Important Exam based Vector Calculus Question Bank
1. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
2. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
3. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
4. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
5. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
6. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
7. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
8. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
9. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
10. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
11. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
12. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
13. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
14. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
15. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
16. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
17. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
18. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
19. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
20. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
21. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
22. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
23. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
24. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
25. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
26. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
27. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
28. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
29. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
30. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
31. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
32. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
33. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
34. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
35. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
36. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
37. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
38. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
39. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
40. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
41. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
42. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
43. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
44. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
45. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
46. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
47. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
48. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
49. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
50. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
51. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
52. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
53. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
54. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
55. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
56. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
57. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
58. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
59. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
60. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
61. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
62. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
63. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
64. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
65. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
66. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
67. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
68. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
69. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
70. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
71. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
72. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
73. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
74. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
75. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
76. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
77. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
78. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
79. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
80. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
81. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
82. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
83. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
84. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
85. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
86. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
87. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
88. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
89. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
90. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
91. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
92. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
93. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
94. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
95. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
96. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
97. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
98. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
99. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
100. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
101. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
102. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
103. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
104. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
105. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
106. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
107. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
108. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
109. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
110. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
111. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
112. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
113. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
114. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
115. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
116. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
117. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
118. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
119. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
120. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
121. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
122. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
123. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
124. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
125. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
126. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
127. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
128. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
129. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
130. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
131. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
132. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
133. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
134. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
135. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
136. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
137. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
138. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
139. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
140. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
141. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
142. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
143. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
144. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
145. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
146. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
147. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
148. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
149. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
150. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
151. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
152. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
153. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
154. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
155. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
156. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
157. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
158. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
159. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
160. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
161. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
162. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
163. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
164. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
165. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
166. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
167. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
168. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
169. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
170. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
171. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
172. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
173. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
174. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
175. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
176. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
177. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
178. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
179. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
180. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
181. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
182. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
183. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
184. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
185. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
186. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
187. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
188. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
189. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
190. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
191. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
192. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
193. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
194. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
195. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
196. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
197. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
198. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
199. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
200. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
201. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
202. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
203. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
204. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
205. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
206. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
207. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
208. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
209. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
210. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
211. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
212. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
213. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
214. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
215. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
216. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
217. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
218. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
219. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
220. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
221. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
222. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
223. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
224. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
225. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
226. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
227. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
228. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
229. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
230. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
231. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
232. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
233. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
234. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
235. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
236. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
237. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
238. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
239. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
240. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
241. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
242. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
243. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
244. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
245. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
246. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
247. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
248. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
249. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
250. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
251. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
252. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
253. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
254. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
255. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
256. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
257. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
258. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
259. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
260. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
261. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
262. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
263. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
264. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
265. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
266. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
267. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
268. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
269. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
270. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
271. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
272. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
273. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
274. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
275. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
276. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
277. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
278. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
279. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
280. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
281. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
282. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
283. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
284. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
285. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
286. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
287. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
288. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
289. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
290. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
291. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
292. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
293. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
294. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
295. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
296. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
297. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
298. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
299. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
300. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
301. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
302. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
303. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
304. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
305. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
306. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
307. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
308. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
309. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
310. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
311. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
312. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
313. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
314. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
315. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
316. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
317. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
318. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
319. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
320. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
321. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
322. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
323. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
324. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
325. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
326. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
327. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
328. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
329. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
330. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
331. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
332. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
333. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
334. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
335. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
336. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
337. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
338. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
339. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
340. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
341. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
342. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
343. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
344. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
345. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
346. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
347. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
348. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
349. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
350. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
351. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
352. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
353. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
354. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
355. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
356. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
357. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
358. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
359. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
360. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
361. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
362. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
363. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
364. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
365. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
366. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
367. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
368. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
369. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
370. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
371. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
372. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
373. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
374. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
375. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
376. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
377. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
378. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
379. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
380. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
381. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
382. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
383. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
384. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
385. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
386. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
387. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
388. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
389. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
390. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
391. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
392. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
393. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
394. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
395. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
396. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
397. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
398. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
399. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
400. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
401. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
402. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
403. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
404. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
405. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
406. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
407. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
408. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
409. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
410. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
411. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
412. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
413. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
414. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
415. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
416. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
417. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
418. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
419. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
420. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
421. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
422. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
423. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
424. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
425. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
426. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
427. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
428. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
429. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
430. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
431. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
432. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
433. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
434. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
435. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
436. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
437. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
438. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
439. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
440. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
441. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
442. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
443. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
444. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
445. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
446. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
447. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
448. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
449. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
450. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
451. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
452. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
453. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
454. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
455. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
456. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
457. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
458. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
459. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
460. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
461. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
462. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
463. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
464. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
465. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
466. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
467. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
468. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
469. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
470. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
471. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
472. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
473. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
474. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
475. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
476. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
477. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
478. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
479. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
480. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
481. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
482. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
483. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
484. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
485. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
486. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
487. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
488. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
489. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
490. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
491. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
492. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
493. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
494. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
495. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
496. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
497. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
498. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
499. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
500. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
501. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
502. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
503. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
504. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
505. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
506. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
507. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
508. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
509. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
510. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
511. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
512. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
513. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
514. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
515. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
516. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
517. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
518. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
519. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
520. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
521. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
522. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
523. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
524. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
525. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
526. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
527. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
528. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
529. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
530. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
531. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
532. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
533. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
534. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
535. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
536. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
537. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
538. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
539. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
540. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
541. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
542. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
543. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
544. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
545. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
546. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
547. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
548. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
549. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
550. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
551. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
552. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
553. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
554. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
555. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
556. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
557. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
558. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
559. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
560. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
561. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
562. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
563. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
564. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
565. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
566. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
567. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
568. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
569. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
570. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
571. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
572. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
573. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
574. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
575. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
576. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
577. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
578. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
579. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
580. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
581. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
582. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
583. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
584. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
585. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
586. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
587. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
588. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
589. Find: 𝛻 ⋅ (3𝑥2+ 2𝑦𝑧)
Answer: 6𝑥 + 2𝑧
590. Verify if 𝑭(𝑥, 𝑦, 𝑧)=(2𝑥𝑦 + 𝑧, 𝑥2+𝑦𝑧,𝑥𝑧) is conservative and find 𝑓(𝑥, 𝑦, 𝑧) if it is.
Answer: Conservative, 𝑓(𝑥, 𝑦, 𝑧)= 𝑥2𝑦 + 𝑥𝑦𝑧 + 1
2𝑧2+ 𝐶
591. Calculate 𝛻 × (𝑥𝑦,𝑦𝑧,𝑧𝑥).
Answer: (𝑧 − 𝑦, 𝑥 − 𝑧, 𝑦 − 𝑥)
592. Evaluate ∫(2𝑦, 𝑥, 𝑧)
𝐶⋅ 𝑑𝒓 where 𝐶 is 𝒓(𝑡)=(𝑡2, 𝑡, 𝑡3) for 𝑡 ∈ [0,1].
Answer: 11
6
593. Find the surface integral of (𝑥2, 𝑦2, 𝑧2) over 𝑧 = 1 − 𝑥 − 𝑦, inside the triangle (0,0,0),
(1,0,0), (0,1,0).
Answer: 1
24
594. Use Stokes’ Theorem to evaluate ∮(𝑦2, 𝑧2, 𝑥2)
𝐶⋅ 𝑑𝒓 where 𝐶 is the intersection of 𝑥 +
𝑦 + 𝑧 = 1 with 𝑥2+ 𝑦2= 1.
Answer: 4𝜋
595. Apply the Divergence Theorem to calculate ∭𝛻
𝑉⋅(𝑥2, 𝑦2, 𝑧2) 𝑑𝑉 over the region
bounded by 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 1.
Answer: 1
2
