■
710
10.1
1–2
■
CHAPTER 10 VECTOR FUNCTIONS
Exercises
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3–4
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冓
■
2t
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■
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●
冔
■
y 苷 t,
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●
●
12. r共t兲 苷 具t 3, t 2 典
13. r共t兲 苷 具t, cos 2t, sin 2t典
14. r共t兲 苷 具1 t, 3t, t典
●
15. r共t兲 苷 具sin t, 3, cos t典
18. r共t兲 苷 sin t i sin t j s2 cos t k
■
■
■
■
■
■
7. x 苷 t,
y 苷 1兾共1 t 2 兲,
■
■
■
■
■
■
■
■
■
■
■
y 苷 t sin t, z 苷 t lies on the cone z 2 苷 x 2 y 2, and use this
fact to help sketch the curve.
■
20. Show that the curve with parametric equations x 苷 sin t,
y 苷 cos t, z 苷 sin 2t is the curve of intersection of the
surfaces z 苷 x 2 and x 2 y 2 苷 1. Use this fact to help
sketch the curve.
z 苷 et
y 苷 t 2,
■
19. Show that the curve with parametric equations x 苷 t cos t,
■
z 苷 sin 4t
6. x 苷 t,
; 21–24
z 苷 t2
y 苷 et sin 10t,
■ Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and
viewpoints that reveal the true nature of the curve.
z 苷 et
9. x 苷 cos t,
y 苷 sin t,
z 苷 sin 5t
21. r共t兲 苷 具sin t, cos t, t 2 典
10. x 苷 cos t,
y 苷 sin t,
z 苷 ln t
22. r共t兲 苷 具t 4 t 2 1, t, t 2 典
z
I
●
17. r共t兲 苷 t 2 i t 4 j t 6 k
■ Match the parametric equations with the graphs
(labeled I–VI). Give reasons for your choices.
8. x 苷 et cos 10t,
●
11. r共t兲 苷 具t 4 1, t典
5–10
5. x 苷 cos 4t,
●
16. r共t兲 苷 t i t j cos t k
ln t
4. lim arctan t, e ,
tl
t
■
■
Find the limit.
3. lim 具cos t, sin t, t ln t典
tl0
■
●
■ Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.
t2
i sin t j ln共9 t 2兲 k
2. r共t兲 苷
t2
■
●
11–18
Find the domain of the vector function.
1. r共t兲 苷 具 t 2, st 1, s5 t 典
■
●
23. r共t兲 苷 具 t 2, st 1, s5 t 典
z
II
24. r共t兲 苷 具sin t, sin 2t, sin 3t典
■
■
■
■
■
■
■
■
■
■
■
■
; 25. Graph the curve with parametric equations
x
y
x
z
III
x 苷 共1 cos 16t兲 cos t, y 苷 共1 cos 16t兲 sin t,
z 苷 1 cos 16t. Explain the appearance of the graph by
showing that it lies on a cone.
y
z
IV
; 26. Graph the curve with parametric equations
x 苷 s1 0.25 cos 2 10t cos t
y 苷 s1 0.25 cos 2 10t sin t
y
x
z 苷 0.5 cos 10t
x
z
V
y
z
VI
Explain the appearance of the graph by showing that it lies
on a sphere.
27. Show that the curve with parametric equations x 苷 t 2,
y 苷 1 3t, z 苷 1 t 3 passes through the points (1, 4, 0)
and (9, 8, 28) but not through the point (4, 7, 6).
x
28–30
■ Find a vector function that represents the curve of
intersection of the two surfaces.
y
y
x
28. The cylinder x 2 y 2 苷 4 and the surface z 苷 xy
■
■
■
■
■
■
■
■
■
■
■
■
■
29. The cone z 苷 sx 2 y 2 and the plane z 苷 1 y
■
◆
SECTION 10.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
(d) lim 关u共t兲 v共t兲兴 苷 lim u共t兲 lim v共t兲
30. The paraboloid z 苷 4x 2 y 2 and the parabolic cylinder
tla
y 苷 x2
■
■
■
■
■
■
■
■
■
■
■
■
■
2
; 32. Try to sketch by hand the curve of intersection of the
parabolic cylinder y 苷 x 2 and the top half of the ellipsoid
x 2 4y 2 4z 2 苷 16. Then find parametric equations for
this curve and use these equations and a computer to graph
the curve.
33. Suppose u and v are vector functions that possess limits as
t l a and let c be a constant. Prove the following properties of limits.
(a) lim 关u共t兲 v共t兲兴 苷 lim u共t兲 lim v共t兲
tla
tla
tla
(b) lim cu共t兲 苷 c lim u共t兲
tla
tla
(c) lim 关u共t兲 ⴢ v共t兲兴 苷 lim u共t兲 ⴢ lim v共t兲
tla
10.2
tla
tla
tla
but it doesn’t reveal the whole story. Use the parametric
equations
lar cylinder x y 苷 4 and the parabolic cylinder z 苷 x .
Then find parametric equations for this curve and use these
equations and a computer to graph the curve.
2
tla
34. The view of the trefoil knot shown in Figure 7 is accurate,
; 31. Try to sketch by hand the curve of intersection of the circu2
711
x 苷 共2 cos 1.5t兲 cos t
y 苷 共2 cos 1.5t兲 sin t
z 苷 sin 1.5t
to sketch the curve by hand as viewed from above, with
gaps indicating where the curve passes over itself. Start by
showing that the projection of the curve onto the xy-plane
has polar coordinates r 苷 2 cos 1.5t and 苷 t, so r
varies between 1 and 3. Then show that z has maximum and
minimum values when the projection is halfway between
r 苷 1 and r 苷 3.
; When you have finished your sketch, use a computer to
draw the curve with viewpoint directly above and compare
with your sketch. Then use the computer to draw the curve
from several other viewpoints. You can get a better impression of the curve if you plot a tube with radius 0.2 around
the curve. (Use the tubeplot command in Maple.)
Derivatives and Integrals of Vector Functions
●
●
●
●
●
●
●
Later in this chapter we are going to use vector functions to describe the motion of
planets and other objects through space. Here we prepare the way by developing the
calculus of vector functions.
Derivatives
The derivative r of a vector function r is defined in much the same way as for realvalued functions:
dr
r共t h兲 r共t兲
苷 r共t兲 苷 lim
hl0
dt
h
1
if this limit exists. The geometric significance of this definition is shown in Figure 1.
l
If the points P and Q have position vectors r共t兲 and r共t h兲, then PQ represents the
z
z
r(t+h)-r(t)
rª(t)
Q
P
P
r(t)
r(t)
r(t+h)
Q
r(t+h)
C
C
0
0
y
x
FIGURE 1
r(t+h)-r(t)
h
(a) The secant vector
y
x
(b) The tangent vector
■
716
10.2
CHAPTER 10 VECTOR FUNCTIONS
Exercises
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■
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■
■
■
■
■
■
■
■
■
■
■
t
■
共1, 1, 1兲
cos t, y 苷 e
t
sin t, z 苷 e ; 共1, 0, 1兲
■
■
■
■
共0, 2, 1兲
■
■
■
■
■
■
■
(a) r共t兲 苷 具t 3, t 4, t 5 典
(c) r共t兲 苷 具cos 3t, sin 3t典
■
■
■
■
■
共兾4, 1, 1兲
■
■
■
■
■
(b) r共t兲 苷 具t 3 t, t 4, t 5 典
curve r共t兲 苷 具sin t, 2 sin t, cos t典 at the points
where t 苷 0 and t 苷 0.5.
(b) Illustrate by graphing the curve and both tangent lines.
27. The curves r1共t兲 苷 具t, t 2, t 3 典 and r2共t兲 苷 具sin t, sin 2t, t典
t 苷 兾3
intersect at the origin. Find their angle of intersection
correct to the nearest degree.
t苷0
28. At what point do the curves r1共t兲 苷 具t, 1 t, 3 t 2 典 and
t 苷 兾4
■
■
Find the derivative of the vector function.
■
■
■
共1, 3, 3兲
26. (a) Find the point of intersection of the tangent lines to the
;
■
■
(1, 1, 1)
25. Determine whether the curve is smooth.
t苷1
10. r共t兲 苷 具cos 3t, t, sin 3t典
■
t苷1
24. x 苷 cos t, y 苷 3e 2t, z 苷 3e2t;
t苷1
9. r共t兲 苷 具 t 2, 1 t, st 典
■
t苷0
23. x 苷 t, y 苷 s2 cos t, z 苷 s2 sin t;
t 苷 兾4
■
■
■ Find parametric equations for the tangent line to the
curve with the given parametric equations at the specified point.
Illustrate by graphing both the curve and the tangent line on a
common screen.
(a) Sketch the plane curve with the given vector equation.
(b) Find r共t兲.
(c) Sketch the position vector r共t兲 and the tangent vector r共t兲
for the given value of t.
■
●
; 23–24
■
■
■
22. x 苷 ln t, y 苷 2st, z 苷 t 2;
■
■
■
2
21. x 苷 e
Explain why these vectors are so close to each other in
length and direction.
9–14
■
20. x 苷 t 1, y 苷 t 1, z 苷 t 1;
r共1.1兲 r共1兲
0.1
8. r共t兲 苷 sec t i tan t j,
●
3
2
function r共t兲 苷 具t , t典 , 0 t 2, and draw the vectors
r(1), r(1.1), and r(1.1) r(1).
(b) Draw the vector r共1兲 starting at (1, 1) and compare it
with the vector
■
■
19. x 苷 t 5, y 苷 t 4, z 苷 t 3;
x
2
j,
●
■ Find parametric equations for the tangent line to the
curve with the given parametric equations at the specified point.
2. (a) Make a large sketch of the curve described by the vector
6. r共t兲 苷 2 sin t i 3 cos t j,
●
19–22
t
5. r共t兲 苷 共1 t兲 i t 2 j,
●
18. If r共t兲 苷 具e 2t, e2t, te 2t 典 , find T共0兲, r共0兲, and r共t兲 ⴢ r共t兲.
1
3. r共t兲 苷 具cos t, sin t典 ,
●
17. If r共t兲 苷 具t, t , t 典 , find r共t兲, T共1兲, r共t兲, and r共t兲 r共t兲.
Q
0
■
■
2
r(4)
■
■
16. r共t兲 苷 4st i t j t k,
■
P
■
■
2
r(4.2)
7. r共t兲 苷 e i e
■
15. r共t兲 苷 cos t i 3t j 2 sin 2t k,
1
2t
●
■ Find the unit tangent vector T共t兲 at the point with the
given value of the parameter t.
r(4.5)
t
●
15–16
R
C
4. r共t兲 苷 具t , t 典 ,
●
14. r共t兲 苷 t a 共b t c兲
y
2
●
13. r共t兲 苷 a t b t 2 c
r共4.2兲 r共4兲
0.2
(c) Write expressions for r共4兲 and the unit tangent
vector T(4).
(d) Draw the vector T(4).
3
●
12. r共t兲 苷 sin1t i s1 t 2 j k
■
3–8
●
11. r共t兲 苷 e t i j ln共1 3t兲 k
(a) Draw the vectors r共4.5兲 r共4兲 and r共4.2兲 r共4兲.
(b) Draw the vectors
and
●
2
1. The figure shows a curve C given by a vector function r共t兲.
r共4.5兲 r共4兲
0.5
●
r2共s兲 苷 具3 s, s 2, s 2 典 intersect? Find their angle of
intersection correct to the nearest degree.
■
29–34
29.
y
1
0
■
Evaluate the integral.
共16t 3 i 9t 2 j 25t 4 k兲 dt
■
◆
SECTION 10.3 ARC LENGTH AND CURVATURE
30.
y
31.
y
1
0
冉
兾4
0
4
冊
冉
st i tet j v共t兲 苷 t i cos t j sin t k, find 共d兾dt兲 关u共t兲 ⴢ v共t兲兴.
42. If u and v are the vector functions in Exercise 41, find
冊
共d兾dt兲 关u共t兲 v共t兲兴.
1
k dt
t2
33.
y 共e i 2t j ln t k兲 dt
34.
y 共cos t i sin t j t k兲 dt
■
41. If u共t兲 苷 i 2t 2 j 3t 3 k and
共cos 2t i sin 2t j t sin t k兲 dt
y
1
40. Prove Formula 6 of Theorem 3.
4
2t
j
k dt
1 t2
1 t2
32.
43. Show that if r is a vector function such that r exists, then
d
关r共t兲 r共t兲兴 苷 r共t兲 r共t兲
dt
t
■
■
■
■
■
44. Find an expression for
■
■
■
■
■
■
[Hint: ⱍ r共t兲 ⱍ
2
36. Find r共t兲 if r共t兲 苷 sin t i cos t j 2t k and
d
关u共t兲 ⴢ 共v共t兲 w共t兲兲兴.
dt
d
1
r共t兲 苷
r共t兲 ⴢ r共t兲.
dt
r共t兲
ⱍ
45. If r共t兲 苷 0, show that
■
35. Find r共t兲 if r共t兲 苷 t 2 i 4t 3 j t 2 k and r共0兲 苷 j.
ⱍ ⱍ
苷 r共t兲 ⴢ r共t兲]
ⱍ
46. If a curve has the property that the position vector r共t兲 is
r共0兲 苷 i j 2 k.
always perpendicular to the tangent vector r共t兲, show that
the curve lies on a sphere with center the origin.
37. Prove Formula 1 of Theorem 3.
47. If u共t兲 苷 r共t兲 ⴢ 关r共t兲 r共t兲兴, show that
38. Prove Formula 3 of Theorem 3.
u共t兲 苷 r共t兲 ⴢ 关r共t兲 r共t兲兴
39. Prove Formula 5 of Theorem 3.
10.3
717
Arc Length and Curvature
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●
●
In Section 6.3 we defined the length of a plane curve with parametric equations x 苷 f 共t兲,
y 苷 t共t兲, a t b, as the limit of lengths of inscribed polygons and, for the case
where f and t are continuous, we arrived at the formula
1
z
b
L 苷 y s关 f 共t兲兴 2 关t共t兲兴 2 dt 苷
a
y 冑冉 冊 冉 冊
dx
dt
b
a
2
dy
dt
2
dt
The length of a space curve is defined in exactly the same way (see Figure 1). Suppose that the curve has the vector equation r共t兲 苷 具 f 共t兲, t共t兲, h共t兲典 , a t b, or,
equivalently, the parametric equations x 苷 f 共t兲, y 苷 t共t兲, z 苷 h共t兲, where f , t, and h
are continuous. If the curve is traversed exactly once as t increases from a to b, then
it can be shown that its length is
0
y
2
x
FIGURE 1
b
L 苷 y s关 f 共t兲兴 2 关t共t兲兴 2 关h共t兲兴 2 dt
a
苷
The length of a space curve is the limit
of lengths of inscribed polygons.
y
b
a
冑冉 冊 冉 冊 冉 冊
dx
dt
2
dy
dt
2
dz
dt
2
dt
Notice that both of the arc length formulas (1) and (2) can be put into the more compact form
3
L苷y
b
a
ⱍ r共t兲 ⱍ dt
◆
SECTION 10.3 ARC LENGTH AND CURVATURE
723
SOLUTION The normal plane at P has normal vector r共兾2兲 苷 具1, 0, 1典 , so an equa-
tion is
冉 冊
2
1共x 0兲 0共y 1兲 1 z ▲ Figure 8 shows the helix and the
osculating plane in Example 7.
苷0
z苷x
or
2
The osculating plane at P contains the vectors T and N, so its normal vector is
T N 苷 B. From Example 6 we have
z
z=_x+π2
2
B
苷
1
1
, 0,
s2
s2
冔
A simpler normal vector is 具1, 0, 1典 , so an equation of the osculating plane is
冉 冊
P
x
冉冊 冓
1
具sin t, cos t, 1典
s2
B共t兲 苷
y
2
1共x 0兲 0共y 1兲 1 z FIGURE 8
苷0
z 苷 x or
2
EXAMPLE 8 Find and graph the osculating circle of the parabola y 苷 x 2 at the origin.
SOLUTION From Example 5 the curvature of the parabola at the origin is 共0兲 苷 2. So
the radius of the osculating circle at the origin is 1兾 苷 12 and its center is (0, 12 ). Its
equation is therefore
y
y=≈
osculating
circle
x 2 ( y 12 ) 苷 14
2
For the graph in Figure 9 we use parametric equations of this circle:
1
2
0
x 苷 12 cos t
x
1
We summarize here the formulas for unit tangent, unit normal and binormal vectors, and curvature.
FIGURE 9
T共t兲 苷
ⱍ
r共t兲
r共t兲
N共t兲 苷
ⱍ
苷
10.3
1–4
■
Exercises
●
●
●
●
●
●
●
●
●
3. r共t兲 苷 s2 t i e t j et k,
■
■
■
●
3
●
●
●
●
●
●
●
●
●
●
●
●
●
●
equations x 苷 cos t, y 苷 sin 3t, z 苷 sin t. Find the total
length of this curve correct to four decimal places.
0t
7–9 ■ Reparametrize the curve with respect to arc length measured from the point where t 苷 0 in the direction of increasing t.
1te
■
B共t兲 苷 T共t兲 N共t兲
ⱍ
ⱍ T共t兲 ⱍ 苷 ⱍ r共t兲 r共t兲 ⱍ
ⱍ r共t兲 ⱍ
ⱍ r共t兲 ⱍ
dT
苷
ds
●
ⱍ
T共t兲
T共t兲
; 6. Use a computer to graph the curve with parametric
0t1
4. r共t兲 苷 t 2 i 2t j ln t k,
■
冟 冟
10 t 10
2. r共t兲 苷 具t 2, sin t t cos t, cos t t sin t典 ,
■
●
Find the length of the curve.
1. r共t兲 苷 具2 sin t, 5t, 2 cos t典 ,
■
y 苷 12 12 sin t
■
■
7. r共t兲 苷 e t sin t i e t cos t j
■
■
■
■
8. r共t兲 苷 共1 2t兲 i 共3 t兲 j 5t k
5. Use Simpson’s Rule with n 苷 10 to estimate the length of
the arc of the twisted cubic x 苷 t, y 苷 t 2, z 苷 t 3 from the
origin to the point 共2, 4, 8兲.
9. r共t兲 苷 3 sin t i 4t j 3 cos t k
■
■
■
■
■
■
■
■
■
■
■
■
■
■
724
CHAPTER 10 VECTOR FUNCTIONS
10. Reparametrize the curve
r共t兲 苷
冉
(b) Estimate the curvature at P and at Q by sketching the
osculating circles at those points.
冊
2
2t
1 i 2
j
t 1
t 1
2
; 28–29
■ Use a graphing calculator or computer to graph both
the curve and its curvature function 共x兲 on the same screen. Is
the graph of what you would expect?
with respect to arc length measured from the point (1, 0) in
the direction of increasing t. Express the reparametrization
in its simplest form. What can you conclude about the
curve?
11–14
28. y 苷 xex
■
■
■
■
■
30.
12. r共t兲 苷 具t 2, sin t t cos t, cos t t sin t典 ,
具 13 t 3, t 2, 2t典
13. r共t兲 苷
■
■
■
■
■
■
■
■
■
■
■
■
■
b
b
x
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
苷
33–34
; 20. Graph the curve with parametric equations
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
27. (a) Is the curvature of the curve C shown in the figure
greater at P or at Q? Explain.
ⱍ
ⱍ
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
Find the vectors T, N, and B at the given point.
(1, 23 , 1)
■
■
■
■
共1, 0, 1兲
■
■
■
■
■
■ Find equations of the normal plane and osculating
plane of the curve at the given point.
■
origin.
■
37. x 苷 2 sin 3t, y 苷 t, z 苷 2 cos 3t;
38. x 苷 t, y 苷 t , z 苷 t ;
2
■
■
■
■
■
共0, , 2兲
共1, 1, 1兲
3
■
■
■
■
■
■
■
■
; 39. Find equations of the osculating circles of the ellipse
9x 2 4y 2 苷 36 at the points 共2, 0兲 and 共0, 3兲. Use a graphing calculator or computer to graph the ellipse and both
osculating circles on the same screen.
P
C
of the parabola
; 40. Find 1equations of the osculating circles
1
y 苷 2 x 2 at the points 共0, 0兲 and (1, 2 ). Graph both osculating circles and the parabola.
1
Q
0
■
37–38
26. Find an equation of a parabola that has curvature 4 at the
y
■
x᝽y
᝽᝽ y᝽x᝽᝽
关x᝽ 2 y᝽ 2 兴 3兾2
36. r共t兲 苷 具e t, e t sin t, e t cos t典 ,
■
25. y 苷 e x
■
■
y 苷 t t2
2
■
At what point does the curve have maximum
curvature? What happens to the curvature as x l ?
■
■
y 苷 e t sin t
35. r共t兲 苷 具t 2, 3 t 3, t 典,
5兾2
■
24. y 苷 ln x
■
35–36
23. y 苷 4x
22. y 苷 cos x
■
■
Use the formula in Exercise 32 to find the curvature.
34. x 苷 1 t 3,
Use Formula 11 to find the curvature.
■
■
33. x 苷 e t cos t,
z 苷 t 2
y 苷 4t 3兾2
and find the curvature at the point 共1, 4, 1兲.
21. y 苷 x
■
where the dots indicate derivatives with respect to t.
(0, 1, 1).
3
■
metric curve x 苷 f 共t兲, y 苷 t共t兲 is
■
19. Find the curvature of r共t兲 苷 具s2 t, e t, et 典 at the point
x苷t
■
x
32. Use Theorem 10 to show that the curvature of a plane para-
point (1, 0, 0).
■
■
Use Theorem 10 to find the curvature.
■
24–25
■
a
18. Find the curvature of r共t兲 苷 具e t cos t, e t sin t, t典 at the
■
■
y
■
17. r共t兲 苷 sin t i cos t j sin t k
21–23
■
a
14. r共t兲 苷 具t 2, 2t, ln t典
16. r共t兲 苷 t i t j 共1 t 2 兲 k
■
■
31.
y
t0
15. r共t兲 苷 t 2 i t k
■
■
■ Two graphs, a and b, are shown. One is a curve
y 苷 f 共x兲 and the other is the graph of its curvature function
y 苷 共x兲. Identify each curve and explain your choices.
11. r共t兲 苷 具2 sin t, 5t, 2 cos t典
15–17
■
30–31
(a) Find the unit tangent and unit normal vectors T共t兲 and N共t兲.
(b) Use Formula 9 to find the curvature.
■
■
29. y 苷 x 4
1
x
41. At what point on the curve x 苷 t 3, y 苷 3t, z 苷 t 4 is the nor-
mal plane parallel to the plane 6x 6y 8z 苷 1?
◆
SECTION 10.4 MOTION IN SPACE
CAS
(c) r 苷 关s 2共s兲3 兴 T 关3 ss 共s兲2 兴 N
共s兲3 B
42. Is there a point on the curve in Exercise 41 where the oscu-
lating plane is parallel to the plane x y z 苷 1? (Note:
You will need a CAS for differentiating, for simplifying,
and for computing a cross product.)
(d) 苷
43. Show that the curvature is related to the tangent and nor-
共r r兲 ⴢ r
r r 2
ⱍ
dT
苷 N
ds
r共t兲 苷 具a cos t, a sin t, bt典
ⱍ
ⱍ
where a and b are positive constants, has constant curvature
and constant torsion. [Use the result of Exercise 47(d).]
44. Show that the curvature of a plane curve is 苷 d兾ds ,
where is the angle between T and i ; that is, is the angle
of inclination of the tangent line.
49. The DNA molecule has the shape of a double helix (see
Figure 3 on page 707). The radius of each helix is about
10 angstroms (1 Å 苷 108 cm). Each helix rises about 34 Å
during each complete turn, and there are about 2.9 10 8
complete turns. Estimate the length of each helix.
45. (a) Show that dB兾ds is perpendicular to B.
(b) Show that dB兾ds is perpendicular to T.
(c) Deduce from parts (a) and (b) that dB兾ds 苷 共s兲N for
some number 共s兲 called the torsion of the curve. (The
torsion measures the degree of twisting of a curve.)
(d) Show that for a plane curve the torsion is 共s兲 苷 0.
50. Let’s consider the problem of designing a railroad track to
make a smooth transition between sections of straight track.
Existing track along the negative x-axis is to be joined
smoothly to a track along the line y 苷 1 for x 1.
(a) Find a polynomial P 苷 P共x兲 of degree 5 such that the
function F defined by
46. The following formulas, called the Frenet-Serret formulas,
are of fundamental importance in differential geometry:
1. dT兾ds 苷 N
2. dN兾ds 苷 T B
3. dB兾ds 苷 N
(Formula 1 comes from Exercise 43 and Formula 3 comes
from Exercise 45.) Use the fact that N 苷 B T to deduce
Formula 2 from Formulas 1 and 3.
再
0
if x 0
F共x兲 苷 P共x兲 if 0 x 1
1
if x 1
47. Use the Frenet-Serret formulas to prove each of the follow-
ing. (Primes denote derivatives with respect to t. Start as in
the proof of Theorem 10.)
(a) r 苷 sT 共s兲2 N
(b) r r 苷 共s兲3 B
Motion in Space
r(t+h)-r(t)
h
rª(t)
Q
z
P
●
r(t+h)
1
C
O
FIGURE 1
y
●
●
;
●
●
is continuous and has continuous slope and continuous
curvature.
(b) Use a graphing calculator or computer to draw the graph
of F .
●
●
●
●
●
●
●
●
●
●
●
In this section we show how the ideas of tangent and normal vectors and curvature can
be used in physics to study the motion of an object, including its velocity and acceleration, along a space curve. In particular, we follow in the footsteps of Newton by
using these methods to derive Kepler’s First Law of planetary motion.
Suppose a particle moves through space so that its position vector at time t is r共t兲.
Notice from Figure 1 that, for small values of h, the vector
r(t)
x
ⱍ
48. Show that the circular helix
mal vectors by the equation
10.4
725
r共t h兲 r共t兲
h
approximates the direction of the particle moving along the curve r共t兲. Its magnitude
measures the size of the displacement vector per unit time. The vector (1) gives the
average velocity over a time interval of length h and its limit is the velocity vector v共t兲
at time t :
◆
SECTION 10.4 MOTION IN SPACE
733
Writing d 苷 h 2兾c, we obtain the equation
ed
1 e cos r苷
12
In Appendix H it is shown that Equation 12 is the polar equation of a conic section
with focus at the origin and eccentricity e. We know that the orbit of a planet is a
closed curve and so the conic must be an ellipse.
This completes the derivation of Kepler’s First Law. We will guide you through the
derivation of the Second and Third Laws in the Applied Project on page 735. The
proofs of these three laws show that the methods of this chapter provide a powerful
tool for describing some of the laws of nature.
10.4
Exercises
●
●
●
●
●
●
●
●
●
●
●
●
1. The table gives coordinates of a particle moving through
space along a smooth curve.
(a) Find the average velocities over the time intervals [0, 1],
[0.5, 1], [1, 2], and [1, 1.5].
(b) Estimate the velocity and speed of the particle at t 苷 1.
t
x
y
z
0
0.5
1.0
1.5
2.0
2.7
3.5
4.5
5.9
7.3
9.8
7.2
6.0
6.4
7.8
3.7
3.3
3.0
2.8
2.7
●
●
t苷1
4. r共t兲 苷 具st, 1 t 典,
t苷1
t
5. r共t兲 苷 e i e j,
■
●
●
●
●
●
t 苷 兾6
t苷0
■
■
■
3
■
■
■
■
■
■
■
■
■
■ Find the velocity, acceleration, and speed of a particle
with the given position function.
9. r共t兲 苷 具t, t 2, t 3 典
10. r共t兲 苷 具2 cos t, 3t, 2 sin t典
11. r共t兲 苷 s2 t i e t j et k
12. r共t兲 苷 t sin t i t cos t j t 2 k
■
■
■
■
■
■
■
■
■
■
■
■
13–14
■ Find the velocity and position vectors of a particle that
has the given acceleration and the given initial velocity and
position.
v共0兲 苷 i j,
14. a共t兲 苷 10 k,
r(2)
■
■
15–16
■
■
r共0兲 苷 0
v共0兲 苷 i j k,
■
■
■
■
r共0兲 苷 2 i 3 j
■
■
■
■
■
■
(a) Find the position vector of a particle that has the given
acceleration and the given initial velocity and position.
; (b) Use a computer to graph the path of the particle.
r(1.5)
x
15. a共t兲 苷 i 2 j 2t k,
v共0兲 苷 0,
16. a共t兲 苷 t i t j cos 2t k,
Find the velocity, acceleration, and speed of a particle
with the given position function. Sketch the path of the particle
and draw the velocity and acceleration vectors for the specified
value of t.
■
■
■
■
■
■
r共0兲 苷 i k
v共0兲 苷 i k,
2
■
●
9–12
■
3–8
●
; 8. r共t兲 苷 t i t j t k, t 苷 1
r(2.4)
2
●
t苷0
6. r共t兲 苷 sin t i 2 cos t j,
13. a共t兲 苷 k,
1
●
3. r共t兲 苷 具t 2 1, t典 ,
2
y
2
●
7. r共t兲 苷 sin t i t j cos t k,
position vector r共t兲 at time t.
(a) Draw a vector that represents the average velocity of the
particle over the time interval 2 t 2.4.
(b) Draw a vector that represents the average velocity over
the time interval 1.5 t 2.
(c) Write an expression for the velocity vector v(2).
(d) Draw an approximation to the vector v(2) and estimate
the speed of the particle at t 苷 2.
1
●
t
2. The figure shows the path of a particle that moves with
0
●
■
■
■
■
r共0兲 苷 j
■
■
17. The position function of a particle is given by
r共t兲 苷 具t 2, 5t, t 2 16t典 . When is the speed a minimum?
■
734
■
CHAPTER 10 VECTOR FUNCTIONS
18. What force is required so that a particle of mass m has the
constant heading and a constant speed of 5 m兾s, determine
the angle at which the boat should head.
position function r共t兲 苷 t 3 i t 2 j t 3 k?
19. A force with magnitude 20 N acts directly upward from the
xy-plane on an object with mass 4 kg. The object starts at
the origin with initial velocity v共0兲 苷 i j. Find its
position function and its speed at time t.
20. Show that if a particle moves with constant speed, then the
velocity and acceleration vectors are orthogonal.
21. A projectile is fired with an initial speed of 500 m兾s and
angle of elevation 30. Find (a) the range of the projectile,
(b) the maximum height reached, and (c) the speed at
impact.
22. Rework Exercise 21 if the projectile is fired from a position
29–32
■ Find the tangential and normal components of the
acceleration vector.
30. r共t兲 苷 t i t 2 j 3t k
29. r共t兲 苷 共3t t 3 兲 i 3t 2 j
31. r共t兲 苷 cos t i sin t j t k
32. r共t兲 苷 t i cos 2t j sin 2t k
■
■
■
■
■
■
■
■
■
■
■
■
■
2
33. The magnitude of the acceleration vector a is 10 cm兾s . Use
the figure to estimate the tangential and normal components
of a.
y
200 m above the ground.
a
23. A ball is thrown at an angle of 45 to the ground. If the ball
lands 90 m away, what was the initial speed of the ball?
24. A gun is fired with angle of elevation 30. What is the
muzzle speed if the maximum height of the shell is 500 m?
25. A gun has muzzle speed 150 m兾s. Find two angles of eleva-
tion that can be used to hit a target 800 m away.
26. A batter hits a baseball 3 ft above the ground toward the
center field fence, which is 10 ft high and 400 ft from home
plate. The ball leaves the bat with speed 115 ft兾s at an angle
50 above the horizontal. Is it a home run? (In other words,
does the ball clear the fence?)
; 27. Water traveling along a straight portion of a river normally
flows fastest in the middle, and the speed slows to almost
zero at the banks. Consider a long stretch of river flowing
north, with parallel banks 40 m apart. If the maximum
water speed is 3 m兾s, we can use a quadratic function as a
basic model for the rate of water flow x units from the west
3
bank: f 共x兲 苷 400
x共40 x兲.
(a) A boat proceeds at a constant speed of 5 m兾s from a
point A on the west bank while maintaining a heading
perpendicular to the bank. How far down the river on
the opposite bank will the boat touch shore? Graph the
path of the boat.
(b) Suppose we would like to pilot the boat to land at the
point B on the east bank directly opposite A. If we
maintain a constant speed of 5 m兾s and a constant
heading, find the angle at which the boat should head.
Then graph the actual path the boat follows. Does the
path seem realistic?
28. Another reasonable model for the water speed of the river in
Exercise 27 is a sine function: f 共x兲 苷 3 sin共 x兾40兲. If a
boater would like to cross the river from A to B with
0
x
34. If a particle with mass m moves with position vector r共t兲,
then its angular momentum is defined as
L共t兲 苷 mr共t兲 v共t兲 and its torque as ␶ 共t兲 苷 mr共t兲 a共t兲.
Show that L共t兲 苷 ␶ 共t兲. Deduce that if ␶ 共t兲 苷 0 for all t,
then L共t兲 is constant. (This is the law of conservation of
angular momentum.)
35. The position function of a spaceship is
冉
r共t兲 苷 共3 t兲 i 共2 ln t兲 j 7 4
t 1
2
冊
k
and the coordinates of a space station are 共6, 4, 9兲. The captain wants the spaceship to coast into the space station.
When should the engines be turned off?
36. A rocket burning its onboard fuel while moving through
space has velocity v共t兲 and mass m共t兲 at time t. If the
exhaust gases escape with velocity ve relative to the rocket,
it can be deduced from Newton’s Second Law of Motion
that
dv
dm
m
苷
ve
dt
dt
m共0兲
ve.
m共t兲
(b) For the rocket to accelerate in a straight line from rest to
twice the speed of its own exhaust gases, what fraction
of its initial mass would the rocket have to burn as fuel?
(a) Show that v共t兲 苷 v共0兲 ln
■
740
CHAPTER 10 VECTOR FUNCTIONS
So the vector equation is
▲ For some purposes the parametric
representations in Solutions 1 and 2 are
equally good, but Solution 2 might be
preferable in certain situations. If we are
interested only in the part of the cone
that lies below the plane z 苷 1, for
instance, all we have to do in Solution 2
is change the parameter domain to
r共x, y兲 苷 x i y j 2sx 2 y 2 k
SOLUTION 2 Another representation results from choosing as parameters the polar
coordinates r and . A point 共x, y, z兲 on the cone satisfies x 苷 r cos , y 苷 r sin ,
and z 苷 2 sx 2 y 2 苷 2r. So a vector equation for the cone is
0 2
0r2
1
r共r, 兲 苷 r cos i r sin j 2r k
where r 0 and 0 2.
Surfaces of Revolution
z
Surfaces of revolution can be represented parametrically and thus graphed using a
computer. For instance, let’s consider the surface S obtained by rotating the curve
y 苷 f 共x兲, a x b, about the x-axis, where f 共x兲 0. Let be the angle of rotation
as shown in Figure 9. If 共x, y, z兲 is a point on S, then
0
y
y=ƒ
ƒ
¨
x
z 苷 f 共x兲 sin Therefore, we take x and as parameters and regard Equations 3 as parametric equations of S. The parameter domain is given by a x b, 0 2.
ƒ
x
EXAMPLE 8 Find parametric equations for the surface generated by rotating the curve
FIGURE 9
z
y 苷 f 共x兲 cos x苷x
3
(x, y, z)
z
y 苷 sin x, 0 x 2, about the x-axis. Use these equations to graph the surface of
revolution.
y
SOLUTION From Equations 3, the parametric equations are
y 苷 sin x cos x苷x
x
z 苷 sin x sin and the parameter domain is 0 x 2, 0 2. Using a computer to plot
these equations and rotate the image, we obtain the graph in Figure 10.
FIGURE 10
We can adapt Equations 3 to represent a surface obtained through revolution about
the y- or z-axis. (See Exercise 28.)
10.5
1–4
■
Exercises
●
●
●
●
●
●
●
●
●
●
Identify the surface with the given vector equation.
2. r共u, v兲 苷 共1 2u兲 i 共u 3v兲 j 共2 4u 5v兲 k
4. r共x, 兲 苷 具x, x cos , x sin 典
■
■
■
■
■
■
■
■
●
●
●
●
●
●
●
5. r共u, v兲 苷 具u 2 1, v 3 1, u v 典,
1 u 1, 1 v 1
3. r共x, 兲 苷 具x, cos , sin 典
■
●
■
●
●
●
●
●
■ Use a computer to graph the parametric surface. Get a
printout and indicate on it which grid curves have u constant
and which have v constant.
1. r共u, v兲 苷 u cos v i u sin v j u k
■
●
; 5–10
2
■
●
■
6. r共u, v兲 苷 具u v, u 2, v 2 典,
1 u 1, 1 v 1
●
◆
SECTION 10.5 PARAMETRIC SURFACES
7. r共u, v兲 苷 具cos 3u cos 3v, sin 3u cos 3v, sin 3v 典 ,
0 u , 0 v 2
17–24
■
741
Find a parametric representation for the surface.
17. The plane that passes through the point 共1, 2, 3兲 and con-
tains the vectors i j k and i j k.
8. r共u, v兲 苷 具cos u sin v, sin u sin v, cos v ln tan共v兾2兲典 ,
0 u 2, 0.1 v 6.2
18. The lower half of the ellipsoid 2x 2 4y 2 z 2 苷 1
9. x 苷 cos u sin 2v,
19. The part of the hyperboloid x 2 y 2 z 2 苷 1 that lies to
y 苷 sin u sin 2v,
10. x 苷 u sin u cos v,
■
■
11–16
■
■
z 苷 sin v
y 苷 u cos u cos v,
■
■
■
■
the right of the xz-plane
z 苷 u sin v
■
■
■
■
20. The part of the elliptic paraboloid x y 2 2z 2 苷 4 that
■
lies in front of the plane x 苷 0
■
Match the equations with the graphs labeled I–VI and
give reasons for your answers. Determine which families of grid
curves have u constant and which have v constant.
21. The part of the sphere x 2 y 2 z 2 苷 4 that lies above the
11. r共u, v兲 苷 cos v i sin v j u k
22. The part of the cylinder x 2 z 2 苷 1 that lies between the
cone z 苷 sx 2 y 2
planes y 苷 1 and y 苷 3
12. r共u, v兲 苷 u cos v i u sin v j u k
23. The part of the plane z 苷 5 that lies inside the cylinder
13. r共u, v兲 苷 u cos v i u sin v j v k
14. x 苷 u 3,
y 苷 u sin v,
x 2 y 2 苷 16
24. The part of the plane z 苷 x 3 that lies inside the cylinder
z 苷 u cos v
y 苷 共1 cos u兲 sin v,
15. x 苷 共u sin u兲 cos v,
x2 y2 苷 1
z苷u
■
16. x 苷 共1 u兲共3 cos v兲 cos 4 u,
y 苷 共1 u兲共3 cos v兲 sin 4 u,
z 苷 3u 共1 u兲 sin v
z
I
CAS
■
■
■
■
■
■
■
■
■
■
■ Use a computer algebra system to produce a graph
that looks like the given one.
26.
3
z
z 0
0
_1
_1
_3
_3
y
y
x
x
z
III
■
z
IV
■
25–26
25.
z
II
■
0
■
■
y
x
0 5
y
■
■
_1
■
■
0
0
1 1
■
■
■
x
■
■
■
; 27. Find parametric equations for the surface obtained by
rotating the curve y 苷 e x, 0 x 3, about the x-axis
and use them to graph the surface.
; 28. Find parametric equations for the surface obtained by
x
rotating the curve x 苷 4y 2 y 4, 2 y 2, about the
y-axis and use them to graph the surface.
y
y
x
29. (a) Show that the parametric equations x 苷 a sin u cos v,
y 苷 b sin u sin v, z 苷 c cos u, 0 u , 0 v 2,
z
V
z
VI
;
represent an ellipsoid.
(b) Use the parametric equations in part (a) to graph the
ellipsoid for the case a 苷 1, b 苷 2, c 苷 3.
; 30. The surface with parametric equations
x 苷 2 cos r cos共兾2兲
y
x
y
x
y 苷 2 sin r cos共兾2兲
z 苷 r sin共兾2兲
■
■
■
■
■
■
■
■
■
■
■
■
■
where 12 r 12 and 0 2, is called a Möbius
742
■
CHAPTER 10 VECTOR FUNCTIONS
strip. Graph this surface with several viewpoints. What is
unusual about it?
z
(x, y, z)
; 31. (a) What happens to the spiral tube in Example 2 (see Figure 5) if we replace cos u by sin u and sin u by cos u?
(b) What happens if we replace cos u by cos 2u and sin u
by sin 2u?
0
;
by rotating about the z -axis the circle in the xz-plane
with center 共b, 0, 0兲 and radius a b. [Hint: Take as
parameters the angles and shown in the figure.]
(b) Use the parametric equations found in part (a) to graph
the torus for several values of a and b.
10
Review
å
¨
32. (a) Find a parametric representation for the torus obtained
x
y
(b, 0, 0)
CONCEPT CHECK
1. What is a vector function? How do you find its derivative
and its integral?
2. What is the connection between vector functions and space
curves?
3. (a) What is a smooth curve?
(b) How do you find the tangent vector to a smooth curve at
a point? How do you find the tangent line? The unit tangent vector?
4. If u and v are differentiable vector functions, c is a scalar,
and f is a real-valued function, write the rules for differentiating the following vector functions.
(a) u共t兲 v共t兲
(b) cu共t兲
(c) f 共t兲u共t兲
(d) u共t兲 ⴢ v共t兲
(e) u共t兲 v共t兲
(f) u共 f 共t兲兲
5. How do you find the length of a space curve given by a
6. (a) What is the definition of curvature?
(b) Write a formula for curvature in terms of r共t兲 and T共t兲.
(c) Write a formula for curvature in terms of r共t兲 and r共t兲.
(d) Write a formula for the curvature of a plane curve with
equation y 苷 f 共x兲.
7. (a) Write formulas for the unit normal and binormal vectors
of a smooth space curve r共t兲.
(b) What is the normal plane of a curve at a point? What is
the osculating plane? What is the osculating circle?
8. (a) How do you find the velocity, speed, and acceleration of
a particle that moves along a space curve?
(b) Write the acceleration in terms of its tangential and normal components.
9. State Kepler’s Laws.
10. What is a parametric surface? What are its grid curves?
vector function r共t兲?
T R U E – FA L S E Q U I Z
Determine whether the statement is true or false. If it is true, explain why.
If it is false, explain why or give an example that disproves the statement.
6. If r共t兲 is a differentiable vector function, then
d
r共t兲 苷 r共t兲
dt
ⱍ
1. The curve with vector equation r共t兲 苷 t 3 i 2t 3 j 3t 3 k is
a line.
2. The curve with vector equation r共t兲 苷 具t, t 3, t 5 典 is smooth.
3. The curve with vector equation r共t兲 苷 具cos t, t 2, t 4 典 is
smooth.
4. The derivative of a vector function is obtained by differenti-
ating each component function.
5. If u共t兲 and v共t兲 are differentiable vector functions, then
d
关u共t兲 v共t兲兴 苷 u共t兲 v共t兲
dt
ⱍ ⱍ
ⱍ
7. If T共t兲 is the unit tangent vector of a smooth curve, then the
ⱍ
ⱍ
curvature is 苷 dT兾dt .
8. The binormal vector is B共t兲 苷 N共t兲 T共t兲.
9. The osculating circle of a curve C at a point has the same
tangent vector, normal vector, and curvature as C at that
point.
10. Different parametrizations of the same curve result in iden-
tical tangent vectors at a given point on the curve.