Explanation Definition used: “A vector field F is said to be conservative if there exists a potential function ϕ such that F = Δ ϕ ”. Theorem used: “Let F = 〈 f , g , h 〉 be a vector field defined on a simply connected domain D of ℝ 3 , where f , g and h have continuous partial derivatives on D then F = 〈 f , g , h 〉 is conservative if and only if ∂ f ∂ y = ∂ g ∂ x , ∂ f ∂ z = ∂ h ∂ x and ∂ g ∂ z = ∂ h ∂ y ”. Theorem used: “Let R in ℝ 3 be an open connected region the F is conservative if and only if ∮ C F · d r = 0 on all simple closed piecewise smooth oriented curves in R”. Calculation: From the given, it is observed that C is the unit circle oriented counterclockwise. The integral ∮ C d s is the length of the curve C , which is 2 π . That is, ∮ C d s = 2 π . Find the value of ∮ C d x . Here, F = 〈 1 , 0 , 0 〉 as it is a unit circle. ∮ C d x = ∮ C 〈 1 , 0 , 0 〉 〈 d x , d y , d z 〉 = ∮ C 〈 1 , 0 , 0 〉 · d r = ∮ C F · d r F = 〈 1 , 0 , 0 〉 where, f = 1 , g = 0 and h = 0 . Then it is clearly observed that, ∂ f ∂ y = ∂ g ∂ x , ∂ f ∂ z = ∂ h ∂ x and ∂ g ∂ z = ∂ h ∂ y

2nd Edition

ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

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Chapter 14.3, Problem 44E

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To evaluate: The values of ∮Cds , ∮Cdx and ∮Cdy , where C is the unit circle oriented counterclockwise.

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Chapter 14.3, Problem 43E

Chapter 14.3, Problem 45E

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The values of ∮ C d s , ∮ C d x and ∮ C d y , where C is the unit circle oriented counterclockwise.

Solution Summary: The author explains that a vector field F=langle f,g,hrangle