Explanation Given information: The vector field is F ( x , y ) = ( − y , x ) ;And the curve specified as the upper half of the unit circle oriented clockwise. Formula Used: The quantity ∂ F 2 ∂ x − ∂ F 1 ∂ y in Green’s theoremis the “circulation per enclosed unit area”. A similar interpretation is valid in R 3 . Consider a plane through a point P with unit normal vector n , and let D be a small domain containing P with boundary curve C . By stoke’s theorem i.e. ∮ c F . d r = ∬ D ( C u r l ( F ) . n ) d s - E q ( 09 ) The vector field c u r l ( F ) is continuous, so its value does not change much on D if D is sufficient small. To first approximation, we can replace c u r l ( F ) by the constant value c u r l ( F ) ( P ) ,giving us the approximation. ∬ D c u r l ( F ) . n d s ≈ ∬ D ( c u r l ( F ) ( p ) . n ) d s ≈ ( c u r l ( F ) ( p ) . n ) A r e a ( D ) _ E q ( 10 ) By E q ( 09 ) and E q ( 10 ) we get ∮ c F . d r = ( c u r l ( F ) ( p ) . n ) A r e a ( D ) _ E q ( 11 ) Stoke’ s Theorem is often written in the form; Let S be a surface and ∂ s the boundary of S surface and F vector field. ∫ ∂ s F

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 17.2, Problem 64E

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The flux of F through S.

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Chapter 17.2, Problem 63E

Chapter 17.2, Problem 65E

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Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

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The flux of F through S .

Solution Summary: The author explains how to calculate the flux of F through S.

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The flux of F through S.

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The flux of F through S .

Solution Summary: The author explains how to calculate the flux of F through S.

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To calculate: The flux of F through S.

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Chapter 17 Solutions

To calculate:

The flux of F through S.