Explanation Given: The line segment from ( 0 , 0 ) to ( 0 , 1 ) . Calculation: (a) Given f ( x , y ) = x from (0,0) to (0,1) Realizing the above function in its parametric form the Then x = t . c + ( 1 − t ) a y = t . d + ( 1 − t ) b x = t . c + ( 1 − t ) a = t .0 + ( 1 − t ) .0 = 0 y = t . d + ( 1 − t ) b = t .1 + ( 1 − t ) .0 = t r I ( t ) → = < 0 , 1 > | r I ( t ) → | = 0 2 + 1 2 = 1 Now the line integral to be evaluated is ∫ c f . d s = ∫ 0 1 f ( x , y ) . | r I ( t ) → | d t Substituting the values in the above equation ⇒ ∫ 0 1 x .1 d t ⇒ ∫ 0 1 0. 1 d t ( ∵ x = 0 ) = 0 (b) Given function is f ( x , y ) = y from (0, 0) to (0, 1) Realizing the above function in its parametric form (0, 0) to (0, 1), we get: x = t . c + ( 1 − t ) a y = t . d + ( 1 − t ) b x = t . c + ( 1 − t ) a = t .0 + ( 1 − t ) .0 = 0 y = t . d + ( 1 − t ) b = t .1 + ( 1 − t ) .0 = t r I ( t ) → = < 0 , 1 > | r I ( t ) → | = 0 2 + 1 2 = 1 Now the line integral to be evaluated is ∫ c f . d s = ∫ 0 1 f ( x , y ) . | r I ( t ) → | d t Substituting the values in the above equation ⇒ ∫ 0 1 y .1 d t ⇒ ∫ 0 1 t . 1 d t ( ∵ y = t ) = [ t 2 2 ] 0 1 = 1 2 (c) Given F = < x , 0 > from (0, 0) to (0, 1) Line segment given is (0, 0) to (0, 1) x = t . c + ( 1 − t ) a y = t . d + ( 1 − t ) b x = t . c + ( 1 − t ) a = t .0 + ( 1 − t ) .0 = 0 y = t . d + ( 1 − t ) b = t .1 + ( 1 − t ) .0 = t F = 〈 y , 0 〉 r ¯ ( t ) = 〈 0 , t 〉 r ¯ 1 ( t ) = 〈 0 , 1 〉 ∫ c F ¯ . T ¯ d s = ∫ 0 1 〈 y , 0 〉 . 〈 0 , 1 〉 d t = ∫ 0 1 〈 t , 0 〉 . 〈 0 , 1 〉 d t ( ∵ y = t ) ⇒ ∫ 0 1 ( t .0 + 0.1 ) d t = 0 (d) Given F = 〈 y , 0 〉 from (0, 0) to (0, 1) Line segment given is (0, 0) to (0, 1) x = t ...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17.2, Problem 2PQ

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Which of the given functions have zero line integral over the vertical segment from the given points

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Chapter 17.2, Problem 1PQ

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Chapter 17 Solutions

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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

Ch. 17.1 - Prob. 7E Ch. 17.1 - Prob. 8E Ch. 17.1 - Prob. 9E Ch. 17.1 - Prob. 10E Ch. 17.1 - Prob. 11E Ch. 17.1 - Prob. 12E Ch. 17.1 - Prob. 13E Ch. 17.1 - Prob. 14E Ch. 17.1 - Prob. 15E Ch. 17.1 - Prob. 16E Ch. 17.1 - Prob. 17E Ch. 17.1 - Prob. 18E Ch. 17.1 - Prob. 19E Ch. 17.1 - Prob. 20E Ch. 17.1 - Prob. 21E Ch. 17.1 - Prob. 22E Ch. 17.1 - Prob. 23E Ch. 17.1 - Prob. 24E Ch. 17.1 - Prob. 25E Ch. 17.1 - Prob. 26E Ch. 17.1 - Prob. 27E Ch. 17.1 - Prob. 28E Ch. 17.1 - Prob. 29E Ch. 17.1 - Prob. 30E Ch. 17.1 - Prob. 31E Ch. 17.1 - Prob. 32E Ch. 17.1 - Prob. 33E Ch. 17.1 - Prob. 34E Ch. 17.1 - Prob. 35E Ch. 17.1 - Prob. 36E Ch. 17.1 - Prob. 37E Ch. 17.1 - Prob. 38E Ch. 17.1 - Prob. 39E Ch. 17.1 - Prob. 40E Ch. 17.1 - Prob. 41E Ch. 17.1 - Prob. 42E Ch. 17.1 - Prob. 43E Ch. 17.1 - Prob. 44E Ch. 17.1 - Prob. 45E Ch. 17.1 - Prob. 46E Ch. 17.1 - Prob. 47E Ch. 17.1 - Prob. 48E Ch. 17.1 - Prob. 49E Ch. 17.1 - Prob. 50E Ch. 17.1 - Prob. 51E Ch. 17.1 - Prob. 52E Ch. 17.1 - Prob. 53E Ch. 17.1 - Prob. 54E Ch. 17.1 - 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Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - Prob. 16E Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.3 - Prob. 19E Ch. 17.3 - Prob. 20E Ch. 17.3 - Prob. 21E Ch. 17.3 - Prob. 22E Ch. 17.3 - Prob. 23E Ch. 17.3 - Prob. 24E Ch. 17.3 - Prob. 25E Ch. 17.3 - Prob. 26E Ch. 17.3 - Prob. 27E Ch. 17.3 - Prob. 28E Ch. 17.3 - Prob. 29E Ch. 17.3 - Prob. 30E Ch. 17.3 - Prob. 31E Ch. 17.3 - Prob. 32E Ch. 17.3 - Prob. 33E Ch. 17.3 - Prob. 34E Ch. 17.3 - Prob. 35E Ch. 17.4 - Prob. 1PQ Ch. 17.4 - Prob. 2PQ Ch. 17.4 - Prob. 3PQ Ch. 17.4 - Prob. 4PQ Ch. 17.4 - Prob. 5PQ Ch. 17.4 - Prob. 6PQ Ch. 17.4 - Prob. 1E Ch. 17.4 - Prob. 2E Ch. 17.4 - Prob. 3E Ch. 17.4 - Prob. 4E Ch. 17.4 - Prob. 5E Ch. 17.4 - Prob. 6E Ch. 17.4 - Prob. 7E Ch. 17.4 - Prob. 8E Ch. 17.4 - Prob. 9E Ch. 17.4 - Prob. 10E Ch. 17.4 - Prob. 11E Ch. 17.4 - Prob. 12E Ch. 17.4 - Prob. 13E Ch. 17.4 - Prob. 14E Ch. 17.4 - Prob. 15E Ch. 17.4 - Prob. 16E Ch. 17.4 - Prob. 17E Ch. 17.4 - Prob. 18E Ch. 17.4 - Prob. 19E Ch. 17.4 - Prob. 20E Ch. 17.4 - 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Which of the given functions have zero line integral over the vertical segment from the given points

Solution Summary: The author determines which of the given functions have zero line integral over the vertical line segment.

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Which of the given functions have zero line integral over the vertical segment from the given points

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Chapter 17 Solutions