7 CHAPTER VECTOR FUNCTIONS. Eercises 2 Find the domin of the vector function.. 2. 3 4 Find the limit. 3. lim cos t, sin t, t ln t t l 4. lim t, e t l rctn 2t, ln t t 5 Mtch the prmetric equtions with the grphs (lbeled I VI). Give resons for our choices. 5. cos 4t, t, 6. t, t 2, 7. t, t 2, 8. e t cos t, e t sin t, 9. cos t, sin t,. cos t, sin t, I rt t 2, st, s5 t rt t 2 t 2 i sin t j ln9 t2 k e t sin 4t t 2 sin 5t ln t II e t 8 Sketch the curve with the given vector eqution. Indicte with n rrow the direction in which t increses.. rt t 4, t 2. rt t 3, t 2 3. rt t, cos 2t, sin 2t 4. rt t, 3t, t 5. 6. 7. rt sin t, 3, cos t rt t i t j cos t k rt t 2 i t 4 j t 6 k 8. rt sin t i sin t j s2 cos t k 9. Show tht the curve with prmetric equtions t cos t, t sin t, t lies on the cone 2 2 2, nd use this fct to help sketch the curve. 2. Show tht the curve with prmetric equtions sin t, cos t, sin 2 t is the curve of intersection of the surfces 2 nd 2 2. Use this fct to help sketch the curve. ; 2 24 Use computer to grph the curve with the given vector eqution. Mke sure ou choose prmeter domin nd viewpoints tht revel the true nture of the curve. 2. rt sin t, cos t, t 2 22. 23. 24. rt sin t, sin 2t, sin 3t rt t 4 t 2, t, t 2 rt t 2, st, s5 t III IV V VI ; 25. Grph the curve with prmetric equtions cos 6t cos t, cos 6t sin t, cos 6t. Eplin the ppernce of the grph b showing tht it lies on cone. ; 26. Grph the curve with prmetric equtions s.25 cos 2 t cos t s.25 cos 2 t sin t.5 cos t Eplin the ppernce of the grph b showing tht it lies on sphere. 27. Show tht the curve with prmetric equtions t 2, 3t, t 3 psses through the points (, 4, ) nd (9, 8, 28) but not through the point (4, 7, 6). 28 3 Find vector function tht represents the curve of intersection of the two surfces. 28. The clinder 2 2 4 nd the surfce 29. The cone s 2 2 nd the plne
SECTION.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS 7 3. The prboloid 4 2 2 nd the prbolic clinder 2 ; 3. Tr to sketch b hnd the curve of intersection of the circulr clinder 2 2 4 nd the prbolic clinder 2. Then find prmetric equtions for this curve nd use these equtions nd computer to grph the curve. ; 32. Tr to sketch b hnd the curve of intersection of the prbolic clinder 2 nd the top hlf of the ellipsoid 2 4 2 4 2 6. Then find prmetric equtions for this curve nd use these equtions nd computer to grph the curve. 33. Suppose u nd v re vector functions tht possess limits s t l nd let c be constnt. Prove the following properties of limits. () lim ut vt lim ut lim vt t l t l t l (b) lim cut c lim ut t l t l (c) lim ut vt lim ut lim vt t l t l t l (d) lim t l ut vt lim ut lim vt t l t l 34. The view of the trefoil knot shown in Figure 7 is ccurte, but it doesn t revel the whole stor. Use the prmetric equtions 2 cos.5t cos t sin.5t 2 cos.5t sin t to sketch the curve b hnd s viewed from bove, with gps indicting where the curve psses over itself. Strt b showing tht the projection of the curve onto the -plne hs polr coordintes r 2 cos.5t nd, so r vries between nd 3. Then show tht hs mimum nd minimum vlues when the projection is hlfw between r nd r 3. ; When ou hve finished our sketch, use computer to drw the curve with viewpoint directl bove nd compre with our sketch. Then use the computer to drw the curve from severl other viewpoints. You cn get better impression of the curve if ou plot tube with rdius.2 round the curve. (Use the tubeplot commnd in Mple.) t.2 Derivtives nd Integrls of Vector Functions Lter in this chpter we re going to use vector functions to describe the motion of plnets nd other objects through spce. Here we prepre the w b developing the clculus of vector functions. Derivtives The derivtive r of vector function r is defined in much the sme w s for relvlued functions: dr rt h rt rt lim h l h if this limit eists. The geometric significnce of this definition is shown in Figure. If the points P nd Q hve position vectors rt nd rt h, then PQ l represents the C P r(t+h)-r(t) r(t) r(t+h) Q C rª(t) P r(t) r(t+h) r(t+h)-r(t) h Q FIGURE () The secnt vector (b) The tngent vector
76 CHAPTER VECTOR FUNCTIONS.2 Eercises. The figure shows curve C given b vector function rt. () Drw the vectors r4.5 r4 nd r4.2 r4. (b) Drw the vectors r4.5 r4.5 nd r4.2 r4.2. 2. 3. 4. rt e t 2 i j ln 3tk rt sin t i s t 2 j k rt t b t 2 c rt t b t c (c) Write epressions for r4 nd the unit tngent vector T(4). (d) Drw the vector T(4). C R 5 6 Find the unit tngent vector Tt t the point with the given vlue of the prmeter t. 5. rt cos t i 3t j 2 sin 2t k, t 6. rt 4st i t 2 j t k, t r(4.5) r(4.2) Q 7. If rt t, t 2, t 3, find rt, T, rt, nd rt rt. 8. If rt e 2t, e 2t, te 2t, find T, r, nd rt rt. 2. () Mke lrge sketch of the curve described b the vector function rt t 2, t, t 2, nd drw the vectors r(), r(.), nd r(.) r(). (b) Drw the vector r strting t (, ) nd compre it with the vector Eplin wh these vectors re so close to ech other in length nd direction. 3 8 () Sketch the plne curve with the given vector eqution. (b) Find rt. (c) Sketch the position vector rt nd the tngent vector rt for the given vlue of t. 3. rt cos t, sin t, 4. rt t 3, t 2, 5. rt t i t 2 j, 6. rt 2 sin t i 3 cos t j, 7. rt e t i e 2t j, t 8. rt sec t i tn t j, r. r. t 4 t r(4) t t 3 t 4 9 4 Find the derivtive of the vector function. 9. rt t 2, t, st. rt cos 3t, t, sin 3t P 9 22 Find prmetric equtions for the tngent line to the curve with the given prmetric equtions t the specified point. 9. t 5, t 4, t 3 ; (,, ) 2. t 2, t 2, t ;,, 2. e t cos t, e t sin t, e t ;,, 22. ln t, 2st, t 2 ;, 2, ; 23 24 Find prmetric equtions for the tngent line to the curve with the given prmetric equtions t the specified point. Illustrte b grphing both the curve nd the tngent line on common screen. 23. t, s2 cos t, s2 sin t; 4,, 24. cos t, 3e 2t, 3e 2t ;, 3, 3 25. Determine whether the curve is smooth. () rt t 3, t 4, t 5 (b) rt t 3 t, t 4, t 5 (c) rt cos 3 t, sin 3 t 26. () Find the point of intersection of the tngent lines to the curve rt sin t, 2 sin t, cos t t the points where t nd t.5. ; (b) Illustrte b grphing the curve nd both tngent lines. 27. The curves r t t, t 2, t 3 nd r 2t sin t, sin 2t, t intersect t the origin. Find their ngle of intersection correct to the nerest degree. 28. At wht point do the curves r t t, t, 3 t 2 nd r 2s 3 s, s 2, s 2 intersect? Find their ngle of intersection correct to the nerest degree. 29 34 Evlute the integrl. 29. 6t 3 i 9t 2 j 25t 4 k
SECTION.3 ARC LENGTH AND CURVATURE 77 3. 3. 32. 33. 34. 4 2t j 2 t t k 2 4 cos 2t i sin 2t j t sin t k 4 st i te t j t 2 k e t i 2t j ln t k cos t i sin t j t k 35. Find rt if rt t 2 i 4t 3 j t 2 k nd r j. 36. Find rt if rt sin t i cos t j 2t k nd r i j 2k. 37. Prove Formul of Theorem 3. 38. Prove Formul 3 of Theorem 3. 39. Prove Formul 5 of Theorem 3. 4. Prove Formul 6 of Theorem 3. 4. If ut i 2t 2 j 3t 3 k nd vt t i cos t j sin t k, find d ut vt. 42. If u nd v re the vector functions in Eercise 4, find d ut vt. 43. Show tht if r is vector function such tht r eists, then d rt rt rt rt d 44. Find n epression for ut vt wt. d 45. If rt, show tht rt rt. rt rt [Hint: rt 2 rt rt] 46. If curve hs the propert tht the position vector rt is lws perpendiculr to the tngent vector rt, show tht the curve lies on sphere with center the origin. 47. If ut rt rt rt, show tht ut rt rt rt.3 Arc Length nd Curvture In Section 6.3 we defined the length of plne curve with prmetric equtions f t, tt, t b, s the limit of lengths of inscribed polgons nd, for the cse where f nd t re continuous, we rrived t the formul L b s f t2 tt 2 b d 2 d 2 The length of spce curve is defined in ectl the sme w (see Figure ). Suppose tht the curve hs the vector eqution rt f t, tt, ht, t b, or, equivlentl, the prmetric equtions f t, tt, ht, where f, t, nd h re continuous. If the curve is trversed ectl once s t increses from to b, then it cn be shown tht its length is FIGURE The length of spce curve is the limit of lengths of inscribed polgons. 2 L b s f t2 tt 2 ht 2 b d 2 d 2 d 2 Notice tht both of the rc length formuls () nd (2) cn be put into the more compct form 3 L b rt
SECTION.3 ARC LENGTH AND CURVATURE 723 Figure 8 shows the heli nd the osculting plne in Emple 7. =_+ π 2 FIGURE 8 osculting circle P = SOLUTION The norml plne t P hs norml vector r2,,, so n eqution is or 2 2 The osculting plne t P contins the vectors T nd N, so its norml vector is T N B. From Emple 6 we hve Bt s2 sin t, cos t, A simpler norml vector is,,, so n eqution of the osculting plne is or 2 2 B 2 s2,, s2 EXAMPLE 8 Find nd grph the osculting circle of the prbol 2 t the origin. SOLUTION From Emple 5 the curvture of the prbol t the origin is 2. So the rdius of the osculting circle t the origin is nd its center is (, 2 ). Its eqution is therefore 2 2 ( 2) 2 4 2 For the grph in Figure 9 we use prmetric equtions of this circle: 2 cos t 2 2 sin t FIGURE 9 We summrie here the formuls for unit tngent, unit norml nd binorml vectors, nd curvture. Tt rt rt dt Nt ds Tt rt Tt Tt rt rt rt 3 Bt Tt Nt.3 Eercises 4 Find the length of the curve.. rt 2 sin t, 5t, 2 cos t, t 2. rt t 2, sin t t cos t, cos t t sin t, t 3. rt s2t i e t j e t k, t 4. rt t 2 i 2t j ln t k, t e 5. Use Simpson s Rule with n to estimte the length of the rc of the twisted cubic t, t 2, t 3 from the origin to the point 2, 4, 8. ; 6. Use computer to grph the curve with prmetric equtions cos t, sin 3t, sin t. Find the totl length of this curve correct to four deciml plces. 7 9 Reprmetrie the curve with respect to rc length mesured from the point where t in the direction of incresing t. 7. rt e t sin t i e t cos t j 8. rt 2t i 3 t j 5t k 9. rt 3sin t i 4t j 3 cos t k
724 CHAPTER VECTOR FUNCTIONS. Reprmetrie the curve with respect to rc length mesured from the point (, ) in the direction of incresing t. Epress the reprmetrition in its simplest form. Wht cn ou conclude bout the curve? 4 () Find the unit tngent nd unit norml vectors Tt nd Nt. (b) Use Formul 9 to find the curvture.. 2. rt 2 sin t, 5t, 2 cos t rt t 2, sin t t cos t, cos t t sin t, t 3. rt 3 t 3, t 2, 2t 4. rt t 2, 2t, ln t 5 7 Use Theorem to find the curvture. 5. rt t 2 i t k 6. rt t i t j t 2 k 7. rt sin t i cos t j sin t k 8. Find the curvture of rt e t cos t, e t sin t, t t the point (,, ). 9. Find the curvture of rt s2t, e t, e t t the point (,, ). ; 2. Grph the curve with prmetric equtions nd find the curvture t the point, 4,. 2 23 Use Formul to find the curvture. 24 25 At wht point does the curve hve mimum curvture? Wht hppens to the curvture s l? 26. Find n eqution of prbol tht hs curvture 4 t the origin. 27. () Is the curvture of the curve C shown in the figure greter t P or t Q? Eplin. P rt 2 t 2 i t 4t 32 24. ln 25. e 2t t 2 j t 2 2. 3 22. cos 23. 4 52 Q C (b) Estimte the curvture t P nd t Q b sketching the osculting circles t those points. ; 28 29 Use grphing clcultor or computer to grph both the curve nd its curvture function on the sme screen. Is the grph of wht ou would epect? 28. e 29. 4 3 3 Two grphs, nd b, re shown. One is curve f nd the other is the grph of its curvture function. Identif ech curve nd eplin our choices. 3. 3. 32. Use Theorem to show tht the curvture of plne prmetric curve f t, tt is where the dots indicte derivtives with respect to t. 33 34 Use the formul in Eercise 32 to find the curvture. 33. e t cos t, 34. t 3, et sin t t t 2 35 36 Find the vectors T, N, nd B t the given point. 35. rt t 2, 2, (, 2 3 t 3, t 3, ) 36. rt e t, e t sin t, e t cos t, 37 38 Find equtions of the norml plne nd osculting plne of the curve t the given point. 37. 2 sin 3t, t, 2 cos 3t; 38. t, t 2, t 3 ; b 2 2 32,,,,,, 2 ; 39. Find equtions of the osculting circles of the ellipse 9 2 4 2 36 t the points 2, nd, 3. Use grphing clcultor or computer to grph the ellipse nd both osculting circles on the sme screen. ; 4. Find equtions of the osculting circles of the prbol t the points nd (, 2 2, 2 ). Grph both osculting circles nd the prbol. 4. At wht point on the curve t 3, 3t, t 4 is the norml plne prllel to the plne 6 6 8? b
SECTION.4 MOTION IN SPACE 725 CAS 42. Is there point on the curve in Eercise 4 where the osculting plne is prllel to the plne? (Note: You will need CAS for differentiting, for simplifing, nd for computing cross product.) 43. Show tht the curvture is relted to the tngent nd norml vectors b the eqution 44. Show tht the curvture of plne curve is, where is the ngle between T nd i; tht is, is the ngle of inclintion of the tngent line. dt ds N dds 45. () Show tht dbds is perpendiculr to B. (b) Show tht dbds is perpendiculr to T. (c) Deduce from prts () nd (b) tht dbds sn for some number s clled the torsion of the curve. (The torsion mesures the degree of twisting of curve.) (d) Show tht for plne curve the torsion is s. 46. The following formuls, clled the Frenet-Serret formuls, re of fundmentl importnce in differentil geometr:. dtds N 2. dnds T B 3. dbds N (Formul comes from Eercise 43 nd Formul 3 comes from Eercise 45.) Use the fct tht N B T to deduce Formul 2 from Formuls nd 3. 47. Use the Frenet-Serret formuls to prove ech of the following. (Primes denote derivtives with respect to t. Strt s in the proof of Theorem.) () r st s 2 N (b) r r s 3 B (c) r s 2 s 3 T 3ss s 2 N s 3 B (d) r r r rr 2 48. Show tht the circulr heli rt cos t, sin t, bt where nd b re positive constnts, hs constnt curvture nd constnt torsion. [Use the result of Eercise 47(d).] 49. The DNA molecule hs the shpe of double heli (see Figure 3 on pge 77). The rdius of ech heli is bout ngstroms ( Å 8 cm). Ech heli rises bout 34 Å during ech complete turn, nd there re bout 2.9 8 complete turns. Estimte the length of ech heli. 5. Let s consider the problem of designing rilrod trck to mke smooth trnsition between sections of stright trck. Eisting trck long the negtive -is is to be joined smoothl to trck long the line for. () Find polnomil P P of degree 5 such tht the function F defined b F P if if if is continuous nd hs continuous slope nd continuous curvture. ; (b) Use grphing clcultor or computer to drw the grph of F..4 Motion in Spce C O FIGURE r(t+h)-r(t) h rª(t) Q P r(t) r(t+h) In this section we show how the ides of tngent nd norml vectors nd curvture cn be used in phsics to stud the motion of n object, including its velocit nd ccelertion, long spce curve. In prticulr, we follow in the footsteps of Newton b using these methods to derive Kepler s First Lw of plnetr motion. Suppose prticle moves through spce so tht its position vector t time t is rt. Notice from Figure tht, for smll vlues of h, the vector rt h rt h pproimtes the direction of the prticle moving long the curve rt. Its mgnitude mesures the sie of the displcement vector per unit time. The vector () gives the verge velocit over time intervl of length h nd its limit is the velocit vector vt t time t:
SECTION.4 MOTION IN SPACE 733 Writing d h 2 c, we obtin the eqution 2 r ed e cos In Appendi H it is shown tht Eqution 2 is the polr eqution of conic section with focus t the origin nd eccentricit e. We know tht the orbit of plnet is closed curve nd so the conic must be n ellipse. This completes the derivtion of Kepler s First Lw. We will guide ou through the derivtion of the Second nd Third Lws in the Applied Project on pge 735. The proofs of these three lws show tht the methods of this chpter provide powerful tool for describing some of the lws of nture..4 Eercises. The tble gives coordintes of prticle moving through spce long smooth curve. () Find the verge velocities over the time intervls [, ], [.5, ], [, 2], nd [,.5]. (b) Estimte the velocit nd speed of the prticle t t. 2. The figure shows the pth of prticle tht moves with position vector rt t time t. () Drw vector tht represents the verge velocit of the prticle over the time intervl 2 t 2.4. (b) Drw vector tht represents the verge velocit over the time intervl.5 t 2. (c) Write n epression for the velocit vector v(2). (d) Drw n pproimtion to the vector v(2) nd estimte the speed of the prticle t t 2. 2 t 2.7 9.8 3.7.5 3.5 7.2 3.3. 4.5 6. 3..5 5.9 6.4 2.8 2. 7.3 7.8 2.7 r(2.4) 2 r(2) r(.5) 3 8 Find the velocit, ccelertion, nd speed of prticle with the given position function. Sketch the pth of the prticle nd drw the velocit nd ccelertion vectors for the specified vlue of t. 3. rt t 2, t, 4. rt st, t, 5. rt e t i e t j, t 6. rt sin t i 2 cos t j, 7. rt sin t i t j cos t k, ; 8. rt t i t 2 j t 3 k, t t 9 2 Find the velocit, ccelertion, nd speed of prticle with the given position function. 9. rt t, t 2, t 3. rt 2 cos t, 3t, 2 sin t. rt s2t i e t j e t k 2. rt t sin t i t cos t j t 2 k 3 4 Find the velocit nd position vectors of prticle tht hs the given ccelertion nd the given initil velocit nd position. 3. t k, v i j, t 6 t t r 4. t k, v i j k, 5 6 () Find the position vector of prticle tht hs the given ccelertion nd the given initil velocit nd position. ; (b) Use computer to grph the pth of the prticle. 5. t i 2j 2t k, v, r i k 6. t t i t 2 j cos 2t k, v i k, r 2i 3j r j 7. The position function of prticle is given b rt t 2, 5t, t 2 6t. When is the speed minimum?
734 CHAPTER VECTOR FUNCTIONS 8. Wht force is required so tht prticle of mss m hs the position function rt t 3 i t 2 j t 3 k? 9. A force with mgnitude 2 N cts directl upwrd from the -plne on n object with mss 4 kg. The object strts t the origin with initil velocit v i j. Find its position function nd its speed t time t. 2. Show tht if prticle moves with constnt speed, then the velocit nd ccelertion vectors re orthogonl. 2. A projectile is fired with n initil speed of 5 ms nd ngle of elevtion 3. Find () the rnge of the projectile, (b) the mimum height reched, nd (c) the speed t impct. 22. Rework Eercise 2 if the projectile is fired from position 2 m bove the ground. 23. A bll is thrown t n ngle of 45 to the ground. If the bll lnds 9 m w, wht ws the initil speed of the bll? 24. A gun is fired with ngle of elevtion 3. Wht is the mule speed if the mimum height of the shell is 5 m? 25. A gun hs mule speed 5 ms. Find two ngles of elevtion tht cn be used to hit trget 8 m w. 26. A btter hits bsebll 3 ft bove the ground towrd the center field fence, which is ft high nd 4 ft from home plte. The bll leves the bt with speed 5 fts t n ngle 5 bove the horiontl. Is it home run? (In other words, does the bll cler the fence?) ; 27. Wter trveling long stright portion of river normll flows fstest in the middle, nd the speed slows to lmost ero t the bnks. Consider long stretch of river flowing north, with prllel bnks 4 m prt. If the mimum wter speed is 3 ms, we cn use qudrtic function s bsic model for the rte of wter flow units from the west bnk: f 3 4 4. () A bot proceeds t constnt speed of 5 ms from point A on the west bnk while mintining heding perpendiculr to the bnk. How fr down the river on the opposite bnk will the bot touch shore? Grph the pth of the bot. (b) Suppose we would like to pilot the bot to lnd t the point B on the est bnk directl opposite A. If we mintin constnt speed of 5 ms nd constnt heding, find the ngle t which the bot should hed. Then grph the ctul pth the bot follows. Does the pth seem relistic? 28. Another resonble model for the wter speed of the river in Eercise 27 is sine function: f 3 sin4. If boter would like to cross the river from A to B with constnt heding nd constnt speed of 5 ms, determine the ngle t which the bot should hed. 29 32 Find the tngentil nd norml components of the ccelertion vector. 29. rt 3t t 3 i 3t 2 j 3. rt t i t 2 j 3t k 3. rt cos t i sin t j t k 32. rt t i cos 2 t j sin 2 t k 33. The mgnitude of the ccelertion vector is cms 2. Use the figure to estimte the tngentil nd norml components of. 34. If prticle with mss m moves with position vector rt, then its ngulr momentum is defined s Lt mrt vt nd its torque s t mrt t. Show tht Lt t. Deduce tht if t for ll t, then Lt is constnt. (This is the lw of conservtion of ngulr momentum.) 35. The position function of spceship is rt 3 ti 2 ln tj 7 4 t 2 k nd the coordintes of spce sttion re 6, 4, 9. The cptin wnts the spceship to cost into the spce sttion. When should the engines be turned off? 36. A rocket burning its onbord fuel while moving through spce hs velocit vt nd mss mt t time t. If the ehust gses escpe with velocit v e reltive to the rocket, it cn be deduced from Newton s Second Lw of Motion tht m dv dm () Show tht vt v ln m. mt ve (b) For the rocket to ccelerte in stright line from rest to twice the speed of its own ehust gses, wht frction of its initil mss would the rocket hve to burn s fuel? v e
74 CHAPTER VECTOR FUNCTIONS For some purposes the prmetric representtions in Solutions nd 2 re equll good, but Solution 2 might be preferble in certin situtions. If we re interested onl in the prt of the cone tht lies below the plne, for instnce, ll we hve to do in Solution 2 is chnge the prmeter domin to r 2 2 So the vector eqution is r, i j 2s 2 2 k SOLUTION 2 Another representtion results from choosing s prmeters the polr coordintes r nd. A point,, on the cone stisfies r cos, r sin, nd 2s 2 2 2r. So vector eqution for the cone is rr, r cos i r sin j 2r k 2 where r nd. Surfces of Revolution =ƒ Surfces of revolution cn be represented prmetricll nd thus grphed using computer. For instnce, let s consider the surfce S obtined b rotting the curve f, b, bout the -is, where f. Let be the ngle of rottion s shown in Figure 9. If,, is point on S, then ƒ ƒ (,, ) 3 f cos f sin Therefore, we tke nd s prmeters nd regrd Equtions 3 s prmetric equtions of S. The prmeter domin is given b b,. 2 FIGURE 9 FIGURE EXAMPLE 8 Find prmetric equtions for the surfce generted b rotting the curve sin, 2, bout the -is. Use these equtions to grph the surfce of revolution. SOLUTION From Equtions 3, the prmetric equtions re sin cos sin sin 2 nd the prmeter domin is 2,. Using computer to plot these equtions nd rotte the imge, we obtin the grph in Figure. We cn dpt Equtions 3 to represent surfce obtined through revolution bout the - or -is. (See Eercise 28.).5 Eercises 4 Identif the surfce with the given vector eqution.. ru, v u cos v i u sin v j u 2 k 2. ru, v 2u i u 3v j 2 4u 5v k 3. r,, cos, sin 4. r,, cos, sin ; 5 Use computer to grph the prmetric surfce. Get printout nd indicte on it which grid curves hve u constnt nd which hve v constnt. 5. 6. ru, v u 2, v 3, u v, u, v ru, v u v, u 2, v 2, u, v
SECTION.5 PARAMETRIC SURFACES 74 7. ru, v cos 3 u cos 3 v, sin 3 u cos 3 v, sin 3 v, u, v 2 8. ru, v cos u sin v, sin u sin v, cos v ln tnv2, u 2,. v 6.2 9. cos u sin 2v, sin u sin 2v, sin v. u sin u cos v, u cos u cos v, u sin v 6 Mtch the equtions with the grphs lbeled I VI nd give resons for our nswers. Determine which fmilies of grid curves hve u constnt nd which hve v constnt.. 2. 3. ru, v cos v i sin v j u k ru, v u cos v i u sin v j u k ru, v u cos v i u sin v j v k 4. u 3, u sin v, u cos v 5. u sin u cos v, cos u sin v, u 6. u3 cos v cos 4u, u3 cos v sin 4u, 3u u sin v I II CAS 7 24 Find prmetric representtion for the surfce. 7. The plne tht psses through the point, 2, 3 nd contins the vectors i j k nd i j k. 8. The lower hlf of the ellipsoid 2 2 4 2 2 9. The prt of the hperboloid 2 2 2 tht lies to the right of the -plne 2. The prt of the elliptic prboloid 2 2 2 4 tht lies in front of the plne 2. The prt of the sphere 2 2 2 4 tht lies bove the cone s 2 2 22. The prt of the clinder 2 2 tht lies between the plnes nd 3 23. The prt of the plne 5 tht lies inside the clinder 2 2 6 24. The prt of the plne 3 tht lies inside the clinder 2 2 25 26 Use computer lgebr sstem to produce grph tht looks like the given one. 25. 26. 3 _3 _3 5 _ III IV ; 27. Find prmetric equtions for the surfce obtined b rotting the curve e, 3, bout the -is nd use them to grph the surfce. V VI ; 28. Find prmetric equtions for the surfce obtined b rotting the curve 4 2 4, 2 2, bout the -is nd use them to grph the surfce. 29. () Show tht the prmetric equtions sin u cos v, b sin u sin v, c cos u, u, v 2, represent n ellipsoid. ; (b) Use the prmetric equtions in prt () to grph the ellipsoid for the cse, b 2, c 3. ; 3. The surfce with prmetric equtions 2 cos r cos2 2 sin r cos2 r sin2 2 where 2 r 2 nd, is clled Möbius
742 CHAPTER VECTOR FUNCTIONS strip. Grph this surfce with severl viewpoints. Wht is unusul bout it? ; 3. () Wht hppens to the spirl tube in Emple 2 (see Figure 5) if we replce cos u b sin u nd sin u b cos u? (b) Wht hppens if we replce cos u b cos 2u nd sin u b sin 2u? 32. () Find prmetric representtion for the torus obtined b rotting bout the -is the circle in the -plne with center b,, nd rdius b. [Hint: Tke s prmeters the ngles nd shown in the figure.] ; (b) Use the prmetric equtions found in prt () to grph the torus for severl vlues of nd b. (b,, ) (,, ) å Review CONCEPT CHECK. Wht is vector function? How do ou find its derivtive nd its integrl? 2. Wht is the connection between vector functions nd spce curves? 3. () Wht is smooth curve? (b) How do ou find the tngent vector to smooth curve t point? How do ou find the tngent line? The unit tngent vector? 4. If u nd v re differentible vector functions, c is sclr, nd f is rel-vlued function, write the rules for differentiting the following vector functions. () ut vt (b) cut (c) f tut (d) ut vt (e) ut vt (f) u f t 5. How do ou find the length of spce curve given b vector function rt? 6. () Wht is the definition of curvture? (b) Write formul for curvture in terms of rt nd Tt. (c) Write formul for curvture in terms of rt nd rt. (d) Write formul for the curvture of plne curve with eqution f. 7. () Write formuls for the unit norml nd binorml vectors of smooth spce curve rt. (b) Wht is the norml plne of curve t point? Wht is the osculting plne? Wht is the osculting circle? 8. () How do ou find the velocit, speed, nd ccelertion of prticle tht moves long spce curve? (b) Write the ccelertion in terms of its tngentil nd norml components. 9. Stte Kepler s Lws.. Wht is prmetric surfce? Wht re its grid curves? TRUE FALSE QUIZ Determine whether the sttement is true or flse. If it is true, eplin wh. If it is flse, eplin wh or give n emple tht disproves the sttement.. The curve with vector eqution rt t 3 i 2t 3 j 3t 3 k is line. 2. The curve with vector eqution rt t, t 3, t 5 is smooth. 3. The curve with vector eqution rt cos t, t 2, t 4 is smooth. 4. The derivtive of vector function is obtined b differentiting ech component function. 5. If ut nd vt re differentible vector functions, then d ut vt ut vt 6. If rt is differentible vector function, then d rt rt 7. If Tt is the unit tngent vector of smooth curve, then the curvture is. dt 8. The binorml vector is Bt Nt Tt. 9. The osculting circle of curve C t point hs the sme tngent vector, norml vector, nd curvture s C t tht point.. Different prmetritions of the sme curve result in identicl tngent vectors t given point on the curve.