Mthemtics or Egieers Prt II (ISE Versio /4-6- Curves i Prmetric descriptio o curves We exted the theory o derivtives d itegrls to uctios whose rge re vectors i isted o rel umers Deiitio : A curve C i is the grph o uctio : I rom itervl I [;] ito such tht ech t i I hs the imge ( ( t ( ( t, ( t,, ( t The compoets i : I IR,t (t, i,,,, i o ( t ( ( t, ( t,, ( t re cotiuous uctios o the itervl I Equtio ( is clled the prmetric equtio o the curve C The poits ( d ( re clled the edpoits o the curve Remrk : I we idetiy ech poit ( (t ( (t, (t, (t [ (t, (t, (t] ( ( t ( ( t, ( t,, ( t with its positio vector [ ( t, ( t,, ( t ], the we c uderstd s vectorvlued uctio Fig Prmetric curve i Exmple : (i Suppose r > is rel umer, the the grph o the curve C give y the prmetric equtio : [ ;] IR, t ( r cost,r sit is circle i o rdius r (Fig (ii Let : IR IR,t (r cost,r sit, ct, r >, c, e the prmetric equtio o curve i The grph o the curve trced y (t s t vries is clled circulr helix (Fig -
Mthemtics or Egieers Prt II (ISE Versio /4-6- (iii Suppose g: I is rel vlued cotiuous uctio o the itervl I The grph o this uctio g is suset o d c e uderstood s curve i with the prmetric equtio ( : IR,t ( t,g( t I Fig Circle i Deiitio : Let I e ope itervl d (,,, : I IR The curve C give y is clled dieretile i ech compoet i : I IR,t (t, i,,,, i is dieretile t t I The vector ( t [ ( t, ( t,, ( t ] is clled tget vector o the curve t the poit (t Fig circulr helix Remrk : (i I ( t [,,], the uit tget vector to is give y the ormul ( T ( t ( t ( t [ ( t,, ( t ] ( ( t + + ( ( t (ii A curve C give y : I is ot ecessrily the grph o ijective uctio I, or t t, ( t ( t d (d,d,,d, the d is poit o itersectio o C At poit o itersectio there re i geerl two dieret tget vectors For exmple, the curve C deied y : with the prmetric equtio t t,t t (Fig 4 hs poit o itersectio t d (,, sice ( ( ( (- (, Fig 4 Poit o itersectio d - The tget vector t (t is give y ( t [ t, t ] Thus ( [,] ( [,] -
Mthemtics or Egieers Prt II (ISE Versio /4-6- Deiitio : Suppose : I is cotiuously dieretile prmeteriztio o curve C The curve C is clled regulr i ( t (,, or ll t I A prmeter vlue t I or t,, is clled sigulr which ( ( Exmple : We cosider Neil s prol give y : IR ( t,t IR, t The grph o the curve give prmetriclly y is the set { IR } x,y x / ( IR ( x,y ±, we get ( (, Thus the oly sigulr poit o Neil s prol is t t Sice ( t ( t, t Fig 5 Neil s prol Deiitio 4: Suppose : I d g: I re cotiuously dieretile prmeteriztios o the regulr curves C d C g I ( t g( t or some t I, the the gle o itersectio ϑ etwee C d C g is deied s t I (4 ( t g ( t ( t g ( t cosϑ, ϑ π Remrk : The gle o itersectio ϑ etwee C d C g is the gle etwee the two tget vectors t the poit o itersectio Fig 6 Agle o itersectio -
Mthemtics or Egieers Prt II (ISE Versio /4-6- Legth o curves Suppose [;] is itervl d C give y :[;] is smooth (ie cotiuously dieretile curve which does ot itersect itsel, tht is, dieret vlues o t [;] determie dieret poits o the grph o Cosider prtitio o [;] give y Let ti ti ti d let ( t < t < t < < t < t t i e the poit o the grph o determied y t i I ( t ( is the legth o the lie segmet ( t ( t i t i roke lie show i Fig is, the the legth L p o the i i ( t ( t k k L p ( t,,t ( tk ( tk tk k k tk Thereore we get or the legth o curve: L ( t,,t ( t dt lim Lp Propositio : I smooth curve C is give prmetriclly y Fig Lie segmets :[;], ( t ( ( t, ( t,, ( t d i C does ot itersect itsel, except possily t the edpoits, the the legth L o C is ( L (t dt Note! ( t ( ( t + + ( ( t Exmple : (i Suppose s > is rel umer We cosider the circulr rc give y : [,s] IR, ( t ( cost,sit, Sice ( t ( sit,cost we get ( t si t+ cos t Hece the rc legth is give y -4
Mthemtics or Egieers Prt II (ISE Versio /4-6- s ( t dt dt s L I prticulr the circumerece o the uit circle equls π s (ii We cosider the cycloid give prmetriclly y ( t ( t sit, cost : IR IR, The cycloid is the locus o poit o the rim o circle o rdius r rollig log stright lie (Fig It ws studied d med y Glileo i 599 Fig cycloid We compute the legth L o the prt o the cycloid elogig to the prmeter vlues t π, tht is the rc ABC i Fig Sice ( t ( cost,sit we get y the dditio theorems: Fig ( t ( cost cost 4 si t + si t Thereore ( t Hece we get: t t si si or t π t L si dt 4 sixdx 8 Remrkle: The legth o the rc ABC is iteger -5
Mthemtics or Egieers Prt II (ISE Versio /4-6- Remrk : There re iiitely my possile prmeteriztios o give curve C Cosider the prt o the uit prol prmetriclly give y :[;], ( t ( t,t (Fig 4 For exmple, ll uctios o the type t t [ ] ( α : ; α IR,α t,, α >, α α Fig 4 Uit prol hve the sme grph Tht is, i you oly look t the poits o the grph i you cot decide which prmeteriztio is resposile or this curve Now rises importt questio: Does the legth o curve C deped o the prmetric equtio? The swer is: O course ot! We will prove this i the ollowig Deiitio :, is dieretile strictly mootoe icresig uctio; ϑ, the we cll ϑ vlid trsormtio I ϑ:[α;β] [;], t ϑ( t ie ( t > Propositio : The legth o curve C is prmeteriztio-ivrit or vlid trsormtios Tht is: I : [;], ( τ ( ( τ, ( τ,, ( τ, is prmeteriztio o curve C d ϑ:[α;β] [;], t ϑ ( t τ, is vlid trsormtio, the ϑ : [ ; ], ( t ( ϑ( t ( ( ϑ( t, ( ϑ( t,, ( ( t is prmeteriztio o C d ϑ ϑ L C ( d t ( τ ϑ t dτ Proo: It is esy to see y the chi rule tht ( t ( ( ϑ( t ( ( ϑ( t ϑ ( t,, ( ϑ( t ( t ϑ ϑ ϑ : Thereore, sice ( t > ( t ( ( ϑ( t ( ϑ ( t + + ( ( ϑ( t ( ϑ ( t ( ϑ( t ( t ϑ ϑ -6
Mthemtics or Egieers Prt II (ISE Versio /4-6- We c coclude y direct sustitutio: L C ( ( ( ϑ t dt ϑ t ϑ ( t dt ( τ dτ Motio I this sectio we itroduce cocepts tht re eeded to study the motio o poit or prticle i To lyze the ehviour o poit, it is ecessry to kow its positio ( x ( t,,x ( t t every time t I we set the time depedig positio ( ( t ( ( t, ( t,, ( t, the s t vries, the edpoit o the positio vector [ ( t, ( t,, ( t ] trces the pth C o the prticle From the precedig sectio, the tget lie to C t the poit P t (t is prllel to ( ( t [ ( t, ( t,, ( t ] d the mgitude o this vector is ( ( t ( ( t + + ( ( t Suppose C is smooth curve i prmetriclly give y :[t ;t ], ( t ( ( t, ( t,, ( t with the strtig poit P correspodig to t t The the rc legth s σ(t o C rom P to P t (t is σ d (4 ( t ( t t The rc legth s σ(t is rel vlued uctio d its esy to see tht σ is σ t t > or regulr curve (this is cosequece o the dieretile with ( ( me vlue theorem or itegrls I other words: ( t is the rte o chge o rc legth with respect to time For this reso we reer to ( t s the speed o the prticle The tget vector ( t, whose mgitude is the speed, is deied s the -7
Mthemtics or Egieers Prt II (ISE Versio /4-6- velocity o the prticle t the time t The vector ( t prticle t the time t is clled the ccelertio o the Remrk : (i Velocity d ccelertio deped o prmeteriztio I ϑ:[α;β] [;] is vlid trsormtio d ϑ ( t ( ϑ( t the ( t ( ϑ( t ϑ ( t ϑ Note: Prmeteriztio determies the speed o the prticle, ut ot the shpe o the curve (ii The rc legth σ d, σ:[;] [;L], ( t ( t is dieretile strictly mootoe icresig uctio Thus the iverse uctio σ - : [;L] [;], σ - (s t, is lso dieretile d strictly mootoe icresig Thereore σ - is vlid trsormtio Exmple : (i For the helix i Exmple (ii prmetriclly give y : [ ; ] IR, t (r cost,r sit, ct, r, >, we get ( t ( rsit,rcost,c ( t r c + Thereore the rc legth s is give y t ( t r + c d t r c s + Hece t r s + c, d ormul or the helix with the rc legth s s prmeter is : [; r + c ], ( s s s cs ϕ r cos,r si, r + c r + c r + c -8
Mthemtics or Egieers Prt II (ISE Versio /4-6- (ii The positio o prticle movig i the ple is give y r : [;], ( t ( t,t r The velocity d the ccelertio t the time t re ( t [, t] d r ( t [,] r I prticulr, t t ( [,] d r ( [,] r Fig These vectors d the pth o the prticle re sketched i Figure The speed o the prticle t the poit P(, is ( + 4 5 r Suppose α > is rel umer, the prmeteriztio o the sme curve i Figure is give y t 4 t γ: [;α], ( t, I this cse the trsormtio is deied y ϑ: [;α] [;], ϑ(t t/α The velocity d the ccelertio t the time t re ow 8 t ( t, d ( t, 8 γ Usig this prmeteriztio, the prticle reches the poit P(, ter t α/ Hece the velocity d ccelertio t P(, re 4 8, d γ, Thereore the speed t the poit P(, is ow 4 6 + 5-9
Mthemtics or Egieers Prt II (ISE Versio /4-6- 4 Curvture For my pplictios ivolvig vector-vlued uctios it is coveiet to employ uit tget vectors to curves We cosider curves i prmetriclly give y (4 ( t ( ( t, ( t,, ( t From previous sectios we kow tht ( t [ ( t, ( t,, ( t ] t the poit P t ( ( t, ( t,, ( t I ( t [,,,] is give y (4 ( t ( t ( t T is tget vector to C the uit tget vector to C Propositio 4: Suppose smooth curve C is give y : [;], ( t ( ( t, ( t,, ( t I ( t is costt, the the vector ( t [ ( t, ( t,, ( t ] vector o ( t or every t [;] Proo: By hypothesis or rel umer c Thereore d dt is orthogol to the positio ( t ( t ( t c ( ( t ( t ( t ( t + ( t ( t ( t ( t Hece ( t ( t, tht is ( t d ( t re orthogol Sice i (4 T ( t, it ollows with Propositio 4 tht ( t T ( t or every t We deie (4 N ( t T T which gives uit vector orthogol to the uit tget vector T ( t ( t ( t T is orthogol to Deiitio 4: is clled the uit orml vector to the curve C with The vector N ( t T ( t T ( t prmeteriztio : [; ], ( t ( ( t, ( t,, ( t, ( t (,,, -
Mthemtics or Egieers Prt II (ISE Versio /4-6- simpliyig leds to N ( T ( Fig 4 Uit tget d uit orml vector Exmple 4: Let C e the curve prmetriclly give y ( t ( t,t, t 4 t + d ( t (, t It is esy to veriy / ( 4 t + T t Applyig (4 d 4 t + The T ( t ( t, T tht ( ( 4 t + Note: ( t N ( t ( t t N T 4 t + ( t (, t Thus T (t d N (t re, ideed, orthogol As we metioed eore there re iiitely my wys to represet curve prmetriclly Sometimes it is coveiet to use rc legth s prmeter Suppose ϕ is the rc legth prmeteriztio o curve C with give ϕ s g σ s g t pplies: ( ( prmeteriztio g Sice ( ( d d ϕ ds ds (44 ( s g ( t σ ( s ( s Now ( s d thus ( s ds d ( ( ( s ( s d ds : Thereore we get (rememer ( t g ( t (45 ( s Hece with (44 ds d ( t g ( t d ds (46 ϕ ( s ϕ( s g g ( t ( t, s σ(t Thus ϕ ( s d thereore ϕ ( s [ ϕ ( s, ϕ ( s,, ( s ] is uit tget vector ϕ -
Mthemtics or Egieers Prt II (ISE Versio /4-6- Note: Prmeter trsormtio to rc legth prmeter leds to prmeteriztio o curve with costt speed ϕ ( s Deiitio 4: Let C e curve i give y ( s ( ϕ ( s, ϕ ( s,, ϕ ( s prmeter Let T ( s ϕ ( s poit P s ( ϕ ( s, ϕ ( s,, ( s is deied s ϕ (47 K ( s T ( s ϕ where s is the rc legth e the uit tget vector The curvture K(s t the We give ow ormuls tht c e used to id the curvture o curves i d o spce curves i Propositio 4: Let C e curve i give prmetriclly y ( t ( ( t, ( t, where the compoet uctios d re two times cotiuously dieretile The the curvture K(t t P t ( ( t, ( t is (48 K( t ( t ( t ( t ( t ( ( t + ( ( t I prticulr the grph o twice cotiuously dieretile uctio : I hs the curvture (49 K( x ( x + ( ( x Proo: Suppose ϕ is trsormtio o to rc legth prmeter; ie: ( ( s ( s ϕ With (44 d (45 pplies: (4 ( ( t ϕ s, ( t ( t, d ϕ ( s ( t Sice ( t ( k t k ( t ollows: ( t k ( t ( ( t ( t ( ( s + -
Mthemtics or Egieers Prt II (ISE Versio /4-6- (4 ( ( s ( t ( ( ( ( ( ( ( k t 4 k t s t t k Thereore we get with (4 d (4: (4 ( s Now or : (4 ϕ ( Hece ϕ ( t ( t ( t kk 4 k ( t ( t ( ( t ( t ( t ( t [ ( t, ( t ] s 4 (44 K( t ϕ ( ( t ( t 4 ( t ( t ( t ( t ( t ( t ( t ( t ( t ( t Remrk 4: I the curvture K t poit P o curve C is ot zero, the the circle o rdius /K whose cetre lies o the cocve side o C d which hs the sme tget lie t P s C is clled the circle o curvture or P Exmple 4: Let C e the curve prmetriclly give y ( t ( t, t K ( t, t The the curvture is t 6 t t 6 t / 4 [( t + ( t ] / ( 4 t + 9t I t /, the P, 4 8 K 96 5 The poit correspodig to t / hs coordites d the rdius o curvture ρ t tht poit is 5 ρ K / ( 96 Propositio 4: Let curve C e give y ( t ( ( t, ( t, ( t, where the compoet uctios re two times cotiuously dieretile The curvture K(t t the poit P ( ( t, ( t, ( t o C is -
Mthemtics or Egieers Prt II (ISE Versio /4-6- (48 K( t ( t ( t ( t Exmple 4: We compute the curvture o the twisted cuic x t, y t, z t t the poit P(t,t,t t t,t, t the we get: I we set ( ( K 4 ( ( 9 t + 9 t + t 4 ( + 4 t + 9 t / / Note: Formul (48 is oly vlid i, sice we use the vector product -4
Mthemtics or Egieers Prt II (ISE Versio /4-6- 5 Ple curves d polr coordites I curve i is ot give i prmetric represettio, the polr coordites my e very helpul (5 Polr coordites: The rectgulr coordites (x,y d polr coordites (r,ϕ o poit P(x,y re relted s ollows: (i x r cos ϕ, y r siϕ (ii y t ϕ, x r + x y, x rccos r π rccos ϕ x r udeied, i,y,y < r y The prmetric equtio i polr coordites o curve i is give y (5 ( ϕ r ( ϕ( cosϕ, siϕ, r P(x,y with the polr gle ϕ, α ϕ β, s prmeter d r rel vlued uctio o ϕ ϕ Fig 5 Polr coordites Fig 5 Descrtes le x Exmple 5: The set o the solutios o the equtio (5 x + y xy i two ukows determies curve i kow s Descrtes le (Fig 5 We use polr coordites to id prmetric represettio or the curve deied y equtio (5 By settig x r cosϕ d y r siϕ we get: r d thereore ϕ + r ϕ r ϕ ϕ cos si cos si cosϕ siϕ r ( ϕ cos ϕ + si ϕ -5
Mthemtics or Egieers Prt II (ISE Versio /4-6- Hece prmetric equtio o Descrtes le is give y cosϕ siϕ, ϕ π cos ϕ + si ϕ (54 ( ϕ ( cosϕ, siϕ Asymptote o the curve is the lie y -x - It is lso possile to get prmetric equtio or Descrtes le which is ot i polr orm; mely: (55 g ( t, + + t t t t, t \{-} I Figure 5 ud 54 you c see two spshots rom the geertio o the curve Propositio 5: Suppose C is curve i give y prmetric equtio i polr coordites; ie: ( ϕ r ( ϕ( cosϕ, siϕ, α ϕ β The the rc legth L o C is Fig 5 Grph o g(t, -8 t (56 L ( r ( + ( r ( ϕ ϕ d ϕ Proo: Sice ( ϕ ( r ( ϕ cosϕ r ( ϕ siϕ,r ( ϕ siϕ r ( ϕ cosϕ + it is esy to veriy tht ( ϕ ( r ( ϕ + ( r ( ϕ Fig 54 Grph o g(t, - t d thus (56 ollows immeditely Exmple 5: We cosider the curve C give y the prmetric equtio i polr coordites ϕ ϕ ϕ ϕ ϕ (57 ( ( cos,si, >, < This curve is clled the Archimede spirl (Figure 55 d 56-6
Mthemtics or Egieers Prt II (ISE Versio /4-6- The rc legth ter oe circultio is π L ϕ + d ϕ Fig 55 Archimede spirl < ϕ π d thereore ( + 4 + l( + 4 L + Propositio 5: Suppose C is smooth curve i give prmetriclly y ( t ( ( t, ( t, t I Fig 56 Archimede spirl < ϕ Let ech stright lie rom origi cut the curve oly oce the the re o the regio eclosed y tht curve d the stright lies rom origi to the edpoits o the curve, the so clled sector re (Figure 58, is give y A dt (58 ( ( t ( t ( t ( t I the Curve C hs piecewise smooth prmeteriztio i polr coordites r r(ϕ, α ϕ β, the pplies: Fig 57 Archimede spirl < ϕ (59 A r ( ϕ d ϕ Exmple 5: (i The cotet o the ellipse x cos(t, y si(t, t π is Fig 58 Sector re A ( cos t+ si t dt A -7
Mthemtics or Egieers Prt II (ISE Versio /4-6- (ii The re eclosed y the Archimede spirl ter oe circultio (Figure 55 is give y 4 ( ϕ d A ϕ More geerlly th the sector re ormul is the ollowig result Propositio 5: The re eclosed y piecewise cotiuously dieretile closed curve C with t t, t, t, which does ot itersect itsel is prmeteriztio ( ( ( ( A dt (5 ( ( t ( t ( t ( t I this cse (59 is lso vlid or polr coordites prmeteriztio Exmple 54: (i The steroid i the ollowig Figure 59 is prmetriclly give y ( t ( Rcos t+,rsi t+, t Fig 59 Asteroid, R The re eclosed y the steroid is -8
Mthemtics or Egieers Prt II (ISE Versio /4-6- A ( Rcos t+ Rsi t cost+ ( Rsi t+ Rcos t sit dt Thus 8 A R The steroid is the locus o poit o the rim o circle o rdius r R/4 rollig log the ier lie o the circle o rdius R (Figure 5 Sice R 4r pplies, A R 6r 8 Fig 5 Geertig steroid (ii A prmeteriztio o the lemiscte (Fig 5 is cost ( t (,sit, t + si t The right loop o the lemiscte i polr coordites is give y g( ϕ cos ϕ ( cosϕ,siϕ, ϕ 4 4 Thereore the re o oe loop o the lemiscte is Fig 5 Lemiscte / 4 / 4 A r ( d cos( d ϕ ϕ ϕ ϕ / 4 / 4 Thereore the re eclosed y the lemiscte is A The lemiscte is lso kow s Eight Curve -9
Mthemtics or Egieers Prt II (ISE Versio /4-6- 6 Ple curves d solids o revolutio I the ollowig we cosider solids o revolutio i geerted y ple curves with prmeteriztio [ ; ] IR, ( t ( x( t,y( t : I y ( t throughout [;], we c use rgumets similr to tht give i the previous sectios to compute the re o the surce geerted y revolvig give curve C out the x-xis respectively the y-xis i x ( t (see Figure 6 d 6 Fig 6 Revolvig C out the x-xis Propositio 6: Let smooth curve C give y [ ; ] IR, ( t ( x( t,y( t : d suppose C does ot itersect itsel, except possily t the poits ( d ( I y ( t, or t [;], the the re S o the surce o revolutio otied y revolvig C out the x-xis is (6 S y( t ( x ( t + ( y ( t x dt Fig 6 Revolvig C out the y-xis I x ( t, or t [;], the the re S o the surce o revolutio otied y revolvig C out the y-xis is (6 S x( t ( x ( t + ( y ( t y dt Remrk 6: Suppose the curve C is prmetriclly give y : [ ; ] IR, ( t ( x( t,y( t prmetric dieretil o rc legth is deied s dx dt dy dt (6 ds ( dx + ( dy + dt ( x ( t + ( y ( t dt The -
Mthemtics or Egieers Prt II (ISE Versio /4-6- Usig (6 we c write (6 d (6 s (64 S y( t ds d x( t x y S ds Exmple 6: (i Veriy tht the surce re o sphere o rdius r is 4πr Solutio: I C is the upper hl o the circle x + y r, the the sphericl surce my e otied e revolvig C out the x-xis A prmeteriztio o C is give y : [ ; ] IR, ( t ( r cos t,r si t π Applyig ormul (6 d usig the idetity si ( t + cos ( t π, we get: ( r si t r si t + r cos t dt r sit dt 4 S x r (ii We compute the re o the surce show i Figure 6 The geertig curve C is prmetriclly give y Usig ormul (6 we hve: [ ; ] IR, ( t ( t,cos t : π π S y π t + si t dt 5 85 Remrk 6: The ormuls (6 d (6 c e exteded to the cse i which y(t or x(t is egtive or some t [;] y replcig the uctio y(t tht precedes ds y y(t (or x(t y x(t I the meridi o solid o revolutio is prmetriclly give s ple curve C, the we c esily compute the Volume o this solid geerted y revolvig C out the x- xis or the y-xis usig the ollowig Propositio 6: Let smooth curve C give y [ ; ] IR, ( t ( x( t,y( t : -
Mthemtics or Egieers Prt II (ISE Versio /4-6- d suppose C is the grph o Crtesi uctio with respect to the xis o revolutio, the pplies: (i The volume V x o the solid o revolutio otied y revolvig C out the x-xis is (65 V ( y( t x ( t x dt (ii The volume V y o the solid o revolutio otied y revolvig C out the y-xis is (66 V ( x( t y ( t y dt Corollry 6: Suppose the cotiuously dieretile Crtesi uctio g : [ ; ] IR, y g( x the meridi o ple curve C The : [ ; ] IR, ( x ( x,g( x prmeteriztio o C d with Propositio 6 we hve: (67 ( ( x g x V dx d (68 V x g ( x y dx is, is Exmple 6: The upper hl o the steroid (see Figure 59 or 5 is prmetriclly give y ( t ( Rcos t,rsi t, t, R > The volume o the solid geerted y revolvig this curve out the x-xis is Fig 6 Asteroid revolved out the x-xis V x π R R π si si 7 6 t R cos t cos t dt t ( si t dt Thus we get with the ollowig Lemm 6: -
Mthemtics or Egieers Prt II (ISE Versio /4-6- V x R 9 R R 8 5 π 4 7 5 si π 7 t dt R si t dt R [ cost] R 6 7 5 8 5 π si π 5 t dt sit dt Computig the lst trigoometric itegrls we used Lemm 6: Let m d e positive itegers, the pplies: m (i si ( x cos ( x (ii si ( x si dx m ( x cos ( x ( + m m + + m + ( x cos( x + si dx dx si ( x si m ( x cos ( x dx -