Can you help with 17 and 20 please? Exercises 5.4359integrals of F along the curves c1(t) = (cos t, sin t, 0), 0

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Can you help with 17 and 20 please?

Exercises 5.4
359
integrals of F along the curves c1(t) = (cos t, sin t, 0), 0 <I <1, and c2(t) = (cos t, – sint,0), 0 <
t <T, joining (1, 0) and (-1,0). Explain why this does not contradict Theorem 5.8.
16. Let F(x, y) = yi + xj.
(a) Compute J F·ds along the circular path from (1, 0) counterclockwise to (0, –1), then along the
y-axis from (0, –1) to (0, 2), and then along the straight line from (0, 2) to (1,0).
(b) Show that F(x, y) is a gradient vector field and use this fact to check your answer in (a).
Exercises 17 to 21: Determine whether F is a gradient vector field, and if so, specify its domain U
and find all functions f such that F = V f.
17. F=(4x² – 4y² + x)i + (7xy + ln y)j
%3D
18. F = (3x² lIn x +x?)i + x³y-'j
19. F= 2x In yi + (2y +x²/y)j
20. F= (yz + e* sin z)i + (xz + y² – e' )j + (xy +e* cos z)k
%3D
21. F= y cos (xy)i + (x cos (xy) – z sin y)j+ cos yk
%3D
Exercises 22 to 24: Evaluate the following integrals:
(3,3,1)
22.
(4xy – 2xy²z?) dx + (2x² – 2x²yz²)dy – 2x²y²z dz
(0,1,0)
(3/ה ,2/ה , ח) .
23.
cos x tan zdx +dy + sin x sec² z dz
(0,0,0)
(2,2,2)
24.
x-²dx + z'dy+ yz-² dz
25. Provide omitted detail in the proof of Theorem 5.7: parametrize the path e in Figure 5.36(b) to
obtain the formula (5.18). Then show that af/əx(x, y) = F¡(x, y).
(1,2,1)
26. Let F(x, y, z) = (x, y, z)/(x² + y² + z²)/?, Show that curl F = 0. Show that V(x² + y² +
1/2 = -F (see Example 2.40 in Section 2.4).
(1+x)ye*i – xe'j.

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