1. Let Si be the piece of the cylinder y? + 22 = 1 that extends from r = 0 to r = 1, oriented away from the r-axis. Let F (x, y, z) = (x, y, z). %3D (a) Explain why we cannot use the Divergence Theorem to calculate the flux of F through S1. (b) Let's change the question so that we can use the Divergence Theorem; then, we'll relate our answer back to the original question. i. Add a caps Stop (r = 1) and Spot (r = 0) to the cylinder Si to form a soup can, which we'll called Scan- How should Stop and Spot be oriented so that Scan is oriented outward? (Draw a sketch.) ii. What is the relationship between the the fluxes of F through the four surfaces S1, Stop, Sbot and Scan? iii. Verify that we can use the Divergence Theorem to compute the flux of F out of Scan- Then, use the Divergence Theorem to compute this flux. iv. Compute the flux of F through Stop and through Spot directly. (HINT: check whether F i is constant; if so, there is no need to integrate.) v. Find the flux of F through S1 using your answers to (1(b)ii), (1(b)iii) and (1(b)iv).
1. Let Si be the piece of the cylinder y? + 22 = 1 that extends from r = 0 to r = 1, oriented away from the r-axis. Let F (x, y, z) = (x, y, z). %3D (a) Explain why we cannot use the Divergence Theorem to calculate the flux of F through S1. (b) Let's change the question so that we can use the Divergence Theorem; then, we'll relate our answer back to the original question. i. Add a caps Stop (r = 1) and Spot (r = 0) to the cylinder Si to form a soup can, which we'll called Scan- How should Stop and Spot be oriented so that Scan is oriented outward? (Draw a sketch.) ii. What is the relationship between the the fluxes of F through the four surfaces S1, Stop, Sbot and Scan? iii. Verify that we can use the Divergence Theorem to compute the flux of F out of Scan- Then, use the Divergence Theorem to compute this flux. iv. Compute the flux of F through Stop and through Spot directly. (HINT: check whether F i is constant; if so, there is no need to integrate.) v. Find the flux of F through S1 using your answers to (1(b)ii), (1(b)iii) and (1(b)iv).
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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Transcribed Image Text:1. Let Si be the piece of the cylinder y? + z2 = 1 that extends from r = 0 to x = 1,
oriented away from the r-axis. Let F(x, y, z) = (x, y, z).
(a) Explain why we cannot use the Divergence Theorem to calculate the flux of F
through S1.
(b) Let's change the question so that we can use the Divergence Theorem; then, we'll
relate our answer back to the original question.
i. Add a caps Stop (x = 1) and Spot (x = 0) to the cylinder Sı to form a soup
can, which we'll called Scan. How should Stop and Spot be oriented so that
Scan is oriented outward? (Draw a sketch.)
ii. What is the relationship between the the fluxes of F through the four surfaces
S1, Stop, Spot and Scan?
iii. Verify that we can use the Divergence Theorem to compute the flux of F
out of Scan- Then, use the Divergence Theorem to compute this flux.
iv. Compute the flux of F through Stop and through Spot directly. (HINT: check
whether F ñ is constant; if so, there is no need to integrate.)
v. Find the flux of F through S1 using your answers to (1(b)ii), (1(b)iii) and
(1(b)iv).
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