ENGI 4430 Parametric Vector Functions Page 2-01

Transcription

1 ENGI 4430 Parametric Vectr Functins Page -01. Parametric Vectr Functins (cntinued) Any nn-zer vectr r can be decmpsed int its magnitude r and its directin: r rrˆ, where r r 0 Tangent Vectr: dx dy dz dr T If the parameter t is time, then the tangent vectr is als the velcity vectr, d r dr v, whse magnitude is the speed v. T The unit tangent is T dr dr Arc Length In In : s x y 3 : s x y z s x y z t t t t dx dy dz dr The vectr d r pints in the directin f the tangent T t the curve defined parametrically by t r r. dr dr dr dr dr T T [unit tangent = rate f change f r with respect t distance travelled alng the curve.] [Hwever, arc length s is rarely a cnvenient parameter.]

2 ENGI 4430 Parametric Vectr Functins Page -0 Example.01 (a) Find the arc length alng the curve defined by T r 1 1 t 4 t sin t 4 cs t sin t, frm the pint where t = 0 t the pint where t = 4π. (b) Find the unit tangent T. dx 1 1 cs 1 cs sin sin 4 4 (a) t t t t dy 1 sin t sin t cs t sin t cs t 4 4 dz cst dr sin t sin t cs t cs t T dr sin 4 t sin t cs t cs t sin t sin t cs t cs t 1sin t cs t 1 1 L 1 t Therefre L 4 (b) T dr dr dr But 1 t "fr all t". Therefre T sin t sin t cst cst [Nte that, in this example, t itself is the arc length frm the pint 0, 1 4, 0, (where t = 0). There are very few curves fr which the parameterizatin is this cnvenient!]

3 ENGI 4430 Parametric Vectr Functins Page -03 Arc Length fr a Plar Curve Fr a plar curve defined by r f x f cs y f sin. Using the abbreviatins,, the parameter is and r f, r f, c cs, s sin, dx dy rc rs and rs rc d d dx dy r c rrcs r s r s rrsc d d r c s r s c r d dr rc Therefre the arc length L alng the plar curve r f frm t is dr L r d d Example.0 Find the length L f the perimeter f the cardiid r 1 cs 0, r 1 cs 1 c dr sin s d dr r d 1 c c s c 1c

4 ENGI 4430 Parametric Vectr Functins Page -04 Example.0 (cntinued) But 1 cs x cs x. Set x x dr r cs d L cs d cs d 0 0 Using symmetry in the hrizntal axis, L cs d 4 cs d 4 sin Therefre the perimeter f the cardiid curve is L = 8. Nte: Fr and, cs cs, nt cs 0 cs 0! d Example.03 Find the arc length alng the spiral curve r ae a 0, frm t. dr r ae ae d ae ae ae d L d a e d a e d L a e e

5 ENGI 4430 Parametric Vectr Functins Page -05 Overview f Nrmal and Binrmal: Fr a curve in, there is a unique nrmal line at every pint where the unit tangent vectr is defined: There are nly tw chices fr the unit nrmal vectr. N is chsen t pint in the directin in which the curve is turning. This definitin fails wherever the unit tangent des nt exist. This definitin als fails at all pints f inflexin and wherever the curve is a straight line. Hwever, fr a curve in 3, there is a nrmal plane at each pint, nt a nrmal line. Frm the infinite number f different directins fr a nrmal vectr, we need t identify a unique directin fr a principal nrmal. Again, we shall chse the directin in which the unit tangent vectr is turning at each pint. T 1 T T 1 Using the prduct rule f differentiatin, T T 0 T 0 d r T 0 T (Nte: a unit vectr can never be zer). Select the principal nrmal N t be the directin in which the unit tangent is, instantaneusly, changing: N N and N are therefre parallel t d T. The unit principal nrmal then fllws: N

6 ENGI 4430 Parametric Vectr Functins Page -06 The unit binrmal is at right angles t bth tangent and principal nrmal: B cnstant B TN the curve lies entirely in ne plane (the plane defined by T ˆ and N). ˆ The three vectrs frm an rthnrmal set (they are mutually perpendicular and each ne is a unit vectr; that is, magnitude = 1). All three unit vectrs are undefined where the curve suddenly reverses directin, (such as at a cusp). Curvature: The derivative f the unit tangent with respect t distance travelled alng a curve is the principal nrmal vectr, N. Its directin is the unit principal nrmal N. Its magnitude is a measure f hw rapidly the curve is turning and is defined t be the curvature: dr N N N 1 The radius f curvature is. The radius f curvature at a pint n a curve is the radius f the circle which best fits the curve at that pint. Example.01 (cntinued) s (c) Find the curvature at any pint fr which s 0, fr the curve r s s sin s cs s sin s where s is the arc length frm the pint 0, 1 4, 0. T Frm part (b): sin s T sin scs s cs s

7 ENGI 4430 Parametric Vectr Functins Page -07 Example.01 (c) (cntinued) sin s cs s sin s N cs s sin s cs s sin s sin s s 1 sin s sin s It then fllws that the unit principal nrmal vectr at every pint is sin s 1 N N cs s 1 sin s sin s A Maple plt f this curve is available frm the demnstratin files sectin f the web site,

8 ENGI 4430 Parametric Vectr Functins Page -08 Anther frmula fr curvature: Let d r and d r r s r, then d r d r T r s T r st s (prduct rule f differentiatin) But N s 3 rr st s T s N 0 s B rr s 3, but s r. Therefre r r 3 r If the parameter t is time, then the curvature frmula is als va 3 v where v v acceleratin. is the speed (scalar), the vectr v is the velcity and the vectr a is the

9 ENGI 4430 Parametric Vectr Functins Page -09 Example.04 Find the curvature and the radius f curvature fr the helix, given in parametric frm by x = cs t, y = sin t, z = t. Assume SI units. Let c = cs t, s = sin t. c s c r s c s v r a r t 1 0 ˆi s c s s r r ˆj c s c c kˆ 1 0 s c 1 rr s c 1 r s c 1 rr 1 3 r Therefre and 1 1 m cnstant 1 m

10 ENGI 4430 Parametric Vectr Functins Page -10 Surfaces f Revlutin Cnsider a curve in the x-y plane, defined by the equatin y = f (x). If it is swept nce arund the line y = c, then it will generate a surface f revlutin. At any particular value f x, a thin crss-sectin thrugh that surface, parallel t the y-z plane, will be a circular disc f radius r, where r f x c Let us nw view the circular disc face-n, (s that the x axis and the axis f rtatin are bth pinting directly ut f the page and the page is parallel t the y-z plane). Let (x, y, z) be a general pint n the surface f revlutin. Frm this diagram, ne can see that r yc z Therefre, the equatin f the surface generated, when the curve y = f (x) is rtated nce arund the axis y = c, is y c z f x c Special case: When the curve y = f (x) is rtated nce arund the x axis, the equatin f the surface f revlutin is r y z f x y z f x

11 ENGI 4430 Parametric Vectr Functins Page -11 Example.05 Find the equatin f the surface generated, when the parabla the x axis. y 4ax is rtated nce arund The slutin is immediate: the equatin f the surface is y z 4ax, 4 f x ax which is the equatin f an elliptic parablid, (actually a special case, a circular parablid). A Maple wrksheet fr this surface is available frm the demnstratin files sectin f the ENGI 4430 web site. The Curved Surface Area f a Surface f Revlutin Fr a rtatin arund the x axis, the curved surface area swept ut by the element f arc length s is apprximately the prduct f the circumference f a circle f radius y with the length s. A y s Integrating alng a sectin f the curve y = f (x) frm x = a t x = b, the ttal curved surface area is xb A f x xa Fr a rtatin f y = f (x) abut the axis y = c, the curved surface area is xb A f x c f x c dx xa a dx Therefre and b b A f x c 1 f x dx a dy 1 dx dx

12 ENGI 4430 Parametric Vectr Functins Page -1 Example.06 Find the curved surface area f the circular parablid generated by rtating the prtin f the y 4cx c 0 x a 0 x b a abut the x axis. parabla frm t A xb y xa b a y dx dx d d [Chain rule:] dx dx y 4cx y 4cx y y 4c y c y dy 1 c 1 c x c dx dx y 4cx x A b a c x x c x 1/ dx 4 c x c dx b a Therefre x c 3/ 4 c 3 b a 8 c A b c a c 3 3/ 3/

13 ENGI 4430 Parametric Vectr Functins Page -13 Area Swept Out by a Plar Curve r = f (θ) A Area f triangle 1 r r r sin But the angle is small, s that sin and the increment r is small cmpared t r. Therefre 1 A r A 1 r d Example.07 Find the area f a circular sectr, radius r, angle. r = cnstant, A r d r r Therefre A 1 r Full circle: A r.

14 ENGI 4430 Parametric Vectr Functins Page -14 Example.08 Find the area swept ut by the plar curve r ae ver, (where a 0 and ). The cnditin prevents the same area being swept ut mre than nce. If then ne nee t subtract areas that have been cunted mre than nce [the red area in the diagram] 1 1 e A a e d a Therefre a A e e 4 In general, the area bunded by tw plar curves and is r f and r g and the radius vectrs 1 A f g d

15 ENGI 4430 Parametric Vectr Functins Page -15 Radial and Transverse Cmpnents f Velcity and Acceleratin At any pint P (nt at the ple), the unit radial vectr ˆr pints directly away frm the ple. The unit transverse vectr ˆθ is rthgnal t ˆr and pints in the directin f increasing θ. vectrs frm an rthnrmal basis fr. Unlike the Cartesian î and ĵ, if θ is nt cnstant then ˆr and ˆθ will be variable unit vectrs. Examine ˆr and ˆθ at tw nearby times t and t t: These The magnitudes f the unit vectrs can never change. By definitin, the magnitude f any unit vectr is always 1. Only the directins can change. Examine the difference in the unit radial vectr at these tw times. The tw lnger sides f this issceles triangle are bth 1 r ˆ. The tw larger angles are each equal t /. Clearly the angle between r ˆ t and the incremental vectr r ˆ appraches a right angle as t (and therefre ) appraches zer. ˆ t ˆ t. Therefre the derivative f r is parallel t Sine rule: Als, frm triangle gemetry (the sine rule), rˆ rˆ and r ˆ 1 (unit vectr) sin sin / As t 0, sin and sin / sin / 1 rˆ 1 rˆ drˆ d drˆ d ˆ 1 t t a b c sin A sin B sin C

16 ENGI 4430 Parametric Vectr Functins Page -16 d ˆ ( is parallel t Als, frm triangle gemetry (the sine rule), ˆ ˆ sin sin / Nw examine the difference in the unit transverse vectr at these tw times. The tw lnger sides f this issceles triangle are bth 1. Clearly the angle between ˆ t and the incremental vectr ˆ appraches a right angle as t (and therefre ) appraches zer. Therefre the derivative f ˆ t pints radially inwar r ˆ t ). and ˆ 1 As t 0, sin and sin / sin / 1 ˆ 1 ˆ d ˆ d d ˆ d rˆ 1 t t Therefre the derivatives f the tw [nn-cnstant] plar unit vectrs are drˆ d ˆ and d ˆ d ˆ r Using the verdt ntatin t represent differentiatin with respect t the parameter t, these results may be expressed mre cmpactly as rˆ ˆ and ˆ rˆ The radial and transverse cmpnents f velcity and acceleratin then fllw: r r rˆ v r r rˆ r rˆ r rˆ r ˆ v r and v r radial transverse r rˆ r rˆ ˆ ˆ ˆ r r r a v r r rˆ r ˆ r ˆ r rˆ r r rˆ r r ˆ a radial a transverse 1 d r The transverse cmpnent f acceleratin can als be written as atr r

17 ENGI 4430 Parametric Vectr Functins Page -17 Example.09 A particle fllws the path r, where the angle at any time is equal t the time: t 0. Find the radial and transverse cmpnents f acceleratin. r t r 1 r 0 v r rˆ r ˆ rˆ t ˆ v 1 t ˆ a r r r r r ˆ t rˆ ˆ Therefre a 4 t ar t and atr Example.10 Fr circular mtin arund the ple, with cnstant radius r and cnstant angular velcity, the velcity vectr is purely tangential, v r ˆ, and the acceleratin vectr is ˆ a r r r r r r r which matches the familiar result that centrifugal r centripetal acceleratin radially inward. ˆ ˆ r, directed

18 ENGI 4430 Parametric Vectr Functins Page -18 Tangential and Nrmal Cmpnents f Velcity and Acceleratin d Frm page.01, we knw that the velcity vectr is als the tangent vectr, vt r. dr The [scalar] speed is v v The speed is therefre the derivative with respect t time f distance travelled alng the curve. It fllws that v vt - the speed is als the tangential cmpnent f the velcity vectr. There are n cmpnents f velcity in the nrmal directins. The acceleratin vectr is dv d a v T But dv T v d d T T N N and v v N dv a T v N att an N Therefre the tangential cmpnent f acceleratin is the rate f change f speed (nt velcity) and is psitive if the particle is speeding up, negative if it is slwing dwn and zer if the speed is cnstant. It is als a ˆ T at va. v The nrmal cmpnent f acceleratin (in the directin f the unit principal nrmal) is never negative; bth the curvature and v are nn-negative at all times. There is never any cmpnent f acceleratin in the binrmal directin. dv dv dv Usually it is easiest t evaluate a a a and at first, then t find the nrmal cmpnent (and hence the curvature) frm a a a N T

19 ENGI 4430 Parametric Vectr Functins Page -19 Example.04 (cntinued) Find the tangential and nrmal cmpnents f velcity and acceleratin fr a particle travelling alng the helix, given in parametric frm by x = cs t, y = sin t, z = t. Hence find the curvature. Assume SI units. Frm page.09, cst sin t cst d d t sin t r cst v r sin t v a t 1 0 dv vt v t t a sin cs 1 T 0 The acceleratin is therefre purely nrmal. a t t a a a a cs sin 0 1 N T 1 Therefre t t v T ms and a 0T 1N ms 1 1 an v 1 and v m 1

20 ENGI 4430 Parametric Vectr Functins Page -0 Review f Lines [frm MATH 050] Given a pint,, P x y z knwn t be n a line L, a nn-zer vectr T and vectrs T d abc parallel t the line, r x y z and T OP x y z r, the equatins f the line may be expressed in the vectr frm r r d t t, r, prvided nne f a, b, c are zer, in the symmetric Cartesian frm x x y y z z a b c If, fr example, b = 0, but a and c are nn-zer, then the Cartesian frm becmes y y, x x z z a c Example.11 Find the Cartesian equatin f the line thrugh the pints (1, 3, 5) and (4, 1, 7). The vectr cnnecting the tw pints is an bvius chice fr the line directin vectr. a d b c 7 5 Either pint may be chsen as P. r x 1 y 3 z 5 The equatin f the line is x 1 y 3 z 5 3 4

21 ENGI 4430 Parametric Vectr Functins Page -1 Review f Planes [frm MATH 050] A plane is defined by tw vectrs [ is seen here edge-n ]. Its rientatin is determined by a nn-zer nrmal vectr T n A B C. Its lcatin is determined by the psitin vectr T OP x y z r f any ne pint P knwn t be n the plane. The vectr T r x y z is then the psitin vectr f a pint n the plane if and nly if r, equivalently, r n r n Ax By Cz D 0 where D r n Ax By Cz If u and v are tw nn-zer vectrs nt parallel t each ther and bth parallel t the plane, then a vectr parametric frm f the equatin f the plane is r r su t v, s, t The nrmal vectr t the plane is n u v, frm which the ther tw frms abve fllw. Example.1 Find the Cartesian equatin f the plane thrugh the pints (0, 3, 5), (, 4, 5) and (4, 1, 7) and find the equatin f the nrmal line t that plane that passes thrugh the rigin. The three pints define three vectrs, n pair f which are parallel t each ther but all three f which are parallel t the plane. Any tw f them will serve as vectrs u and v u and v ˆi 1 1 n u v ˆj 1 kˆ

22 ENGI 4430 Parametric Vectr Functins Page - Example.1 (cntinued) Any nn-zer multiple f this vectr will als serve as a nrmal vectr t the plane. Therefre select n 1 6 T. Any f the three pints will serve as P. Chse r T nr The equatin f the plane is n r n r x y 6z 36 We can quickly check that all three given pints satisfy this equatin: 0 + (3) + 6(5) = = 36 + (4) + 6(5) = = ( 1) + 6(7) = = 36 The directin vectr fr the nrmal line is just the nrmal vectr t the plane, d n 1 6 T r 0 (the line passes thrugh the rigin). The equatins f the nrmal line fllw immediately x 0 y 0 z [End f Chapter ]