Explanation Given information: The given field is F = 〈 y 2 z , 1 + 2 x y z , x y 2 〉 . Concept used: We use the concept that a potential function f ( x , y , z ) of the vector field F = 〈 F 1 , F 2 , F 3 〉 exists if it is conservative i.e. c u r l F = 0 and the potential function satisfies f ( x , y , z ) = ∫ f 1 d x , f ( x , y ) = ∫ f 2 d y , f ( x , y ) = ∫ f 3 d z The curl of the function is given by c u r l F = ∇ × F = ( ∂ F 3 ∂ y − ∂ F 2 ∂ z ) i − ( ∂ F 3 ∂ x − ∂ F 1 ∂ z ) j + ( ∂ F 2 ∂ x − ∂ F 1 ∂ y ) k Calculation: As given F = 〈 y 2 z , 1 + 2 x y z , x y 2 〉 , first we check if the potential function exists. Therefore, we find c u r l F = ∇ × F = ( ∂ x y 2 ∂ y − ∂ ( 1 + 2 x y z ) ∂ z ) i − ( ∂ x y 2 ∂ x − ∂ y 2 z ∂ z ) j + ( ∂ ( 1 + 2 x y z ) ∂ x − ∂ y 2 z ∂ y ) k = ( 2 x y − 2 x y ) i − ( y 2 − y 2 ) j + ( 2 y z − 2 y z ) k = 0 Thus, c u r l F = 0

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 17.1, Problem 41E

To determine

To find: The potential function of the given vector field F=〈y2z,1+2xyz,xy2〉 if it exists.

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Chapter 17.1, Problem 40E

Chapter 17.1, Problem 42E

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Chapter 17 Solutions

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The potential function of the given vector field F = 〈 y 2 z , 1 + 2 x y z , x y 2 〉 if it exists.

Solution Summary: The author uses the concept that a potential function f(x,y,z)=xy2z+y exists if it is conservative i.e. curlF

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To find: The potential function of the given vector field F=〈y2z,1+2xyz,xy2〉 if it exists.

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Chapter 17 Solutions