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Question 4 Let H be the part of the hyperboloid with equation x² + y² - z² = 1 located between the planes z = -2 and z = 2. The area of H is A = π(√√2ln(2√2+3) +12). a) Show that H can be parameterized by R(u, v) = √√√1 + v² cos(u) i + √√1+v² sin(u)j +vk with 0 ≤us 2π and -2 ≤ v ≤ 2. a) If the surface H is oriented at the point (1,0,0) by the normal vector i = 1, calculate the flux of the vector field F defined by F (x, y, z) = −yi + xj + z³k through H. a) if G is a vector field of constant norm and perpendicular to the surface H at any point, which is the flow of G through H?

Question 4 Let H be the part of the hyperboloid with equation x² + y² - z² = 1 located between the planes z = -2 and z = 2. The area of H is A = π(√√2ln(2√2+3) +12). a) Show that H can be parameterized by R(u, v) = √√√1 + v² cos(u) i + √√1+v² sin(u)j +vk with 0 ≤us 2π and -2 ≤ v ≤ 2. a) If the surface H is oriented at the point (1,0,0) by the normal vector i = 1, calculate the flux of the vector field F defined by F (x, y, z) = −yi + xj + z³k through H. a) if G is a vector field of constant norm and perpendicular to the surface H at any point, which is the flow of G through H?

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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Question 4 Let H be the part of the hyperboloid with equation x2 + y? - z2 = 1 located between the planes z = -2 and z = 2. The area of H is A = T(V2 ln(2/2+3) + 12). a) Show that H can be parameterized by Ř(u, v) = V1+ v² cos(u)i+ V1+ v² sin(u) j+ v k with 0sus 2n and -2 svs 2. a) If the surface H is oriented at the point (1,0, 0) by the normal vector ñ =1, calculate the flux of the vector field F defined by F(x,y, 2) = -yi + xj + z°k through H. a) if G is a vector field of constant norm and perpendicular to the surface H at any point, which is the flow of G through H ?
Transcribed Image Text:Question 4 Let H be the part of the hyperboloid with equation x2 + y? - z2 = 1 located between the planes z = -2 and z = 2. The area of H is A = T(V2 ln(2/2+3) + 12). a) Show that H can be parameterized by Ř(u, v) = V1+ v² cos(u)i+ V1+ v² sin(u) j+ v k with 0sus 2n and -2 svs 2. a) If the surface H is oriented at the point (1,0, 0) by the normal vector ñ =1, calculate the flux of the vector field F defined by F(x,y, 2) = -yi + xj + z°k through H. a) if G is a vector field of constant norm and perpendicular to the surface H at any point, which is the flow of G through H ?
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