arctan(y=x):TheChainRuleenablesustorelatepartialderivativeswithrespecttoxandytothosewithrespecttorandqandviceversa,e.g.¶f ¶r=¶f ¶x¶x ¶r+¶f ¶y¶y ¶r=x r¶f ¶x+y r¶f ¶y:(5.1)InCalculusII,theruleforchangingcoordinatesinintegralsisalsogiven.Thegeneralruleisthatifwechangecoordinatesfromx;ytou;vwherex=x(u;v),y=y(u;v),thenadxdyinanareaintegralisreplacedbytheJacobiandeterminant¶x ¶u¶x ¶v¶y ¶u¶y ¶vdudv:Ifwedener=(x(u;v);y(u;v);0),differentiatew.r.t.u;vandtakethecross-product,wewillseethattheaboveisequaltodS=¶r ¶u¶r ¶vdudv;64 aswederivedinsection4.2.Forplanepolarcoordinates,replacingu;vwithr;qandcalcuatingthedeterminantabovegivesdS=rdrdq;thiscanalsobeshowngeometricallybyconsideringaninnitesimalquadrilateralwithcornersat(r;q);:::;(r+dr;q+dq)andworkingouttheareafromasketch. Example5.1.TheGaussianintegral(relatedtotheGaussiandistributioninstatistics)ConsidertheintegralZ¥¥Z¥¥e(x2+y2)dxdy=R¥¥ex2dx2:TransformingtopolarcoordinatesgivesZ¥0rer2drZ2p0dq=[1 2er2]¥0[q]2p0=pandhence(accordingtoDr.Saha“themostbeautifulofallintegrals”)Z¥¥ex2dx=p p: Forlateruse,wenowconstructtheunitvectorsinthedirectionsinwhichrandqincreaseatapoint,whichwewilldenoteerandeq.Thesearetangenttothecoordinatelines,whereacoordinatelinemeansacurveonwhichonlyoneofthecoordinatesisvarying,andtheothercoordinatesarexed.Coordinatelinesaregeneralizationsoflinesparalleltothex;y;zaxesinCartesians,butnowtheywon'tbestraightlines(hencethe“curvilinear”inthechaptertitle).Wealreadyknowhowtondthetangentvectorstocoordinatelines,bytakingpartialderivativesofrwithrespecttoeachofr;q;thenallwehavetodoisdividethosebytheirlengthstogetunitvectors.Thusinplanepolarswehaver=rcosqi+rsinqj;soasmallchangedrgivesusachangedrr=¶r ¶rdr=(cosqi+sinqj)dr;¶r ¶r=1)er=¶r ¶r=cosqi+sinqjwhileasmallchangedqgivesusdrq=¶r ¶qdq=(rsinqi+rcosqj)dq;j¶r ¶qj=r)eq=sinqi+cosqj:Soageneralsmalldisplacementbecomesdr=erdr+reqdq:Wewillseethevalueofthislateron;wearenextgoingtoconsiderthree-dimensionalversionsofpolarcoordinates:therearetwocommonversions,rstlycylindricalpolarsandthensphericalpolars.65 5.2CylindricalPolarCoordinatesForcylindricalpolars,weturntheplanepolarsinthex;yplaneintothree-dimensionalcoordinatesbysimplyusingzasthethirdcoordinate(seeFig.5.1).Toavoidconfusionwithothercoordinatesystems,weshallforclarity1renamerasrandqasf,butbewarethatinothercourses,books,andapplicationsoftheseideas,randqwillstillbeused.Thuswehavex=rcosf;y=rsinf;z=z;orr=rcosfi+rsinfj+zk;andquantitiesinanyplanez=constantwillbeasinplanepolars.Thegure5.1showscoordinatelinesforeachofr,fandz;herethecoordinatelineforrisalineofvaryingrandconstantf,z;andlikewisefortheothertwo.Notethatthecoordinatelinesforr,zarestraightlines,whiletheflineisacirclearoundthezaxis.Thomas'sFig.15.37showsanicediagramofsurfacesonwhichoneofthecoordinatesisconstant:theconstant-rsurfaceisacylinderwhoseaxisisthezaxis,whilesurfacesofconstantforconstantzareplanes. Figure5.1:CylindricalpolarcoordinatesrelativetoCartesian,andwithsampler-andf-curvesshown.Thefactthatconstantrgivesacylindergivesthenamecylindricalpolars:thesecoordinatesarenaturalonestousewheneverthereisaprobleminvolvingcylindricalgeometryorsymmetry(forexample,doingasurfaceintegrationoveracylinder,orinphysicscalculatingamagneticeldaroundastraightwire).Togetpartialderivativesincurvilinearcoordinatesweagainusethechainrule(5.1),butnowwiththreetermsontheright.Takingtheplanepolarresults,changingvariablenamesandappendingez=k,theunitvectorsalongthecoordinatelinesareer=cosfi+sinfj;ef=sinfi+cosfj;ez=krespectively.Wecanwritethisinmatrixformas0@erefez1A=0@cosfsinf0sinfcosf00011A0@ijk1A:(5.2) 1Unfortunately,forthesamereasonsofclarity,Thomasadoptsthealternativesolutionofrenamingtwoofthesphericalpolarcoor-dinates.Toavoidconfusionwithpastyears'exampapersIhavekepttothechoiceusedthere,whichisalsotheoneusedinmostbooks.Thomaschooses(r;f;q)fortheusual(r;q;f).Theswapofqandfisparticularlylikelytobeconfusing.66 Itiseasytoseefromtheabovethatthedot-productofanytwoe'sgives1(iftheyarethesame)or0(foranytwodifferentones),liketherulesfori;j;k.Thisimpliesthatthethreee'sareanorthogonaltripleofunitvectors,andalsoimpliesgeometricallythatthecross-productofanytwodifferente'swillbethethirdone.Wecanalsoexpressthispropertyinmatrixnotation:the33matrixabove,callitR,isarotationmatrix,i.e.onesuchthatR1=RT,wheretheRTdenotestranspose.Thiscomesaboutbecausethedot-productofanytwoe'sisgivenbyoneelementofthematrixRRT,andthee'sareanorthogonaltripleifandonlyifRRT=I,theidentitymatrix.2Alsonotethatifwewanti;j;kintermsofthee's,wecanjustmultiplyEq.5.2byR1=RT.Thelengthsof¶r=¶r,¶r=¶fand¶r=¶zarerespectively1,rand1;wecanusethesetogetherwiththee'stondinnitesimalareaelements:e.g.takingasurfacer=constant(acylinder),wecantreatthisasa2-parametersurfacewithf;zastheparameters,sothevectorareaelementforsmallchangesdf;dzisgivenbydS=¶r ¶f¶r ¶zdfdz=refezdfdz=rerdfdz;thiswillbeusefulwhendoingsurfaceintegralsoveracylinder.(Asusual,thereisapotentiallyambiguouschoiceofsignwithvectorareas,duetothesign-ipinchangingorderofacrossproduct;takecarewiththis,e.g.whendoingaproblemcheckthatyourvectorareamatchesthedesireddirection).Whendoingvolumeintegrals,wemayneedthevolumeelementwhichisdV=rdrdfdzfromthescalartripleproduct.5.3SphericalPolarCoordinatesThesearecoordinates(r;q;f),wherermeasuresdistancefromtheorigin,qmeasuresanglefromsomechosenaxis,calledthepolaraxis,andfmeasuresanglearoundthataxis(seeFig5.2.)TorelatethemtoCartesiancoordinatesweusuallyassumethatthez-axisisthepolaraxis.Then,letPbeourchosenpointat(r;q;f),anddropaperpendicularfromPtothezaxismeetingitatQ.ThelineOPisatangleqtothepositivez-axis,soclearlyOQ=z=rcosqandPQ=rsinq.DroppinganotherperpendicularfromPtothexyplane,wegetapointinthexyplaneatdistancersinqfromtheorigin;theninsertingr=rsinqintothecylindricalpolarsinSec.5.2givesus:x=rsinqcosf;y=rsinqsinf;z=rcosq:or,asapositionvectorr=rsinqcosfi+rsinqsinfj+rcosqkHerethefisthesameasthatofcylindricalpolars,whichexplainswhywechosethesameletter.Theinverseoftheserelationsisr=p x2+y2+z2;q=arctan p x2+y2 z!;f=arctany x: 2Rotationmatricesare“special”becausetheypreservelengthsandangles;e.g.ifwetaketwovectorsa,b,writethemascolumnvectors,thentheirscalarproductinmatrixnotationisaTb.ThetwovectorsrotatedbymatrixRareRaandRb.Toconservescalarproduct,wemusthave(Ra)T(Rb)=aTb,andusingthetransposerulethisbecomesaTRTRb=aTb.Forthistoapplyforanytwoa;bwemusthaveRTR=I,theidentitymatrix.67 Coordinatelinesofr,(i.e.linesofconstantqandf),arestraightradiallinesfromtheorigin;coordinate Figure5.2:SphericalpolarcoordinatesrelativetoCartesian,andwithsampler-,qandf-curvesshown.linesofq(constantrandf)aremeridionalsemicircles,i.e.semicirclescentredattheoriginandinaplanecontainingthepolaraxis;andcoordinatelinesoff(constantrandq)arelatitudinalcircles,i.e.circlescentredatapointonthepolaraxisandinaplaneperpendiculartoit.Notehoweverthatwhilerrunsfrom0to¥(liketherofplanepolarsandrofcylindricalpolars)andfrunsfrom0to2p(liketheqofplanepolars),qonlyrunsfrom0top,sinceforanypointPtheanglebetweenOPandthez-axiswon'texceed180degrees=pradians.Thecoordinatelinesofqarestrictlysemi-circles,ratherthancircles.Tomakeacirclewehavetotakethecoordinatelinesofqfortwodifferentf,sayf0andf0+p.Thomas'sFig.15.42showsanicediagramofsurfacesonwhichoneofthecoordinatesisconstant.Youshouldbewareofthefactthatsomeauthors,includingThomas,usedifferentnotation,inparticularswappingthemeaningsofqandfinthedenitionofsphericalpolars.Weshallconsistentlyusetheabovenotationforsphericalpolarcoordinates,whichisthemostcommonone,throughoutthiscourse.Notethattheseagaingeneralizetheplanepolarcoordinates,butthistimethepolarsr;qareinplanescontainingthez(orpolar)axis,ratherthaninplanesperpendiculartoit.Thesphericalpolarcoordinatesareofcoursethenaturalonestousewhenwehaveasphericalgeometry,orpartofasphere.Nowweconstructtheevectorsasbefore:takingpartialderivativesofrabovewithrespecttoeachofthecoordinatesinturn,weget¶r=¶r=sinqcosfi+sinqsinfj+cosqk;¶r=¶q=rcosqcosfi+rcosqsinfjrsinqk¶r=¶f=rsinqsinfi+rsinqcosfj:Thelengthsofthese,bysimpleapplicationsofcos2f+sin2f=1,arerespectively1,r,andrsinq.Dividingthesederivativesbytheirlengthsgivesustheunitvectorser,eqandeftangenttothecoordinatelines,whichwecanwriteas0@ereqef1A=0@sinqcosfsinqsinfcosqcosqcosfcosqsinfsinqsinfcosf01A0@ijk1A:(5.3)Itisstraightforwardtoshowthatagainthedot-productofanytwoe'sis1(iftheyarethesame)or0(ifdifferent);thereforethecross-productofanytwoe'sisthethirdoneandthematrixaboveisagainarotationmatrix.68 Itisalsoworthnotingthater=r=r,asexpectedbysymmetrysinceerisaunitvectorpointingawayfromtheoriginatpointr.Fordoingintegralslateron,thevolumeelementisgivenbythescalartripleproductdV=(¶r=¶r)(¶r=¶q):(¶r=¶f)drdqdf=r2sinqdrdqdf:Theinnitesimalareaelementonasphere(i.e.asurfaceofconstantr)isgivenbydS=(¶r=¶q)(¶r=¶f)dqdf=r2sinqerdqdf:Similarresultsholdforsurfacesofconstantfandofconstantq,butarenotsocommoninpractice;notethattheaboveareaelementonasphereturnsupinmanyexamplesandexamquestions,andiswellworthmemorising. Example5.2.“Earthpolarcoordinates”TodenesphericalpolarsontheEarth,letthepolaraxisbetheEarth'srotationaxis,withzincreasingtotheNorth,lettheequatordenethex;yplane,andlettheprimemeridian(theonethroughGreenwich)bef=0.ThenanypointontheEarth'ssurfacecanbereferredtobythesphericalpolarangles(q;f).Innavigationpeopleuselatitudeandlogitude.LongitudeismeasuredEastorWestfromtheprimemeridianandisintherange(0;180)sotogetfforaplacewithWesterlylongitudewejustsubtractfrom2p=360.Latitudeisdenedtobe0attheequator(whereasq=90=p=2there).Givenalatitude,weneedtosubtractitfrom90ifitisNorthandadditto90ifitisSouth.ForexampleBuenosAires,whichhaslatitude34360S,andlongitude58220W,willhaveEarthpolarcoordinatesq=125;f=302tothenearestdegree. 5.4SomeapplicationsofthesepolarcoordinatesUsingpolar(orcylindrical)coordinatestheareawithinacircleofradiusR,RR0R2p0rdfdr,comesoutimme-diatelyaspR2.UsingsphericalpolarcoordinatesthevolumeofasphereofradiusRisZR0Zp0Z2p0r2sinqdfdqdrwhichevaluatesto4 3pR3.(Rememberthatforafullsphere,therangesofintegrationare0qp,0f2p). Example5.3.Areaofacone:Considertheconicalsurfaceq=q1cutinasphereofradiuss.TheareaisgivenbyintegratingZ2p0dfZs0sinq1rdr=ps2sinq1:Heresistheslantheightofthecone.Thecone'sbase(sayb)willbessinq1.Hencewecanexpresstheslopingareaofaconeneatlyaspsb. Example5.4.WenowreconsiderExample4.5.FindtheuxoftheeldF=zkacrosstheportionofthespherex2+y2+z2=a2intherstoctantwithnormaltakeninthedirectionawayfromtheorigin.69 Becauseofthegeometryofthesurface,itiseasiesttoworkinsphericalpolarcoordinates(r;q;f),sothespherehasr=a.Theunitnormalntothespherethatpointsawayfromtheoriginisjuster,theoutwardradialvectorofunitlength.NowF:er=zk
er=zcosq=rcos2q:using5.3toevaluatek:er=cosq.Anareaelementonthesurfaceofasphereofradiusris(rdq)(rsinqdf)=r2sinqdqdf.Forourgivenspherer=a,soZSF
ndS=Zp=20Zp=20acos2qa2sinqdqdf=a3Zp=20Zp=20cos2qsinqdqdf=p 2a31 3cos3qp=20=p 6a3:Notethattheintegranddidn'tdependonf,sowejustreplacedthedfintegralwithamultiplicationbytherange,here(p=20).Thisisacommonshort-cuttonote. Example5.5.CuttinganappleInhisbook,Matthewsposesagoodproblemforillustratingintegrationusingcurvedcoordinates:“Acylindricalapplecorerofradiusacutsthroughasphericalappleofradiusb.Howmuchoftheappledoesitremove?”Wecanreformulatetheproblemslightly,withoutlosinggenerality,bylettingtheradiusoftheappleequalunityandintroducingsinq1=a=b(i.e.wescaletheproblembyb).Inourrestatedproblemthecorercutsthroughthepeelatq=q1andq=1 2pq1insphericalpolars,i.e.incylindricalpolarsatr=sinq1;z=cosq1;and,ofcourse,atz=cosq1.Wecannowcompletethesolutionofthisproblemin(atleast)fourdifferentways:threeofthesearerelegatedtoanappendix,notgiveninlectures.3Therstwayistointegrateoverzandthenr4pZsinf10rdrZp 1r20dz=4pZsinf10r(1r2)1 2dr=4p 3(1cos3f1): 5.5GeneralOrthogonalCurvilinearCoordinatesThetwosetsofpolarcoordinatesabovehaveafeatureincommon:thethreesetsofcoordinatelinesareorthogonaltooneanotheratallpoints,becausetheirtangentvectorsandcorrespondingunitvectorse'sareorthogonal.(Thisiswheretheorthogonalinthechaptertitlecomesfrom). 3Igiveonlythekeysteps.Somealgebraiclling-inisneeded.Ineachversionwecanshortenthecalculationsbyreplacingthefintegrationwithmultiplicationby2p(sincetheintegranddoesn'tdependonf),andalsodoingtheintegralsonlyforz0,andthendoublingusingsymmetry.70 Generalorthogonalcoordinatesarecoordinatesforwhichthesepropertiesaretrue,i.e.thecoordinatelinesarealwaysmutuallyperpendicularatagivenpoint,thoughtheyaregenerallycurved.Ingeneral,coordinatesneednotbeorthogonal.However,weshallbeconcernedonlywithorthogonalcurvilinearco-ordinates.Cylindricalpolarsandsphericalpolarsaretheonlynon-Cartesiancoordinatesystemsinwhichyouwillbeexpectedtoperformexplicitcalculationsinthiscourse,apartfromsimplesubstitutionsintothegeneralformulae.Suppose(u1;u2;u3)areageneralsetofcoordinates,denedbysomegivenfunctionr(u1;u2;u3).Asbefore,wecalculate¶r=¶u1whichisthetangentvectortoau1line(varyingu1,constantu2;u3).Nextwedenethearc-lengthh1andunitvectore1ash1=¶r ¶u1;e1=¶r ¶u1=h1therefore¶r ¶u1h1e1:Itiseasytocalculatethath21=¶x ¶u12+¶y ¶u12+¶z ¶u12:Likewisedifferentiatingrbyu2;u3,wedenetwomoreunitvectorse2,e3,alongthecoordinatelinesofu2andu3,andassociatedarc-lengthparametersh2andh3.Thisisusefulforseveralreasons:rstly,e1tellsusinwhichdirectionrmoveswithasmallchangeinu1,whileh1du1isthedistancemovedalonge1,andlikewiseforchangesdu2;du3.Wedeneacoordinatesystemtobeorthogonaliffe1,e2ande3aremutuallyorthogonaleverywhere:Coordinates(u1;u2;u3)areorthogonal,¶r ¶u1:¶r ¶u2=¶r ¶u2:¶r ¶u3=¶r ¶u3:¶r ¶u1=0Fororthogonalcoordinates,ageneralsmallchange(du1;du2;du3)inthecoordinatesmeansadisplacementdr=h1du1e1+h2du2e2+h3du3e3;(5.4)whichcorrespondstoadistanceh21du21+h22du22+h23du23
1=2:Also,fororthogonalcoordinatesthedotandcrossproductsofanytwoe'swillobeythesameruleswemetbefore:thereforethematrixRrelating(e1;e2;e3)to(i;j;k)willbearotationmatrix(fromabove)andhavethepropertythatRT=R1.Cartesiancoordinatesareofcourseaspecialsimplecaseoforthogonalcurvilinearcoordinates,inwhichallthecoordinatelinesarestraightlinesandallofh1=h2=h3=1.Sometimesitisconvenienttoreplacethe1,2,3withthelettersofthecoordinates,e.g.incylindricalpolarcoordinates,wewroteer;ef;ez.Therewealreadyfoundhr=1andhz=1,buthf=r,soachangedfcorrespondstomovingadistancerdfalongacirclearoundthezaxis.Insphericalpolarcoordinates,hr=1again,andhq=r.Achangedfinfcorrespondstomovingadistancersinqdf(becausersinqistheradiusoftheparticularlatitudinalcirclearoundthezaxis),sohf=rsinq.Onereasonthatorthogonalcoordinatesaresousefulisthatinanyorthogonalcoordinatesystem(u1;u2;u3),smalldisplacementsalongu1andu2denesmallrectangles,whilesmalldisplacementsalongu1;u2;u3de-nesmallcuboids.Inotherwords,h1h2du1du2isanareaelementnormaltoe3onasurfaceofconstantu3,andh1h2h3du1du2du3isavolumeelement.71 5.6VectoreldsandvectoralgebraincurvilinearcoordinatesScalareldscanofcoursebeexpressedin(orthogonal)curvilinearcoordinates:theyaresimplywrittenasfunctionsf(u1;u2;u3)orforbrevityf(ui).AsyouwillknowfromLinearAlgebra,vectorscanbeexpressedusinganybasisofthevectorspaceconcerned.Thesameistrue,ateachpoint,ofvectorelds.Uptonowwehavealwayschoseni;j;kasourbasisvectors:however,whenusingcurvilinearcoordinateswewillnormallyusetheorthogonalunitvectorsalongthecoordinatelinesasourbasisvectors,andwriteF=F1e1+F2e2+F3e3:Forclarity,wecanusethecoordinatenamesinsteadof1,2,3assubscriptsforthethreecomponents.ThuswemaywriteF=Fxi+Fyj+Fzk=Frer+Ffef+Fzez=Frer+Fqeq+Ffef:toexpressthesamevectorinCartesian,cylindricalpolarandsphericalpolarcoordinates(ofcourseex=iandsooninCartesians).NotethatthesamevectorFwillhavedifferentcomponentsdependingonourchoiceofbasisvectors:supposewearegivenanFwithdenedFx;Fy;Fzabove,butwewanttondFr;Fq;Ff,thenweneedtousethematrixasinEq.5.3toexpressi;j;kintermsofthee's,multiplyoutandcollectintoonetermineache.(Thiseffectivelyturnsintoamatrixmultiplication).Inanyorthogonalcoordinatesystem,thescalar(dot)andvector(cross)productsworkjustasinCartesiancoordinates:w:v=w1v1+w2v2+w3v3(5.5)andwv=e1e2e3w1w3w3v1v2v3;(5.6)butnotethisonlyworksifthevectorsaredenedatthesamepoint,suchasadotproductF
drorF
dSinalineorsurfaceintegral.Wecannotusethesefortwopositionvectorsatwidelyseparatepoints,becausethee'svarywithposition.Vectordifferentiationismorecomplicated,becausetheunitvectorsarenolongerconstant:whenwedifferentiatedavectorinCartesiansF=F1i+F2j+F3kwejustdifferentiatedthecomponents(F1;F2;F3)becausetheunitvectorsareconstant;butingeneralcoor-dinatesthee'sdependonposition,sowehavetousetheproductruleanddifferentiatetheevectorsaswellasthecomponentsFi.Differentiationofthesevectorswithrespecttoavariableotherthanposition(likethederivativesinSec-tion3.1)isstraightforward.Forexampleifpositionrdependsontime,andisgivenincylindricalpolarssor=rer+zez,wejustusetheproductruletogetthetimederivative
r=
rer+r
er+
zez+z
ez:(wheretheover-dotsareshorthandfortimederivative,asiscommon).Thensinceer=cosfi+sinfjfrom(5.2),
er=
f(sinfi+cosfj)=
fef:72 Similarly
ez=0.Substitutingintothepreviousresult,weget
r=
rer+r
fef+
zezforavelocityincylindricalpolarcoordinates.Whendifferentiatingscalarorvectoreldswithrespecttoposition,thekeyoperationsarealwaysgradofascalar,anddivandcurlofavectoreld(thisisbecausethesearetheonlycombinationsthatbehave“sensibly”afterrotations).Inthenextsections,wewillshowhowtocalculatethegrad,divandcurloperatorsingeneralorthogonalcoordinates;thenweapplythosegeneralformulaetothemostcommoncasesofcylindricalpolarsandsphericalpolars.5.7TheGradientOperatorincurvilinearcoordinatesTocalculatethegradientofascalareldV(u1;u2;u3)inorthogonalcurvilinearcoordinates(u1;u2;u3),wegobacktothedenitiondV=ÑV
dr:()forthechangedVcausedbyaninnitesimalpositionchangedr.(Note:heredVistheinnitesimalchangeinscalareldVresultingfromasmallchangedr;itisnotavolumeelement.)WedeneÑV(ÑV)1e1+(ÑV)2e2+(ÑV)3e3,andwewanttondthethreecomponents(ÑV)1etc.Fromthedenitionsoftheunitvectorspreviously,wehavedr=e1h1du1+e2h2du2+e3h3du3,sotheright-handsideof()becomes((ÑV)1e1+(ÑV)2e2+(ÑV)3e3)
(e1h1du1+e2h2du2+e3h3du3)=(ÑV)1h1du1+(ÑV)2h2du2+(ÑV)3h3du3usingtheorthogonalityofthee's.Nowturningtotheleft-handsideofof(),usingTaylor'stheorem(in3dimensions),anddiscardingtermsofsecondandhigherderivatives,wegetdV=¶V ¶u1du1+¶V ¶u2du2+¶V ¶u3du3Thesetwoexpressionsabovemustbeequalforanyarbitrarychangesdu1,du2anddu3.Hencewemusthave(ÑV)1h1=¶V ¶u1;(ÑV)2h2=¶V ¶u2;(ÑV)3h3=¶V ¶u3:Dividingbytheh'sandsubstitutingbackintotheoriginaldenition,inorthogonalcurvilinearcoordinateswehaveÑV=1 h1¶V ¶u1e1+1 h2¶V ¶u2e2+1 h3¶V ¶u3e3:(5.7)ClearlyinCartesiancoordinates,wehaveu1=x;e1=ietcandallthreeh'sare1,sothissimpliestothewell-knownformulafromChapter1.Forageometricalexplanation,the1=hitermstakecareofthearc-lengtheffects,i.e.howfarrmovesforasmallchangeineachcoordinate.Sothe1-componentofÑVrepresentsthechangedVpersmalldistance73 dsinthedirectione1;but,movingadistancedsindirectione1requiresachangedu1=ds=h1incoordinateu1;thereforethe1=hitermsappearingradVabove. Example5.6.WhatisÑVinsphericalpolarcoordinates?EvaluateÑVwhereV=rsinqcosf.Insphericalpolars,(u1;u2;u3)=(r;q;f)andh1=1,h2=r,h3=rsinq.Puttingthoseinto5.7wehaveÑV=¶V ¶rer+1 r¶V ¶qeq+1 rsinq¶V ¶fef:ForthegivenV,¶V=¶r=sinqcosf,¶V=¶q=rcosqcosfand¶V=¶f=rsinqsinf.Hence,usingtheresultabove,ÑV=sinqcosfer+cosqcosfeqsinfef:(InthiscasewecanobservethatV=xandÑV=i,usingthematrixfromEq.5.3,sothisexampleisaloteasierinCartesians;however,manyproblemsinvolvingcircularorsphericalsymmetrydogeteasierinpolarcoordinates). Exercise5.1.WhatisÑVincylindricalpolarcoordinates(r;f;z)?2Exercise5.2.Let(r;q;f)besphericalpolarcoordinates.EvaluateÑfwhere(a)f=f;(b)f=q;(c)f=(rnsinmq):25.8TheDivergenceOperatorincurvilinearcoordinatesNextwewanttocomputeÑ
Finorthogonalcurvilinearcoordinates.Althoughwecoulddirectlycalculatethedivergenceinanycoordinates,usingtheCartesiandenition,thematrixrelatingbasisunitvectors,andthechainrule,theresultscanbefoundwithlesseffortfromtheDivergenceTheorem.TheDivergenceTheoremistrueinallcoordinates(sinceitequatesscalars,whosevaluemustbeindependentofthecoordinates).ThusZVÑ
FdV=ZSF:dS;whereSistheclosedsurfaceenclosingvolumeV.Now,weapplythistoaninnitesimal“cuboid”withonecornerat(u1;u2;u3)andedgescorrespondingtochangesdu1,du2,du3ineachcoordinate;sothishaseightcornersat(u1;u2;u3);(u1+du1;u2;u3);:::(u1+du1;u2+du2;u3+du3).Frombefore,thevolumeofthecuboidisdV=(h1du1)(h2du2)(h3du3).Forasufcientlysmallvolume,wecanapproximateÑ
FasconstantacrossdV,sotheleft-handsidebecomes(Ñ
F)dV=(Ñ
F)(h1h2h3du1du2du3):Nextweconsidertheright-handsideoftheDivergenceTheorem:weneedtotakethesurfaceintegraloverthesixfacesofourcuboid,andaddresults.FirstconsidertheintegralofF:noverthefaceofthecuboidwheretherstcoordinatehasvalueu1+du1.Thisfaceisarectanglewithunitnormal+e1andarea(h2du2)(h3du3),sothesurfaceintegralisapproximately(h2h3du2du3F1)u1+du1;74 wherethesubscriptshowsitisevaluatedatu1+du1.Ontheoppositefaceatu1wehaveunitnormale1(pointingoutwardsi.e.awayfromtherstface),sothesurfaceintegralgivesus(h2h3du2du3F1)u1:Repeatingtheabovefortheotherfourfaceswegetsymmetricalresults;nallysummingthesixtermsandthentakingthelimitasdV!0,weobtainÑ
F=limdu1;du2;du3!01 dV(h2du2h3du3F1)u1+du1(h2du2h3du3F1)u1+(h3du3h1du1F2)u2+du2(h3du3h1du1F2)u2+(h1du1h2du2F3)u3+du3(h1du1h2du2F3)u3:Thougheachpairofbracketslooksthesame,thisisnotzerobecausetheh'sandF'saredifferentonoppositefacesofthecuboid;thersttwotermsgiveusdu1timesthepartialderivative¶=¶u1ofthebracket,andsoonforthenextpairs,sothisgetsustheresultÑ
F=1 h1h2h3¶(h2h3F1) ¶u1+¶(h3h1F2) ¶u2+¶(h1h2F3) ¶u3:(5.8)Note:Inthislaststep,wehavetakensomedu'soutsidethebracketsandcancelledthemwiththeonesindV,butwemustleavetheh'sinsidethedifferentiationsincetheh'sgenerallyvarywithposition.Thiscomesaboutbecauseour“cuboid”maybeslightly“tapering”,sotheareasofoppositefacesarenotexactlyequal;anddifferentiatingthehi'stakescareofthat. Example5.7.WhatisÑ
Fincylindricalpolarcoordinates,whereF=Frer+Ffef+Fzez?Incylindricalpolars,(u1;u2;u3)=(r;f;z)andh1=1,h2=r,h3=1.HenceÑ
F=1 r¶(rFr) ¶r+¶Ff ¶f+¶(rFz) ¶z:Notethatwecanapplytheproductrule,andsince¶r=¶z=0,¶r=¶r=1wegetÑ
F=1 rFr+¶Fr ¶r+1 r¶Ff ¶f+¶Fz ¶z:Note:ItisimportanttonotethatanFrtermhasappearedhere,whichisnotaderivativeofF.Thishasappearedbecausethecoordinatelinesforrhavea“builtindivergence”,theyallradiateoutwardsfromthez-axis,soaeldwithconstantFrhasapositivedivergencetermduetothis.Asafurtherexamplewecannotethatincylindricalpolars,r=rer+0ef+zez.Pluggingincomponents(r;0;z)totheabove,wegetÑ
r=1+1+0+1=3whichagreeswiththeresultinCartesians,asitmust.(Ifwehadjusttaken¶r=¶r+¶z=¶zwewouldhavegotÑ
r=2;clearlywrong). Example5.8.WhatisÑ
Finsphericalpolarcoordinates,whereF=Frer+Fqeq+Ffef?Insphericalpolars,(u1;u2;u3)=(r;q;f)andh1=1,h2=r,h3=rsinq.HenceÑ
F=1 r2sinq¶(r2sinqFr) ¶r+¶(rsinqFq) ¶q+¶(rFf) ¶f: 75 5.9TheCurlOperatorincurvilinearcoordinatesFinallywewantcurl:asbeforewehavecurvilinearcoordinates(u1;u2;u3),andavectoreldF=F1e1+F2e2+F3e3;wewanttocalculateÑF(ÑF)1e1+(ÑF)2e2+(ÑF)3e3;sowewantthe1,2,3componentsoftheabove.Inanalogywiththeprevioussection,weuseStokes'stheoremtoprovideacoordinate-independentde-nitionofÑF:ZS(ÑF)
dS=ZCF
dr;whereSisasurfacespanningtheclosedcurveC.Tocalculatethe1-component(ÑF)1,consideraplanarcurvearoundasmall“rectangle"onasurfaceofconstantu1,withsidesgivenbysmallchangesdu2anddu3.Frompreviousresults,thevectorareaofthisrectangledS=h2du2h3du3e1;nowtaking(ÑF)
dS,the2and3componentsofÑFdisappearsotheLHSofStokes'stheoremisapproximately(ÑF)1h2du2h3du3:NowlookingattheRHSofStokes'stheorem,thelineintegralaroundtheedgeofthesamerectangleisgivenbyaddingthelineintegralsalongthefoursides:thisisapproximately(h2du2F2)u3+(h3du3F3)u2+du2(h2du2F2)u3+du3(h3du3F3)u2;wherethesubscriptsdenotethatthetermisevaluatedatthatvalue,andtwominussignsappearbecauseoppo-sitesidesaretraversedinoppositedirectionsaroundtheclosedrectangle.Equatingthelasttwoexpressions,andtakingthelimitasdu2,du3!0,wehave(ÑF)1=1 h2h3limdu2;du3!0(h3F3)u2+du2(h3F3)u2 du2(h2F2)u3+du3(h2F2)u3 du3=1 h2h3¶(h3F3) ¶u2¶(h2F2) ¶u3:Thisisjustthe1-componentofÑF.Togetthe2-and3-components,wejustrepeatalltheabovefortwomoresmallrectanglesinsurfacesofconstantu2,u3respectively;thislooksthesamebutcyclingthe1/2/3's,andweget(ÑF)2=1 h3h1¶(h1F1) ¶u3¶(h3F3) ¶u1;(ÑF)3=1 h1h2¶(h2F2) ¶u1¶(h1F1) ¶u2:Theseresultscanbewritteninacompact(andmorememorable)formasadeterminant:ÑF=1 h1h2h3h1e1h2e2h3e3¶=¶u1¶=¶u2¶=¶u3h1F1h2F2h3F3:(5.9)Onceagain,inCartesiancoordinatesthissimpliestothewell-knownexpressionfromChapter3.4. Example5.9.WhatisÑFinsphericalpolarcoordinates?76 Insphericalpolarcoordinates(r;q;f)wehaveh1=1,h2=r,h3=rsinq.Hence,usingthedeterminantform:ÑF=1 r2sinqerreqrsinqef¶=¶r¶=¶q¶=¶fFrrFqrsinqFf:orinexpandedformÑF=1 r2sinq¶(rsinqFf) ¶q¶(rFq) ¶fer+1 rsinq¶Fr ¶f¶(rsinqFf) ¶req+1 r¶(rFq) ¶r¶Fr ¶qef:Notethatsincerisindependentofqandf,etc.,wecanforinstancetaketheroutsidethedifferentiationsintheercomponentandcancelitwithanrinthedenominator.Remembertheanswerisacurlsoit'savectoreld.Donotaddallthecomponentstogether,forgettingthevectorseretc(thisisacommonerror).Note:thefullexpressionabovelooksquitedaunting.Howeverinmanyproblemsthismaysimplifyconsiderablyusingsymmetry:forexample,ifagivenproblemissymmetricalaroundthez-axis,thenwewillhaveFf=0and¶Fr=¶f=0and¶Fq=¶f=0,sofourofthesixderivativeswillvanish. Exercise5.3.Showbyexpandingitthatthedeterminantdenitionisequivalenttothefullexpressionsfortheindividualcomponentsgivenabove.2Exercise5.4.WhatisÑFincylindricalpolarcoordinates?Notethatifrandzhavedimensionsoflengthandfisdimensionless(becauseit'sanangle),thenallthetermsintheexpressionforÑFshouldhavethesamedimensions,namelythedimensionsofFdividedbylength.Thisisasimplecheckthatyoushouldmake.2Exercise5.5.Usesphericalpolarcoordinatestoevaluatethedivergenceandcurlofr=r3.[Hint:don'tforgetthatinsphericalpolarcoordinates,thepositionvectorrisequaltorer.]2Exercise5.6.StateStokes'stheorem,andverifyitforthehemisphericalsurfacer=1,z0,withthevectoreldA(r)=(y;x;z).2Exercise5.7.ThevectoreldB(r)=(0;r1;0)incylindricalpolarcoordinates(r;f;z).EvaluateÑB.EvaluatethelineintegralRCB:dr,whereCistheunitcirclez=0,r=1,0f2p.DoesStokes'stheoremapply?2Note:Toconcludethischapter,wewillnotethatmanyappliedmathsorPhysicsproblemsinvolveanexpressionlikeÑ2V,whereVisascalareldandÑ2istheLaplacianoperator,incylindricalorsphericalpolarcoordinates.WecangettheexpressionsforÑ2VinpolarcoordinatesusingrstlythedenitionEq.3.10(recallthiswasÑ2Vdiv(gradV)),andthenusingEq.5.7forgradV,thentakingdivofthatwithEq.5.8.Theresultsareavailableinmosttextbooks;youwillnotbeexpectedtomemorisethose,butyoumightbegiventheminanexamquestionandaskedtocalculatesomething,soit'sworthtakingalookespeciallyifyouaretakingappliedmathscourseslater.77 AppendixOtherwaysofdoingExample5.5areasfollowsThesecondmethodistodividethevolumeremovedintotwoparts:(i)acylinderwithradiussinq1andheightcosq1,and(ii)a`top-slice'.Volume(i),thecylinder,iseasy:2psin2q1cosq1.Togetvolume(ii)weintegrateoverrandthenz4pZ1cosq1dzZp 1z20rdr=2pZ1cosq1(1z2)dz=2p 3(2+cos3q13cosq1):Thesumofvolumes(i)and(ii)is4p 3(1cos3q1)asexpected.Athirdwayalsodividesthevolumeremovedintotwoparts:(i)an`ice-creamcone'orconewithasphericaltop,and(ii)acylinderminuscone.Thevolume(i)is4pZq10sinqdqZ10r2dr=4p 3(1cosq1):Volume(ii),acylinderwithconeremoved,isabitharder:4pZcosq10dzZsinq1ztanq1rdr=2pZcosq10(sin2q1z2tan2q1)dz=4p 3sin2q1cosq1(whichnoticeis2 3ofthevolumeofthecylinder).Againthesumofthevolumesintegratedis4p 3(1cos3q1).Finally,afourthpossibilityistointegrateforthevolumeremainingaftercoring,whichis4pZcosq10dzZp 1z2sinq1rdr=2pZcosq10(1z2sin2q1)dz=4p 3cos3q1:78 SUMMARYOFORTHOGONALCURVILINEARCOORDINATESInorthogonalcurvilinearcoordinates(u1;u2;u3),withcorrespondingunitvectorse1,e2,e3andarc-lengthparametersh1,h2,h3,thegradientofascalareldVisgivenbyÑV=1 h1¶V ¶u1e1+1 h2¶V ¶u2e2+1 h3¶V ¶u3e3;thedivergenceofavectoreldF=F1e1+F2e2+F3e3isgivenbyÑ
F=1 h1h2h3¶ ¶u1(h2h3F1)+¶ ¶u2(h3h1F2)+¶ ¶u3(h1h2F3);andthecurlofthesamevectoreldisgivenbyÑF=1 h1h2h3h1e1h2e2h3e3¶=¶u1¶=¶u2¶=¶u3h1F1h2F2h3F3:Cartesiancoordinates:(u1;u2;u3)(x;y;z);arc-lengthparametersh1=1,h2=1,h3=1.Cylindricalpolarcoordinates:(u1;u2;u3)(r;f;z);arc-lengthparametersh1=1,h2=r,h3=1.Sphericalpolarcoordinates:(u1;u2;u3)(r;q;f);arc-lengthparametersh1=1,h2=r,h3=rsinq. 79