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(6) Let W be the region in R³ inside x² + y² = 1 that is below z = 3-²-² and above √² + y². More precisely, z = V W = {(x, y, z): x² + y² ≤ 1, √√² + y² ≤ x ≤3-2²-y²}. (a) Sketch the region W. (b) Consider a fluid of constant density u(x, y, z) - 5 (measured in g/in³) flowing with velocity v(x, y, z) = (3xy² - e², y³ – x cos(2), z³ - y²) measured in in/s. Use the appropriate version of Stokes theorem to find the total amount of fluid (in the boundary of W flowing in the inward direction after 2 seconds.

(6) Let W be the region in R³ inside x² + y² = 1 that is below z = 3-²-² and above √² + y². More precisely, z = V W = {(x, y, z): x² + y² ≤ 1, √√² + y² ≤ x ≤3-2²-y²}. (a) Sketch the region W. (b) Consider a fluid of constant density u(x, y, z) - 5 (measured in g/in³) flowing with velocity v(x, y, z) = (3xy² - e², y³ – x cos(2), z³ - y²) measured in in/s. Use the appropriate version of Stokes theorem to find the total amount of fluid (in the boundary of W flowing in the inward direction after 2 seconds.

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 W be the region in R³ inside x² + y² = 1 that is below z = 3-²-² and above √² + y². More precisely, z = V W = {(x, y, z): x² + y² ≤ 1, √√² + y² ≤ x ≤3-2²-y²}. (a) Sketch the region W. (b) Consider a fluid of constant density μ(x, y, z) - 5 (measured in g/in³) flowing with velocity v(x, y, z) = (3ry² - e², y³ – x cos(2), z³ - y³) measured in in/s. Use the appropriate version of Stokes theorem to find the total amount of fluid (in grams) that crosses the boundary of W flowing in the inward direction after 2 seconds. (c) What was the sign in your answer and what is the physical interpretation for this sign?
Transcribed Image Text:(6) Let W be the region in R³ inside x² + y² = 1 that is below z = 3-²-² and above √² + y². More precisely, z = V W = {(x, y, z): x² + y² ≤ 1, √√² + y² ≤ x ≤3-2²-y²}. (a) Sketch the region W. (b) Consider a fluid of constant density μ(x, y, z) - 5 (measured in g/in³) flowing with velocity v(x, y, z) = (3ry² - e², y³ – x cos(2), z³ - y³) measured in in/s. Use the appropriate version of Stokes theorem to find the total amount of fluid (in grams) that crosses the boundary of W flowing in the inward direction after 2 seconds. (c) What was the sign in your answer and what is the physical interpretation for this sign?
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