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Consider the vector field F(x, y, z) = (2yz, y sin z, 1 + cos z). (a) Find a vector field G whose curl F. (b) Let S be the half-ellipsoid 4x² + 4y² + z² = 4, z ≥ 0, oriented by the upward normal. Use Stokes's theorem to find ff, F. ds. (c) Find ff F. dS if 5 is the portion of the surface z = 1 - x² - y² above the xy-plane, oriented by the upward normal. (Hint: Take advantage of what you've already done.)

Consider the vector field F(x, y, z) = (2yz, y sin z, 1 + cos z). (a) Find a vector field G whose curl F. (b) Let S be the half-ellipsoid 4x² + 4y² + z² = 4, z ≥ 0, oriented by the upward normal. Use Stokes's theorem to find ff, F. ds. (c) Find ff F. dS if 5 is the portion of the surface z = 1 - x² - y² above the xy-plane, oriented by the upward normal. (Hint: Take advantage of what you've already done.)

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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3.1. Consider the vector field F(x, y, z) = (2yz, y sin z, 1+ cos z). (a) Find a vector field G whose curl is F. (b) Let S be the half-ellipsoid 4x² + 4y² + z² = 4, z ≥ 0, oriented by the upward normal. Use Stokes's theorem to find ffs F. ds. - (c) Find fF.dS if S is the portion of the surface z = 1 – x² − y² above the xy-plane, oriented by the upward normal. (Hint: Take advantage of what you've already done.)
Transcribed Image Text:3.1. Consider the vector field F(x, y, z) = (2yz, y sin z, 1+ cos z). (a) Find a vector field G whose curl is F. (b) Let S be the half-ellipsoid 4x² + 4y² + z² = 4, z ≥ 0, oriented by the upward normal. Use Stokes's theorem to find ffs F. ds. - (c) Find fF.dS if S is the portion of the surface z = 1 – x² − y² above the xy-plane, oriented by the upward normal. (Hint: Take advantage of what you've already done.)
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