1. Math 265
Created by Educational Technology Network

2. Vector Fields
Line Integrals
Independence of Path
Green's Theorem 10 10 10 10 20 20 20 20 30 30 30 30 40 40 40 40

3. Vector Fields – 10 Points QUESTION:
Determine the type of input and output for gradient, divergence, and curl.
ANSWER:
Gradient: Input – scalar, output – vector
Divergence: Input – vector, output – scalar
Curl: Input – vector, output – vector

4. Vector Fields – 20 Points Find 𝑑𝑒𝑙 𝑓 :𝑓 𝑥,𝑦,𝑧 = ln |𝑥𝑦𝑧|
QUESTION:
Find 𝑑𝑒𝑙 𝑓 :𝑓 𝑥,𝑦,𝑧 = ln |𝑥𝑦𝑧|
ANSWER:
1 𝑥 , 1 𝑦 , 1 𝑧

5. Vector Fields – 30 Points Find 𝑑𝑖𝑣 𝐹 and 𝑐𝑢𝑟𝑙 𝐹 :
QUESTION:
Find 𝑑𝑖𝑣 𝐹 and 𝑐𝑢𝑟𝑙 𝐹 :
𝐹 𝑥,𝑦,𝑧 = 𝑥 2 𝑖 −2𝑥𝑦 𝑗 +𝑦 𝑧 2 𝑘
ANSWER:
𝑑𝑖𝑣 𝐹 =2𝑦𝑧
𝑐𝑢𝑟𝑙 𝐹 = 𝑧 2 ,0,−2𝑦

6. Vector Fields – 40 Points Find 𝑔𝑟𝑎𝑑 𝑑𝑖𝑣 𝐹 :
QUESTION:
Find 𝑔𝑟𝑎𝑑 𝑑𝑖𝑣 𝐹 :
𝐹 𝑥,𝑦,𝑧 = 𝑒 𝑥 cos 𝑦 , 𝑒 𝑥 sin 𝑦 ,𝑧
ANSWER:
2 𝑒 𝑥 ,− sin 𝑦 ,0

7. Line Integrals – 10 Points
QUESTION:
What is 𝑑𝑠 equal to?
ANSWER:
𝑥 ′ 𝑡 𝑦 ′ 𝑡 2 𝑑𝑡

8. Line Integrals – 20 Points
QUESTION:
𝐶 𝑥 3 +𝑦 𝑑𝑠 ; C is the curve 𝑥=3𝑡, 𝑦= 𝑡 3 ,0≤𝑡≤1
ANSWER:
14(2 2 −1)≈25.598

9. Line Integrals – 30 Points
QUESTION:
𝐶 𝑥+2𝑦 𝑑𝑥+ 𝑥−2𝑦 𝑑𝑦 ;C is the line segment from (1,1) to (3,-1)
ANSWER:

10. Line Integrals – 40 Points
QUESTION:
Find the work done by F: 𝐹 𝑥,𝑦 = 𝑥+𝑦,𝑥−𝑦 ; C is the quarter ellipse, 𝑥= a cos 𝑡 , 𝑦=𝑏 sin 𝑡 , 0≤𝑡≤2𝜋
ANSWER:
𝑎 2 + 𝑏 2 −2

11. Independence of Path – 10 Points
QUESTION:
What determines if 𝐶 𝐹 ∙𝑑 𝑟 is independent of path?
ANSWER:
𝐹 is conservative or a gradient vector field. ( 𝑀 𝑦 = 𝑁 𝑦 )

12. Independence of Path – 20 Points
QUESTION:
Is F conservative? 𝐹 𝑥,𝑦 = 10𝑥−7𝑦 𝑖 −(7𝑥−2𝑦) 𝑗
ANSWER:
Yes

13. Independence of Path – 30 Points
QUESTION:
Is F conservative? 𝐹 𝑥,𝑦 = −2𝑥 𝑥 2 + 𝑧 2 𝑖 −( −2𝑧 𝑥 2 + 𝑧 2 ) 𝑘
ANSWER: No

14. Independence of Path – 40 Points
QUESTION:
Find a function for which 𝐹 𝑥,𝑦 = 10𝑥−7𝑦 𝑖 −(7𝑥−2𝑦) 𝑗 is the gradient.
ANSWER:
𝑓 𝑥,𝑦 =5 𝑥 2 −7𝑥𝑦+ 𝑦 2 +𝐶

15. Green's Theorem – 10 Points
QUESTION:
𝐶 2𝑥𝑦𝑑𝑥+ 𝑦 2 𝑑𝑦 ; C is the closed curve formed by 𝑦= 𝑥 2 , 𝑦= 𝑥
ANSWER:
− ≈4.2667

16. Green's Theorem – 20 Points
QUESTION:
𝐶 𝑥𝑦𝑑𝑥+ 𝑥+𝑦 𝑑𝑦 ; C is the triangle with vertices (0,0), (2,0), (2,3)
ANSWER: −1

17. Green's Theorem – 30 Points
QUESTION:
Find the flux of 𝐹 = 𝑥 2 + 𝑦 2 ,2𝑥𝑦 across the boundary of the square with vertices at (0,0), (0,1), (1,1), (1,0)
ANSWER: 2

18. Green's Theorem – 40 Points
QUESTION:
Find the work done by 𝐹 = 𝑥 2 + 𝑦 2 𝑖 −2𝑥𝑦 𝑗 moving clockwise around the square with vertices (0,0), (0,1), (1,1), (1,0)
ANSWER: 2