Explanation Given information : F = 〈 x z , x y , y z 〉 , where C is the rectangle with vertices ( 0 , 0 , 0 ) , ( 0 , 0 , 2 ) , ( 3 , 0 , 2 ) and ( 3 , 0 , 0 ) , oriented counter clockwise as viewed from the positive y-axis. Stokes’Theorem: Let S be a surface and let F be a vector field whose components have continuous partial derivatives on an open region containing S. Then ∮ C F . d r = ∬ S c u r l F . d S Calculation: F = 〈 x z , x y , y z 〉 (Given) Parameterization of each line is given by C 1 : r 1 ( t ) = ( 3 t , 0 , 0 ) 0 ≤ t ≤ 1 C 2 : r 2 ( t ) = ( 0 , 0 , 2 − 2 t ) 0 ≤ t ≤ 1 C 3 : r 3 ( t ) = ( 3 − 3 t , 0 , 2 ) 0 ≤ t ≤ 1 C 4 : r 4 ( t ) = ( 3 , 0 , 2 t ) 0 ≤ t ≤ 1. F ( r 1 ( t ) ) . r 1 ' ( t ) = 0 F ( r 2 ( t ) ) . r 2 ' ( t ) = 0 F ( r 3 ( t ) ) = 〈 6 − 6 t , 0 , 0 〉 F ( r 4 ( t ) ) = 〈 6 t , 0 , 0 〉 F ( r 1 ( t ) ) = 〈 0 , 0 , 0 〉 F ( r 3 ( t ) )

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 18.2, Problem 13E

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To evaluate: ∮CF.dr using Stokes’ theorem by finding the flux of curl(F) across an appropriate surface.

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Chapter 18.2, Problem 12E

Chapter 18.2, Problem 14E

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Chapter 18 Solutions

CALCULUS (CLOTH)

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∮ C F . d r using Stokes’ theorem by finding the flux of curl( F ) across an appropriate surface.

Solution Summary: The author evaluates Stokes' theorem by finding the flux of curl( F ) across an appropriate surface.

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To evaluate: ∮CF.dr using Stokes’ theorem by finding the flux of curl(F) across an appropriate surface.

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Chapter 18 Solutions