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Calculate the circulation, fo F. dr, in two ways, directly and using Stokes' Theorem. The vector field F = 8yi - 8aj and C is the boundary of S, the part of the surface z = 4 - x² - y² above the ay-plane, oriented upward. Note that C is a circle in the ay-plane. Find a r(t) that parameterizes this curve. r(t) = <2 cos 1,2 sin 1,0 > with 0 St≤ 2pi (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 Sc F. dr = f Evaluate your integral to find the circulation: fcF-dr=-64pi dt, where a and b are the endpoints you gave above. Using Stokes' Theorem, we equate foF. dr = fg curl F. dÃ. Find curl F = Noting that the surface is given by z = 4-x² - y², find dà = dy da. With R giving the region in the xy-plane enclosed by the surface, this gives Ss curl F. dà = SR dy dx. Evaluate this integral to find the circulation: ScFdr fg curl F. dà =

Calculate the circulation, fo F. dr, in two ways, directly and using Stokes' Theorem. The vector field F = 8yi - 8aj and C is the boundary of S, the part of the surface z = 4 - x² - y² above the ay-plane, oriented upward. Note that C is a circle in the ay-plane. Find a r(t) that parameterizes this curve. r(t) = <2 cos 1,2 sin 1,0 > with 0 St≤ 2pi (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 Sc F. dr = f Evaluate your integral to find the circulation: fcF-dr=-64pi dt, where a and b are the endpoints you gave above. Using Stokes' Theorem, we equate foF. dr = fg curl F. dÃ. Find curl F = Noting that the surface is given by z = 4-x² - y², find dà = dy da. With R giving the region in the xy-plane enclosed by the surface, this gives Ss curl F. dà = SR dy dx. Evaluate this integral to find the circulation: ScFdr fg curl F. dà =

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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Calculate the circulation, foF. dr, in two ways, directly and using Stokes' Theorem. The vector field F = 8yi – 8x and C' is the boundary of S, the part of the surface z = 4 - 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) = < 2 cos t,2 sin t,0 > with 0 <t< 2pi (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 Sc F. dr = So Evaluate your integral to find the circulation: SF. dr = −64pi dt, where a and b are the endpoints you gave above. Using Stokes' Theorem, we equate fF. dr = fg curl F · dÃ. Find curl F = z = 4x² - y², find Noting that the surface is given by a dà = dy dx. With R giving the region in the xy-plane enclosed by the surface, this gives Ss curl F. dà = SR dy dx. Evaluate this integral to find the circulation: ScF. dr = fg curl F · dà =
Transcribed Image Text:Calculate the circulation, foF. dr, in two ways, directly and using Stokes' Theorem. The vector field F = 8yi – 8x and C' is the boundary of S, the part of the surface z = 4 - 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) = < 2 cos t,2 sin t,0 > with 0 <t< 2pi (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 Sc F. dr = So Evaluate your integral to find the circulation: SF. dr = −64pi dt, where a and b are the endpoints you gave above. Using Stokes' Theorem, we equate fF. dr = fg curl F · dÃ. Find curl F = z = 4x² - y², find Noting that the surface is given by a dà = dy dx. With R giving the region in the xy-plane enclosed by the surface, this gives Ss curl F. dà = SR dy dx. Evaluate this integral to find the circulation: ScF. dr = fg curl F · dà =
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