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Exercises 5.4 359 integrals of F along the curves c1(t) = (cos t, sin t, 0), 0

Exercises 5.4 359 integrals of F along the curves c1(t) = (cos t, sin t, 0), 0

Advanced Engineering Mathematics
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.
Transcribed Image Text: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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