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

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ISBN: 9781319379421

Author: Rogawski

Publisher: MAC HIGHER

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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

CALCULUS 4E (HC) W/ ACHIEVE ACCESS

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E

Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E Ch. 16.1 - Prob. 11E Ch. 16.1 - Prob. 12E Ch. 16.1 - Prob. 13E Ch. 16.1 - Prob. 14E Ch. 16.1 - Prob. 15E Ch. 16.1 - Prob. 16E Ch. 16.1 - Prob. 17E Ch. 16.1 - Prob. 18E Ch. 16.1 - Prob. 19E Ch. 16.1 - Prob. 20E Ch. 16.1 - Prob. 21E Ch. 16.1 - Prob. 22E Ch. 16.1 - Prob. 23E Ch. 16.1 - Prob. 24E Ch. 16.1 - Prob. 25E Ch. 16.1 - Prob. 26E Ch. 16.1 - Prob. 27E Ch. 16.1 - Prob. 28E Ch. 16.1 - Prob. 29E Ch. 16.1 - Prob. 30E Ch. 16.1 - Prob. 31E Ch. 16.1 - Prob. 32E Ch. 16.1 - Prob. 33E Ch. 16.1 - Prob. 34E Ch. 16.1 - Prob. 35E Ch. 16.1 - Prob. 36E Ch. 16.1 - Prob. 37E Ch. 16.1 - Prob. 38E Ch. 16.1 - Prob. 39E Ch. 16.1 - Prob. 40E Ch. 16.1 - Prob. 41E Ch. 16.1 - Prob. 42E Ch. 16.1 - Prob. 43E Ch. 16.1 - Prob. 44E Ch. 16.1 - Prob. 45E Ch. 16.1 - Prob. 46E Ch. 16.1 - Prob. 47E Ch. 16.1 - Prob. 48E Ch. 16.1 - Prob. 49E Ch. 16.1 - Prob. 50E Ch. 16.1 - Prob. 51E Ch. 16.1 - Prob. 52E Ch. 16.1 - Prob. 53E Ch. 16.1 - Prob. 54E Ch. 16.1 - 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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.