Explanation Given information: The vector field is given as, F ( x , y ) = 〈 − y x 2 + y 2 , x x 2 + y 2 〉 C R is a circle oriented counterclockwise with the origin as the center and radius is R . Formula used: Let r ( t ) be a parameterization of a curve C for a ≤ t ≤ b . If f ( x , y , z ) and r ' ( t ) are continuous, then ∫ C f ( x , y ) d s = ∫ a b f ( r ( t ) ) ‖ r ' ( t ) ‖ d t Calculation: A circle with radius of R and origin as the center is given by, x 2 + y 2 = R 2 We parameterize the curve with respect to t where 0 ≤ t ≤ 2 π r ( t ) = ( R cos t , R sin t ) The vector field F can be obtained as, F ( x , y ) = ( − y x 2 + y 2 ) i + ( x x 2 + y 2 ) j F ( r ( t ) ) = ( − R sin t R 2 sin 2 t + R 2 cos 2 t ) i + ( R cos t R 2 sin 2 t + R 2 cos 2 t ) j = ( − R sin t R 2 ) i + ( R cos t R 2 ) j F ( r ( t ) ) = ( − sin t R ) i + ( cos t R ) j Also, the derivative of r ( t ) should be calculated

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 17.2, Problem 61E

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That the value of I=∫CRFdr is independent from the radius R of the circle.

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Chapter 17.2, Problem 60E

Chapter 17.2, Problem 62E

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Chapter 17 Solutions

Calculus - Standalone book

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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That the value of I = ∫ C R F d r is independent from the radius R of the circle.

Solution Summary: The author explains that the value of I=displaystyle intC_RFdr is independent from the radius R of the circle.

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To show:

That the value of I=∫CRFdr is independent from the radius R of the circle.

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Chapter 17 Solutions

That the value of I = ∫ C R F d r is independent from the radius R of the circle.

Solution Summary: The author explains that the value of I=displaystyle intC_RFdr is independent from the radius R of the circle.

Videos

To show: That the value of I=∫CRFdr is independent from the radius R of the circle.

Want to see the full answer?

Chapter 17 Solutions

To show:

That the value of I=∫CRFdr is independent from the radius R of the circle.