Explanation Given information: The given vector field is F = 〈 y 2 z , 1 + 2 x y z , x y 2 〉 . Formula used: Potential function of a vector field F = 〈 F 1 , F 2 , F 3 〉 will exist if curl F = ∇ × F = 0 Thus, F is conservative. ∇ × F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F 1 F 2 F 3 | Potential function for the vector field F = 〈 F 1 , F 2 , F 3 〉 is given by a function f such that F = 〈 F 1 ( x , y , z ) , F 2 ( x , y , z ) , F 3 ( x , y , z ) 〉 = ∇ f Where ∇ f = 〈 f x , f y , f z 〉 Calculation: Consider the given vector field. F = 〈 y 2 z , 1 + 2 x y z , x y 2 〉 Here, F 1 ( x , y , z ) = y 2 z , F 2 ( x , y , z ) = 1 + 2 x y z and F 3 ( x , y , z ) = x y 2 Now check whether F is conservative or not. ∇ × F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z y 2 z 1 + 2 x y z x y 2 | = i [ ∂ ∂ y ( x y 2 ) − ∂ ∂ z ( 1 + 2 x y z ) ] − j [ ∂ ∂ x ( x y 2 ) − ∂ ∂ z ( y 2 z ) ] + k [ ∂ ∂ x ( 1 + 2 x y z ) − ∂ ∂ y ( y 2 z ) ] = i [ 2 x y − 2 x y ] − j [ y 2 − y 2 ] + k [ 2 y z − 2 y z ] = 0 Therefore, F is conservative vector field

4th Edition

ISBN: 9781319055905

Author: FRANZOSA

Publisher: VST

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Chapter 16.1, Problem 41E

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To find:

A potential function for the given vector field if it exists.

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Chapter 16.1, Problem 40E

Chapter 16.1, Problem 42E

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Chapter 16 Solutions

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A potential function for the given vector field if it exists.

Solution Summary: The author explains the formula used to find a potential function for the given vector field.

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To find: A potential function for the given vector field if it exists.

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Chapter 16 Solutions

To find:

A potential function for the given vector field if it exists.