1-2 Convert from rectangular to cylindrical coordinates.
1. (a) (2∓, 2, -4) (b) (-5. 5. 6)
(c) (0, 2, 0) (d) (4, -4∓, 6)
2. (a) (−, -−, 1) (b) (-2, 0, 3)
(c) (-4, 4, -7) (d) (2, -2, 4)
3-4 Convert from cylindrical to rectangular coordinates.
3. (a) (6, π/3, 3) (b) (1, 3π/4, -2)
(c) (5, 0, 4) (d) (7, π, -9)
4. (a) (6, 5π/3, 7) (b) (2, π/2, 3)
(c) (3, π/2, 5) (d) (4, π/2, -1)
5-6 Convert from rectangular to spherical coordinates.
5. (a) (2, 2∓, -4) (b) (1, -1, −)
(c) (0, 3∓, 3) (d) (-4∓, 4, 0)
6. (a) (6, -6, 6∓) (b) (1, -∓, -2)
(c) (0, 3, 0) (d) (∓, 1, 2∓)
7-8 Convert from spherical to rectangular coordinates.
7. (a) (4, π/6, π/3) (b) (7, 0, π/2)
(c) (1, π, 0) (d) (2, 3π/2, π/2)
8. (a) (1, 2π/3, π/2) (b) (3, 7π/4, 5π/6)
(c) (4, π/6, π/4) (d) (4, π/2, π/3)
9-10 Convert from cylindrical to spherical coordinates.
9. (a) (2∓, π/3, 6) (b) (1, π/4, -1)
(c) (2, 3π/4, 0) (d) (6, 1, -2∓)
10. (a) (4, 5π/6, 4) (b) (3, 0, -3)
(c) (5, π/2, 12) (d) (6, π, 2)
11-12 Convert from spherical to cylindrical coordinates.
11. (a) (5, π/4, π/3) (b) (2, 5π/6, π)
(c) (3, 0, 0) (d) (4, π/6, π/2)
12. (a) (6, π/2, 0) (b) (6, 0, π/4)
(c) (−, 3π/4, π) (d) (5, 2π/3, 5π/6)
13. Use a CAS or a programmable calculating utility to set up the conversion formulas in Table 11.8.1, and then use the CAS or calculating utility to solve the problems in Exercises 1, 3, 5, 7, 9, and 11.
14. Use a CAS or a programmable calculating utility to set up the conversion formulas in Table 11.8.1, and then use the CAS or calculating utility to solve the problems in Exercises 2, 4, 6, 8, 10, and 12.
15-18 True–False Determine whether the statement is true or false. Explain your answer.
15. In cylindrical coordinates for a point, r is the distance from the point to the z-axis.
16. In spherical coordinates for a point, ρ is the distance from the point to the origin.
17. The graph of θ = θ0 in cylindrical coordinates is the same as the graph of θ = θ0 in spherical coordinates.
18. The graph of r = f(θ) in cylindrical coordinates can always be obtained by extrusion of the polar graph of r = f(θ) in the xy-plane.
19-26 An equation is given in cylindrical coordinates. Express the equation in rectangular coordinates and sketch the graph.
19. r = 3
20. θ = π/4
21. z = r^2
22. z = r cos θ
23. r = 4 sin θ
24. r = 2 sec θ
25. r^2 + z^2 = 1
26. r^2 cos 2θ = z
27-34 An equation is given in spherical coordinates. Express the equation in rectangular coordinates and sketch the graph.
27. ρ = 3
28. θ = π/3
29. φ = π/4
30. ρ = 2 sec φ
31. ρ = 4 cos φ
32. ρ sin φ = 1
33. ρ sin φ = 2 cos θ
34. ρ - 2 sin φ cos θ = 0
35-46 An equation of a surface is given in rectangular coordinates. Find an equation of the surface in (a) cylindrical coordinates and (b) spherical coordinates.
35. z = 7
36. y = 3
37. z = 3x^2 + 3y^2
38. z = ∓(3x^2 + 3y^2)
39. x^2 + y^2 = 9
40. x^2 + y^2 - 6y = 0
41. x^2 + y^2 + z^2 = 4
42. z^2 = x^2 - y^2
43. 2x + 3y + 4z = 2
44. x^2 + y^2 - z^2 = 4
45. x^2 = 16 - z^2
46. x^2 + y^2 + z^2 = 2z