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Let v = (r*z, 3 – 2ryz – 3y + r²y, 3z – 2²z) be the velocity field of a fluid. Compute the flux of v across the surface x² + y² + z² = 4 where y > 0 and the surface is oriented away from the origin. HINT: Call the surface in this problem S1. Sį is "open" and does not enclose a 3D region, so Divergence Theorem cannot be used directly to calculate the flux across S1. Instead, try "capping" the S, with a disk S2. Then the surface formed by combining S1 and S2 is a "closed" surface S which does enclose a 3D region. Use the fact that F- dS F- dS + F- dS F - dS by instead calculating | F - dS (using Divergence Theorem) and calculating and calculate F- dS (using the original formula).

Let v = (r*z, 3 – 2ryz – 3y + r²y, 3z – 2²z) be the velocity field of a fluid. Compute the flux of v across the surface x² + y² + z² = 4 where y > 0 and the surface is oriented away from the origin. HINT: Call the surface in this problem S1. Sį is "open" and does not enclose a 3D region, so Divergence Theorem cannot be used directly to calculate the flux across S1. Instead, try "capping" the S, with a disk S2. Then the surface formed by combining S1 and S2 is a "closed" surface S which does enclose a 3D region. Use the fact that F- dS F- dS + F- dS F - dS by instead calculating | F - dS (using Divergence Theorem) and calculating and calculate F- dS (using the original formula).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 18T
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Let v = (x*z, 3 – 2xyz across the surface x? + y? + z? = 4 where y > 0 and the surface is oriented away from the origin. 3y + x?y, 3z – 2²z) be the velocity field of a fluid. Compute the flux of v HINT: Call the surface in this problem S1. Si is "open" and does not enclose a 3D region, so Divergence Theorem cannot be used directly to calculate the flux across S1. Instead, try "capping" the S1 with a disk S2. Then the surface formed by combining S1 and S2 is a "closed" surface S which does enclose a 3D region. Use the fact that F. ds F· dS + F. dS F. dS by instead calculating F. dS (using Divergence Theorem) and calculating and calculate F. dS (using the original formula).
Transcribed Image Text:Let v = (x*z, 3 – 2xyz across the surface x? + y? + z? = 4 where y > 0 and the surface is oriented away from the origin. 3y + x?y, 3z – 2²z) be the velocity field of a fluid. Compute the flux of v HINT: Call the surface in this problem S1. Si is "open" and does not enclose a 3D region, so Divergence Theorem cannot be used directly to calculate the flux across S1. Instead, try "capping" the S1 with a disk S2. Then the surface formed by combining S1 and S2 is a "closed" surface S which does enclose a 3D region. Use the fact that F. ds F· dS + F. dS F. dS by instead calculating F. dS (using Divergence Theorem) and calculating and calculate F. dS (using the original formula).
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