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

4th Edition

ISBN: 9781319055905

Author: FRANZOSA

Publisher: VST

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 16.3, Problem 32E

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The potential function for the vortex field is g(x,y)=−tan−1xy.

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Chapter 16.3, Problem 31E

Chapter 16.3, Problem 33E

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Chapter 16 Solutions

EBK CALCULUS: EARLY TRANSCENDENTALS

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