Explanation Given: F → = 〈 y z cos ( x y z ) , x z cos ( x y z ) , x y cos ( x y z ) 〉 is a vector field. Formula used: If f is a potential function for the vector field F → then F → = ∇ → f Where, ∇ → = 〈 ∂ ∂ x , ∂ ∂ y , ∂ ∂ z 〉 Curl of a vector is given by ∇ → × F → If Curl F → = 0 F → is conservative then potential function exists. If Curl F → ≠ 0 mean F → is not conservative then potential function does not exist. Calculation: Checking for existence of the potential function by taking Curl of the given potential function, F → = 〈 y z cos ( x y z ) , x z cos ( x y z ) , x y cos ( x y z ) 〉 we get- = ∇ → × F → = | i ^ j ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z y z cos ( x y z ) x z cos ( x y z ) x y cos ( x y z ) | = i ^ [ ( x cos ( x y z ) − x 2 y z sin ( x y z ) ) − ( x cos ( x y z ) − x 2 y z sin ( x y z ) ) ] − j ^ [ ( y cos ( x y z ) − x y 2 z sin ( x y z ) ) − ( y cos ( x y z ) − x y 2 z sin ( x y z ) ) ] + k ^ [ ( z cos ( x y z ) − x y z 2 sin ( x y z ) ) − ( ( z cos ( x y z ) − x y z 2 sin ( x y z ) ) ) ] = 0 Thus, Curl F → = 0 . This mean F → is conservative...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17.1, Problem 47E

To determine

Potential function for the vector field $\vec{F} = 〈 y z \cos (x y z), x z \cos (x y z), x y \cos (x y z) 〉$ by inspection or show that one does not exist.

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Chapter 17.1, Problem 46E

Chapter 17.1, Problem 48E

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Chapter 17 Solutions

CALCULUS (CLOTH)

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Potential function for the vector field F → = 〈 y z cos ( x y z ) , x z cos ( x y z ) , x y cos ( x y z ) 〉 by inspection or show that one does not exist.

Solution Summary: The author explains that the potential function for the vector field F exists and is f = Sin (x y z) + c.

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Potential function for the vector field $\vec{F} = 〈 y z \cos (x y z), x z \cos (x y z), x y \cos (x y z) 〉$ by inspection or show that one does not exist.

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Chapter 17 Solutions

Potential function for the vector field F → = 〈 y z cos ( x y z ) , x z cos ( x y z ) , x y cos ( x y z ) 〉 by inspection or show that one does not exist.

Solution Summary: The author explains that the potential function for the vector field F exists and is f = Sin (x y z) + c.

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Potential function for the vector field F→=〈yzcos(xyz),xzcos(xyz),xycos(xyz)〉 by inspection or show that one does not exist.

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Chapter 17 Solutions

Potential function for the vector field $\vec{F} = 〈 y z \cos (x y z), x z \cos (x y z), x y \cos (x y z) 〉$ by inspection or show that one does not exist.