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8. Let F = (4xyz + ay², 2r²z+ bry, 2r2y), where a and b are real-valued constants, be the velocity field of a fluid. (a) Find div F. the sphere (b) Use the Divergence Theorem to evaluate fF.dS, where S is the surface of the portion of radius R with y 20 and z 20. All three surfaces of the solid are included in S, and S is oriented outward. Your answer should contain a and b. (c) Is the net flow into the surface, out of the surface, or zero? Why? 1) Use curl F to determine all values of a and b for which F is conservative. If a and b have these values, how does this effect for your answers for (b) and (c) (if at all)?

8. Let F = (4xyz + ay², 2r²z+ bry, 2r2y), where a and b are real-valued constants, be the velocity field of a fluid. (a) Find div F. the sphere (b) Use the Divergence Theorem to evaluate fF.dS, where S is the surface of the portion of radius R with y 20 and z 20. All three surfaces of the solid are included in S, and S is oriented outward. Your answer should contain a and b. (c) Is the net flow into the surface, out of the surface, or zero? Why? 1) Use curl F to determine all values of a and b for which F is conservative. If a and b have these values, how does this effect for your answers for (b) and (c) (if at all)?

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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8. Let F = (4xyz + ay2, 2x²z+ bry, 2x2y), where a and b are real-valued constants, be the velocity field of a fluid. Find div F. (a) (b) Use the Divergence Theorem to evaluate fF.dS, where S is the surface of the portion of the sphere S of radius R with y 20 and z> 0. All three surfaces of the solid are included in S, and S is oriented outward. Your answer should contain a and b. (c) Is the net flow into the surface, out of the surface, or zero? Why? (d) Use curl F to determine all values of a and b for which F is conservative. If a and b have these values, how does this effect for your answers for (b) and (c) (if at all)?
Transcribed Image Text:8. Let F = (4xyz + ay2, 2x²z+ bry, 2x2y), where a and b are real-valued constants, be the velocity field of a fluid. Find div F. (a) (b) Use the Divergence Theorem to evaluate fF.dS, where S is the surface of the portion of the sphere S of radius R with y 20 and z> 0. All three surfaces of the solid are included in S, and S is oriented outward. Your answer should contain a and b. (c) Is the net flow into the surface, out of the surface, or zero? Why? (d) Use curl F to determine all values of a and b for which F is conservative. If a and b have these values, how does this effect for your answers for (b) and (c) (if at all)?
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