Page 854 - Advanced_Engineering_Mathematics o'neil

834 Answers to Selected Problems

CHAPTER ELEVEN VECTOR DIFFERENTIAL CALCULUS
Section 11.1 Vector Functions of One Variable

1. ( f (t)F(t)) =−12sin(3t)i + 12t[2cos(3t) − 3t sin(3t)]j

+8[cos(3t) − 3t sin(3t)]k
3. (F × G) = (1 − 4sin(t))i − 2tj − (cos(t) − t sin(t))k

3
2
3
5. ( f (t)F(t)) = (1 − 8t )i + (6t cosh(t) − (1 − 2t )sinh(t))j

2 t
3
t
+(−6t e + e (1 − 2t ))k
t
7. te (2 + t)(j − k)
9. (a) F(t) = sin(t)i + cos(t)j + 45tk for 0 ≤ t ≤ 2π,

F (t)= cos(t)i − sin(t)j + 45k
√
(b) s(t) = 2026t
1 √ √ √
(c) F(t(s)) = √ sin s/ 2026 i + cos s/ 2026 + 45s/ 2026 k
2026
2

11. (a) F(t) = t (2i + 3j + 4k) for 1 ≤ t ≤ 3, F (t) = 2t(2i + 3j + 4k)
√
2
(b) s(t) = 29(t − 1)
s

(c) F(t(s)) = 1 + √ (2i + 3j + 4k)
29
Section 11.2 Velocity and Curvature
√
2
1. v(t) = 3i + 2tk,v(t) = 9 + 4t ,a(t) = 2k,
4t 6 6 6
a T = √ ,κ = ,a N =
2 1/2
2 1/2
2 3/2
9 + 4t 2 (9 + 4t ) (9 + 4t ) (9 + 4t )
3. v(t) = 2i − 2j + k,v(t) = 3,a(t) = O
κ = 0,a T = a N = 0
√
−t
−t
−t
5. v(t) =−3e (i + j − 2k),v(t) = 3 6e ,a(t) = 3e (i + j − 2k)
√
−t
κ = 0,a T =−3 6e ,a N = 0
√
7. v(t) = 2cosh(t)j − 2sinh(t)k,v(t) = 2 cosh(2t),
a(t) = 2sinh(t)j − 2cosh(t)k
1
κ = ,
2(cosh(2t)) 3/2
√ √
a T = 2sinh(2t)/ cosh(2t),a N = 2/ cosh(2t)

2
2
2
2
9. v(t) = 2t(αi + βj + γ k), v(t) = 2|t| α + β + γ , a N = 0, κ = 0, a T = 2σ α + β + γ ,where σ equals 1 if t ≥ 0,
2
2
and −1if t < 0
Section 11.3 Vector Fields and Streamlines
1. x = x, y = 1/(x + c), z = e x+k ; x = x, y = 1/(x − 1), z = e x−2
x 2
3. x = x, y = e (x − 1) + c, x =−2z + k;
1
2
2
x
x = x, y = e (x − 1) − e , z = (12 − x )
2
z
5. x = c, y = y,2e = k − sin(y);
√
x = 3, y = y, z = ln 1 + 2/4 − (1/2)sin(y)
7. There are many such vector fields. One whose streamlines are circles about the origin in the plane z = 0isgiven by
1 1
F(x, y, z) =− i + j.
x y
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
October 14, 2010 17:50 THM/NEIL Page-834 27410_25_Ans_p801-866