Explanation No work done in displacement from a point A to another point in the field and back to A and the mechanical energy is conserved. i.e., ∮ F ⋅ d R = 0 , the field is said to be conservative. Every irrotational field is conservative. c u r l F = 0 , then vector field is irrotational. The field is irrotational the vector F can be expressed as the gradient of a scalar potential ϕ i.e., F = g r a d ϕ ⇒ F = ∇ ϕ Given: F ( x , y , z ) = ( x y z , 1 2 x 2 z , 2 z 2 y ) is the vector point function. Formula used: F = F 1 i + F 2 j + F 3 k Curl of a continuously differentiable vector point function F is c u r l F = ∇ × F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F 1 F 2 F 3 | Calculation: The components of the vector F are F 1 = x y z , F 2 = 1 2 x 2 z , F 3 = 2 z 2 y c u r l F = ∇ × F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z x y z 1 2 x 2 z 2 z 2 y | ,expanding 3 × 3 determinant by 1 st row = i ( ∂ ∂ y ( 2 z 2 y ) − ∂ ∂ z ( 1 2 x 2 z ) ) − j ( ∂ ∂ x ( 2 z 2 y ) − ∂

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Differential Equations

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Chapter 17, Problem 22CRE

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Whether the vector field is conservative, if so find potential function.

Solution Summary: The author explains that if the vector field is conservative, the mechanical energy is conserved. If c u r l F = 0, then it is irrotational.

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Whether the vector field is conservative, if so find potential function.

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Chapter 17 Solutions