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Calculate the circulation, SF. dr, in two ways, directly and using Stokes' Theorem. The vector field F = 4y - 4x and C is the boundary of S, the part of the surface z = 9x² - y²2 above the xy-plane, oriented upward. Note that C is a circle in the xy-plane. Find a F(t) that parameterizes this curve. r(t) with

Calculate the circulation, SF. dr, in two ways, directly and using Stokes' Theorem. The vector field F = 4y - 4x and C is the boundary of S, the part of the surface z = 9x² - y²2 above the xy-plane, oriented upward. Note that C is a circle in the xy-plane. Find a F(t) that parameterizes this curve. r(t) with

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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Calculate the circulation, SF. dr, in two ways, directly and using Stokes' Theorem. The vector field F = 4yi — 4x and C is the boundary of S, the part of the surface z = 9 — x² - y² above the xy-plane, oriented upward. Note that C is a circle in the xy-plane. Find a r(t) that parameterizes this curve. r(t) = with <t< (Note that answers must be provided for all three of these answer blanks to be able to determine correctness of the parameterization.) With this parameterization, the circulation integral is ScF¹ · dr = Sº | dt, where a and b are the endpoints you gave above. Evaluate your integral to find the circulation: SF. dr = Using Stokes' Theorem, we equate SF · dỡ = ſçcurl ♬ · dÃ. Find curl F Noting that the surface is given by z = 9 - x² - y²2, find d.à = dy dx. With R giving the region in the xy-plane enclosed by the surface, this gives Ss ₁ curl F. d.à = SR dy dr. Evaluate this integral to find the circulation: ScF·dr = f curl F · dà : = -0
Transcribed Image Text:Calculate the circulation, SF. dr, in two ways, directly and using Stokes' Theorem. The vector field F = 4yi — 4x and C is the boundary of S, the part of the surface z = 9 — x² - y² above the xy-plane, oriented upward. Note that C is a circle in the xy-plane. Find a r(t) that parameterizes this curve. r(t) = with <t< (Note that answers must be provided for all three of these answer blanks to be able to determine correctness of the parameterization.) With this parameterization, the circulation integral is ScF¹ · dr = Sº | dt, where a and b are the endpoints you gave above. Evaluate your integral to find the circulation: SF. dr = Using Stokes' Theorem, we equate SF · dỡ = ſçcurl ♬ · dÃ. Find curl F Noting that the surface is given by z = 9 - x² - y²2, find d.à = dy dx. With R giving the region in the xy-plane enclosed by the surface, this gives Ss ₁ curl F. d.à = SR dy dr. Evaluate this integral to find the circulation: ScF·dr = f curl F · dà : = -0
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