Explanation Given: F → = 〈 y z 2 , x z 2 , 2 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 exist. 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 2 , x z 2 , 2 x y z 〉 we get- = ∇ → × F → = | i ^ j ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z y z 2 x z 2 2 x y z | = i ^ ( 2 x z − 2 x z ) − j ^ ( 2 y z − 2 y z ) + k ^ ( z 2 − z 2 ) = 0 Thus, Curl F → = 0 . This mean F → is conservative. Thus the potential function exist...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17.1, Problem 45E

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Potential function for the vector field $\vec{F} = 〈 y z^{2}, x z^{2}, 2 x y z 〉$ by inspection or show that one does not exist.

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Chapter 17.1, Problem 44E

Chapter 17.1, Problem 46E

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Chapter 17 Solutions

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Potential function for the vector field F → = 〈 y z 2 , x z 2 , 2 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 = x y z 2.

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Potential function for the vector field $\vec{F} = 〈 y z^{2}, x z^{2}, 2 x y z 〉$ by inspection or show that one does not exist.

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Potential function for the vector field $\vec{F} = 〈 y z^{2}, x z^{2}, 2 x y z 〉$ by inspection or show that one does not exist.