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6. Let F (x,y, z) = (x+ y,–2y + z, 8z – x) and let U be the sphere where x2 + y? + z² = 25 oriented with outward-facing normal vectors. Calculate Sfy F ·d S . (In case it is helpful, a sphere of radius R has volume AR³ and surface area 4AR².)

6. Let F (x,y, z) = (x+ y,–2y + z, 8z – x) and let U be the sphere where x2 + y? + z² = 25 oriented with outward-facing normal vectors. Calculate Sfy F ·d S . (In case it is helpful, a sphere of radius R has volume AR³ and surface area 4AR².)

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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6. Let F (x,y,z) = (x + y, –2y + z, 8z – x) and let U be the sphere where x? + y? + z² with outward-facing normal vectors. Calculate fSy F ·d S. (In case it is helpful, a sphere of radius R has volume TR3 and surface area 4¤R².) = 25 oriented 7. Let C be the curve consisting on the line segment from (2,1) to (4,4), and then the line segment from (4, 4) to (–1,3). Let F (x,y) = (2y + 2x, 2x – 1). Calculate ſc F ·d7. 8. Let f(x,y) = x² – 8x + 2y² + 4y +7. a. Find and classify all critical points of f. b. Does f has a global maximum on the region where x2 + y? < 4? Justify your answer, being sure to mention any theorems that you use (you do not need to find the global maximum). 9. Find an equation for the plane containing the points (3,1,4), (–1, –1, –1), and (0,2,2).
Transcribed Image Text:6. Let F (x,y,z) = (x + y, –2y + z, 8z – x) and let U be the sphere where x? + y? + z² with outward-facing normal vectors. Calculate fSy F ·d S. (In case it is helpful, a sphere of radius R has volume TR3 and surface area 4¤R².) = 25 oriented 7. Let C be the curve consisting on the line segment from (2,1) to (4,4), and then the line segment from (4, 4) to (–1,3). Let F (x,y) = (2y + 2x, 2x – 1). Calculate ſc F ·d7. 8. Let f(x,y) = x² – 8x + 2y² + 4y +7. a. Find and classify all critical points of f. b. Does f has a global maximum on the region where x2 + y? < 4? Justify your answer, being sure to mention any theorems that you use (you do not need to find the global maximum). 9. Find an equation for the plane containing the points (3,1,4), (–1, –1, –1), and (0,2,2).
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