Math 265 Created by Educational Technology Network. www.edtechnetwork.com 2009

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

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

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

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

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

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

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

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

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

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 )

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

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

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

Green's Theorem – 10 Points QUESTION: 𝐶 2𝑥𝑦𝑑𝑥+ 𝑦 2 𝑑𝑦 ; C is the closed curve formed by 𝑦= 𝑥 2 , 𝑦= 𝑥 ANSWER: − 64 15 ≈4.2667

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

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

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