Explanation Definition used: “A vector field F is said to be conservative if there exists a potential function φ such that F = ∇ 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 ”. Calculation The given vector field is F = 〈 y , x , 1 〉 . Here, f = y , g = x and h = 1 . Obtain the value of ∂ f ∂ y = ∂ g ∂ x , ∂ f ∂ z = ∂ h ∂ x , and ∂ g ∂ z = ∂ h ∂ y . Take partial derivative of the function f with respect to y and obtain ∂ f ∂ y . ∂ f ∂ y = ∂ ∂ y ( y ) = 1 Similarly, obtain the derivative of f with respect to z and ∂ f ∂ z . ∂ f ∂ z = ∂ ∂ z ( y ) = 0 Take partial derivative of the function g with respect to x and obtain ∂ g ∂ x . ∂ g ∂ x = ∂ ∂ x ( x ) = 1 Obtain the derivative of the function g with respect to z and obtain ∂ g ∂ z . ∂ g ∂ z = ∂ ∂ z ( x ) = 0 Again, obtain the derivative of the function h with respect to y and obtain ∂ h ∂ y . ∂ h ∂ y = ∂ ∂ y ( 1 ) = 0 Obtain the derivative of the function h with respect to x and obtain ∂ h ∂ x . ∂ h ∂ x = ∂ ∂ x ( 1 ) = 0 Thus, it is proved that ∂ f ∂ y = ∂ g ∂ x , ∂ f ∂ z = ∂ h ∂ x and ∂ g ∂ z = ∂ h ∂ y . Therefore, the given vector field F = 〈 y , x , 1 〉 is conservative on ℝ 3

Calculus: Early Transcendentals, 2nd Edition

2nd Edition

ISBN: 9780321965165

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0…

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Chapter 14.3, Problem 20E

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Whether the given vector field is conservative or not and obtain the potential function if it is conservative on ℝ3 .

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Chapter 14.3, Problem 19E

Chapter 14.3, Problem 21E

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Chapter 14 Solutions

Calculus: Early Transcendentals, 2nd Edition

Ch. 14.1 - How is a vector field F = f, g, h used to describe...Ch. 14.1 - Sketch the vector field F = x, y.Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Interpret the gradient field of the temperature...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...

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Whether the given vector field is conservative or not and obtain the potential function if it is conservative on ℝ 3 .

Solution Summary: The author explains that the given vector field is conservative and the potential function is phi =xy+z+k.

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Whether the given vector field is conservative or not and obtain the potential function if it is conservative on ℝ3 .

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Chapter 14 Solutions