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1 Chapter(1.((Vector(Calculus(SpecialTopic) 1.1.Review Vectordot(inner)productisdefinedby A B = ( îa 1 + ĵa 2 + ˆkA 3 ) ( îb 1 + ĵb 2 + ˆkB 3 ) where î, ĵ, ˆk = ( A 1, A 2, A 3 ) ( B 1, B 2, B 3 ) = A 1 B 1 + A 2 B 2 + A 3 B 3, (1.1) ( ) areunitvectorsinthedirectionof x, y, z ( ),respectively.notethat dotproductoftwovectorsresultsinascalar.ifwerotatethecoordinatessothat A and B lieinthe xy planeand A alignswiththe x axisasshowninfigure1.1 below,then A B = A, 0, 0 ( ) B 1, B 2, 0 ( ) = AB 1. (1.2) Thus,thedotproductmaybeinterpretedasmultiplicationofproectionof B onto A withthemagnitudeof A. Figure1.1.Twovectors A and B lyinginthe xy planewith A alignedalongthe x axis. Vectorcrossproductisdefinedby A B = ( îa 1 + ĵa 2 + ˆkA 3 ) ( îb 1 + ĵb 2 + ˆkB 3 ) = ( A 1, A 2, A 3 ) ( B 1, B 2, B 3 ) = ( A 2 B 3 A 3 B 2, A 3 B 1 A 1 B 3, A 1 B 2 A 2 B 1 ) = î ( A 2B 3 A 3 B 2 ) + ĵ ( A 3B 1 A 1 B 3 ) + ˆk ( A 1 B 2 A 2 B 1 ), (1.3) wherethecrossproductoftwovectorsresultsinavector.theresulting directionofavectorcrossproductcanbedeterminedbythehandghandruleas depictedinfigure1.2below: 1

2 Figure1.2.Vectorproductoftwovectorsanheresultingdirectionaccording torightghandrule.wrapyourrighthandfrom A to B asindicatedby circulationarrowinthefigure.then,thedirectionofyourthumbistheresulting directionof A B. 1.2.TheDel( )Operator The operator,alsocallehegradientoperator,isavectordifferential operatordefinedby = î x + ĵ y + ˆk z = # x, y, (, (1.4) z ' whereitmeansdifferentiationinthreedifferentdirectionsorthogonaltoone another.notethat isavectorandadifferentialoperator.severaldifferent operationsarepossible.forexample, canbemultipliedwithascalarfunction yielding # T = x, y, # (T = T z ' x, T y, T ( = z ' î T x + ĵ T y T + ˆk. (1.5) z ThisparticularoperationiscalledgradientofT.Sinceitisasimpleproduct betweenavectorandascalar,theresultisavector.wecanalsoimagineadot productbetween andavector.thatis, u = x, y, ' ) u, v, w z ( ( ) = u x v y w, (1.6) z whichiscalleddivergenceof u.sinceitisadotproductbetweentwovectors, theresultisascalar.yetwecanformacrossproductbetween andavector, i.e., u = x, y, ' ) u, v, w z ( ( ) = î # w y v (+ z ' ĵ # u z w # v (+ ˆk x ' x u ( y ' 2

3 # = w y v z, u z w x, v x u (. (1.7) y ' Thisproductiscalledgradientof u anheresultisavector. Acommonmistakethatstudentsmakeistodropdot( )orcross( )in theoperationsabove.notethat u iscompletelydifferentfrom u. Specifically, u # = î x + ĵ y + ˆk ( îu + ĵv + ˆkw z ' ( ) u v w u v w u v w = îî + îĵ + î ˆk + ĵî + ĵĵ + ĵ ˆk + ˆkî + ˆkĵ + ˆk ˆk x x x y y y z z z,(1.8) whichisnotascalarasin(1.6)buteachtermin(1.8)representstwodistinct directions.suchaquantityiscalleda(twogdimensional)tensor.wewill introduceafewmoretensorquantitieslateron. 1.3.IndicialNotation Itisoftenconvenienttoexpressvectorsanensorsintermsofindices. Suchanexpressioniscalledindicialnotation.Forexample, u meansthe th componentofvector u.likewise, i = meansthei thcomponentof. Similarly, T notationsuchas ( ) i isthei thcomponentof T.Thereisaspecialkindofindicial u = u,. (1.9) Inordertodiscriminate(1.9)fromausualtwoGdimensionaltensoru,weput commabetweenthetwoindices.ofcourse,theleftghandsideof(1.9)isalways fineandisclearerinmeaning;thus,theleftghandsideispreferredasmuchas possible.thereisaspecialrulewhenidenticalindexappearstwiceinthesame expression.thisruleiscallehesummation/convention:whenidenticalindex appearsrepeatedlyinthesameexpression,summationovertherepeatingindex isassumed.thatis, u 3 u = u, = = u 1 u 2 u 3. (1.10) x =1 Inotherwords,repeatingindeximpliesadotproductwithrespecttothatindex. Thus,(1.10)canberewritteninavectornotationas u 3 u = u, = = u 1 u 2 u 3 = u. (1.11) x =1 Notethatthereoccursacontractionbetweentherepeatingindex.Thismeans thatthedimensiontherepeatingindexrepresentsdisappear.notein(1.11)that 3

4 thefinalproduceisascalar;thedimensiontheindex referstodisappearsasa resultofcontraction. Letusnowconsideramorecomplicatedproblem ),whichisadot productbetweenavector( )andatensor ).Inindicialnotation, ) = ) k = ) 1k ) 2k ) 3k. (1.12) Asaresultofcontraction,thereisonlyonedimensionleft,whichisthedirection oftheresultingvectorproduct.infullcomponentform,(1.12)canbewrittenas ( ) = î u + ĵ # + ˆk # ) 11 ) 12 ) 13 ) 21 ) 22 ) 23 = î " u 1 u 1 u 1 ' # + ĵ " u 2 u 2 u 2 ' # + ˆk " u 3 u 3 u 3 ' # ' ) 31 ) ( ) 32 ( ' ) 33 ( ' = 2 u. (1.13) Therearetwointerestingtensorsthatweneeousefrequentlyinclass. TheyareKroneckerdeltaδ i andpermutationtensorε ik : # δ i = " # and 1, if i = 0, otherwise " 1, if (i,, k) is in cyclic order ε ik = # 1, if (i,, k) is in acyclic order 0, otherwise, (1.14). (1.15) 4

5 Inindicialnotation,crossproductoftwovectorsisgivenby A B = ε ik A B k, (1.16) whereindices and k disappearleavingonly i.thus, i denotestheresulting directionof A B.Infullcomponentform(1.16)iswrittenas A B = ε ik A B k = ε 123 A 2 B 3 +ε 132 A 3 B 2 +ε 231 A 3 B 1 +ε 213 A 1 B 3 +ε 312 A 1 B 2 +ε 321 A 2 B 1 = î ( A B A B ) + ĵ ( A B A B ) + ˆk ( A 1 B 2 A 2 B 1 ), (1.17) whichisidenticalwith(1.3).similarly,curlofavectorisgivenby u = ε ik ( u k ) = ε 123 u 3 +ε 132 u 2 +ε 231 u 1 +ε 213 u 3 +ε 312 u 2 +ε 321 u 1 = î # u 3 u 2 (+ ' # u 1 u 3 (+ ˆk # u 2 u 1 (, ' ' (1.18) 1.4.Theε δ Theorem Wewillstudysomedetailedproblemsofvectorcalculusinthenext section.wewillusetheε δ theoremfrequentlyinmanyderivations.the theoremhastheform: ε ik = ε ik ε lmk = δ il δ m δ im δ l. (1.19) Notethespecificsequenceoftheindicesinthetheorem.Wewillnotprovethe theoreminarigorousmanner,butwillverifythetheorem. For i = 1,therearetwononGzerotermswhen (1) = 2, k = 3,l =1,and m = 2 forwhich ε ik =1, =1,δ il =1,δ m =1,δ im = 0,δ l = 0 ε ik =1,δ il δ m δ im δ l =1, (1.20) (2) = 2, k = 3,l = 2,and m =1forwhich ε ik =1, = 1,δ il = 0,δ m = 0,δ im =1,δ l =1 ε ik = 1,δ il δ m δ im δ l = 1, (1.21) (3) = 3, k = 2,l = 3,and m =1forwhich ε ik = 1, =1,δ il = 0,δ m = 0,δ im =1,δ l =1 ε ik = 1,δ il δ m δ im δ l = 1, (1.22) 5

6 (4) = 3, k = 2,l =1,and m = 3forwhich ε ik = 1, = 1,δ il =1,δ m =1,δ im = 0,δ l = 0 ε ik =1,δ il δ m δ im δ l =1. (1.23) Noteinallothercaseswithi = 1,boththeleftanherighthandsidesarezero. Thisverificationprocedurecanberepeatedfor i = 2,andi = VectorCalculus Inthissection,wewillderivesomeofvectorcalculusidentities.First showthat ( ) = ( A ) B + ( B ) A + A ( B ) + B ( A ). (1.24) A B NotethattheleftGhandsideof(1.24)isgradientofascalarfunction,whichisa vector.likewise,therightghandsideof(1.24)isavector.letuswritethethird termontherightghandsideinanindicialform: A B ( ) = ε ik A B ( ) k = ε ik A B m x l. (1.25) Byusingtheε δ theorem,wehave Similarly, Therefore, A ( B B ) = ε ik A m B = ( δ il δ m δ im δ l ) A m B = A B A i.(1.26) x l x l B ( A A ) = ε ik B m A = ( δ il δ m δ im δ m A l )B = B A B i.(1.27) x l x l A ( B ) + B ( A B ) = A B A i A + B A B i Arearrangementyields A B = ( A B ) B A i A B i = ( A B ) ( A ) B ( B ) A.(1.28) ( ) = ( A ) B + ( B ) A + A ( B ) + B ( A ), (1.29) whichisthedesiredresult. Wewillusetwoverywellknownresultsrepeatedly.Namely, and T = 0 (curl/gradientofascalarisalwayszero), (1.30) ( u ) = 0 (divergence/curlofavectorisalwayszero). (1.31) 6

7 Equation(1.30)canberewritteninanindicialformas T ' T = ε ik ) x k ( = ε 123 whereweused " T # '+ε 132 " T +ε 231 # " T +ε 312 # '+ε 213 '+ε 321 " T ' # " T ' # " T ' # = î # 2 T 2 T (+ ' ĵ # 2 T 2 T (+ ˆk # 2 T 2 T ( ' ' = ( 0, 0, 0), (1.32) 2 T = 2 T. (1.33) Equation(1.31)canberewritteninanindicialformas ( u ) = ( u ) = i u (( ε k ik x ' i x * ' * = ε ki )) 2 u k, (1.34) whichisascalaruponcontractionini, and k.incomponentform, ( u ) 2 u = ε 1 2 u 123 +ε ε u 2 +ε u 2 +ε u 3 +ε u 3 # 2 u = 1 2 u 1 # (2 u 2 2 u 2 # (2 u 3 2 u 3 ( ' ' ' = 0, (1.35) where(1.33)hasbeenused.derivationofsomeofimportantidentitiesisleftas ahomework. 1.6.CurvilinearCoordinateSystem Animportantreasonwhyweusevectorortensornotationisthatitdoes notvaryforspecificcoordinatesystems.often,however,weneeowrite equationsgoverningfluidmotionsforaspecificcoordinatesystem.therefore, 7

8 wewilldevelopageneralizedexpressionforimportantvectorexpressionsina curvilinear/coordinate/system.inacurvilinearcoordinatesystem,three coordinatesintersectatarightangle.atypicalexampleiscartesiancoordinates, inwhich (x, y, z) coordinatesareperpendiculartoeachother.inacurvilinear coordinatesystem,gradient,divergence,curl,andlaplacianareexpressed respectivelyas T = ê1 T + ê2 T + ê3 T = h 3 A 1 = h 3 A 1 h 3 ( ) # T, h 3 A 2 A ê 1 ê 2 h 3 ê 3 1 = h 3 A 1 A 2 h 3 A 3 ( ) 2 T = T T, ( ) T (, (1.36) h 3 ' ( ) A 3 ' ), (1.37) ( ' ) ) ), (1.38) ) ) ( 1 * h 3 T ' ) h 3 T ' ) T '-, )/,(1.39) h 3 + ( ( h 3 (. where (x 1, x 2, x 3 ) areindependentvariableswiththerespectiveunitvectors (ê 1,ê 2,ê 3 ) and (,, h 3 )arecalledscale/factors.scalefactorsarefactors convertingindependentvariablesintolengths(dimensionoflength).wewill discussthisinmoredetailbelow. InCartesiancoordinates(Figure1.3),independentvariables,unitvectors, andscalefactorsaregivenby Figure1.3.Cartesiancoordinateswithindependentvariables(x, y, z) withunit vectors(î, ĵ, ˆk). 8

9 (x 1, x 2, x 3 ) = (x, y, z), (ê 1,ê 2,ê 3 ) = (î, ĵ, ˆk), (,, h 3 ) = (1,1,1), (1.40) Notethat (dx, dy, dz) denotethelengthincrementsinthe (x, y, z) directions. Thus,thescalefactorsareunityinallthreedirections.Thus, # T = T x, T y, T (, (1.41) z ' A = A x x A y y A ' z ), (1.42) z ( A = î A z y A ( y ' *+ z ) ĵ A x z A ( z ' *+ ˆk A y x ) x A ( x ' *, (1.43) y ) # 2 T = 2 T x 2 T 2 y 2 T (. (1.44) 2 z 2 ' Figure1.4.Cylindricalcoordinateswithindependentvariables(r,θ, z) withunit vectors(ˆr, θ, ˆ ẑ). Incylindricalcoordinates(Figure1.4),independentvariables,unit vectors,andscalefactorsaregivenby (x 1, x 2, x 3 ) = (r,θ, z), (ê 1, ê 2, ê 3 ) = ( ˆr, ˆ θ, ẑ), (,, h 3 ) = (1, r,1), (1.45) where (r,θ, z) areradius,azimuthalangle,andelevation,respectively.notethat thescalefactorinthedirectionofθ is r sincethearclengthchangeisgivenby ds = rdθ.wewilldiscussthisinmoredetailinclass.asaresultof(1.45), gradient,divergence,andcurlarewrittenrespectivelyas 9

10 # T = T r, T r θ, T (, (1.46) dz ' A = 1 r A = 1 r ( ra r ) r ( ) A θ θ ra ' z ), (1.47) z ( ˆr rθ ˆ ' ẑ ) ) r θ z ). (1.48) ) A r ra θ A z () Figure1.5.Sphericalcoordinateswithindependentvariables φ,θ, r ( ) withunit ( ). vectors ˆφ, ˆ θ, ˆr Finally,insphericalcoordinates(Figure1.5),independentvariables,unit vectors,andscalefactorsaregivenby ( x 1, x 2, x 3 ) = ( φ,θ,r), ( ), ( ê 1,ê 2,ê 3 ) = ˆφ, θ, ˆ ˆr (,, h 3 ) = ( r cosθ,r,1), (1.49) where ( φ,θ, r) arelongitude(east),latitude(north),andradius(altitude), respectively.notethatthescalefactorinthelongitudedirectionis dl = r cosθdφ whereasthescalefactorinthelatitudedirectionis ds = rdθ.componentforms ofthegradient,divergence,andcurlaregivenrespectivelyby # T T = r cosθ φ, T r θ, T (, (1.50) r ' 10

11 A 1 = r 2 cosθ ( ra φ ) φ ( r cosθ A ) θ θ ( ) r2 cosθ A ' r ), (1.51) r () r cosθ ˆφ r ˆ ' A θ ˆr ) 1 ) = ). (1.52) r 2 cosθ φ θ r ) r cosθ A φ ra θ A r ) ( Letusnowtrytounderstanhemeaningofscalefactorsindetail.Inthe directionoflatitude(orazimuthalangle),thearclengthassociatedwith dθ is givenby(seefigure1.6) ds = rdθ, (1.53) where r isthedistancefromthecenterofthecircle.therefore,scalefactor associatedwithlatitude(orazimuthalangle)is r.thearclengthassociated with dφ isgivenby(seefigure1.6) dl = r cosθdφ, (1.54) where r cosθ istheradiusofthelongitudecircleatlatitudeθ.notethatthe radiusofthesmallcirclevariesas cosθ.therefore,scalefactorinthedirection oflongitudeapproacheszeroasthepolesareapproached.verifythisinfigure 1.5. Figure1.6.(left)Arclengthchangeduetolatitudechangeand(right)arclength changeduetolongitudechangeatlatitudeθ. 1.7.IntegralTheorems Weoftenencounterintegrationofphysicalquantitiesinavolume element.letusconsider ( f )dv, (1.55) V 11

12 whichmeasuresthenetamountofflux f leavingthevolumeelementv. Equation(1.55)canberewrittenas ( f )dv = V = lim dx,dy,dz 0 = lim dx,dy,dz 0 + lim # dx,dy,dz 0 + lim dx,dy,dz 0 f x x f y y f ( z ' *dv V z ) f x (x + dx) f x (x) dx ( f x (x + dx) f x (x))dydz ( f y (y + dy) f y (y))dzdx ( f z (z + dz) f z (z))dxdy + f y (y + dy) f y (y) dy + f z (z + dz) f z (z) dz (dxdydz ' = ˆn f da, (1.56) A where A isthesurfaceofthevolumev and ˆn isthenormalunitvectoroneach surface.equation(1.56)saysthatthenetamountoffluxleavingthevolumev is equaltothesumofallfluxesleavingthesurface A ofthevolumev.this theoremiscallehegauss /theorem.inasimilarmanner,wecanshowthat and G dv = ˆnG da V A, (1.57) f dv = ˆn f da. (1.58) V A Wealsoencounteranintegrationoftheform ( f ) ˆn da, (1.59) A whichrepresentstheamountof f passingthroughthesurfacearea A with thenormalvector ˆn. Usingindicialnotation,(1.59)canberewrittenas ( f f ) ˆn da = ε k ik n i da, (1.60) A A wherewenotethat x and x k aretwoindependentvariablesnormaltothe directionof ˆn (inthedirectionof x i ).Then,(1.60)canberewrittenas(see Figure1.7) ( f ) ˆn da = ε ik lim A dx,dx k 0 f k (x + dx ) f k (x ) dx n i dx dx k, Wewilldiscusstheconceptof f inchapter4. 12

13 = ε ik n i lim dx,dx k 0 ( f k (x + dx ) f k (x ))dx k = f d l. (1.61) C Figure1.7.Graphicalillustrationof(1.61). ThistheoremiscalleheStokes /theorem.notethatthesurfaceelement A may notnecessarilybeaflatplane;itcanbeacurvedplane. AnotherimportantintegraltheoremistheLeibnitztheorem.Letus considerintegrationof f (x, t)overarange a(t), b(t) ( ).Then,thetimerateof changeofsuchanintegralisgivenby d b(t) a(t) F(x, t)dx = lim 0 = lim 0 b(t+) b(t) F(x,t + )dx F(x, t)dx a(t+) a(t) b(t) F(x,t + ) F(x,t) dx a(t) ( b(t + ) b(t) ) + f (b(t + ),t + ) ( a(t + ) a(t) ) f (a(t),t). (1.62) Figure1.8.Integrationoffunction f (x, t)overatimegvaryingrange a(t), b(t) ( ). 13

14 Thus, d b(t) F(x, t)dx = a(t) b(t) a(t) F(x, t) t dx + F(b(t),t) db(t) F(a(t),t) da(t).(1.63) Equation(1.63)tellsusthatthevariationoftheintegrationintimeisduetothe temporalchangein F(x, t) aswellasthetemporalchangeintheintegration bounds. LetusconsiderasmallmaterialvolumeV * consistingofidenticalfluid particles. Thismaterialvolumehasasurfacearea A *.Overasmalltime interval,thecontrolvolumeundergoesageometrictransformationasshown infigure1.9.then, d F( x,t)dv F( x,t + )dv F( x,t)dv V = lim * (t+) V * (t) V * (t) 0 = lim 0 = V * (t) ( ) ( F( x,t + ) F( x,t) ) dv + F(x,t) V * (t + ) V * (t) F( x,t) dv + F( x,t) v ˆndA V * (t) t. (1.64) A * (t) Notethat v ˆn da isvolumesweptbyasmallareaelement da perunittime. Thus,byintegrating v ˆndA overthewholesurfacearea A *,wewillhavechange inthevolumev *. Figure1.9.AgeometricaldepictionofacontrolvolumeV * (t) havingasurface area A * (t) thatmovesatthespeedof v duringasmalltimeinterval. ( ( Amaterialvolumeisanimaginaryvolumeconsistingofidenticalfluid particles.likewiseamaterialsurfaceisanimaginarysurfaceconsistingof identicalfluidparticles. 14