Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - 16.3 The Fundamental Theorem for Line Integrals - 16.3 Exercises - Page 1134: 3


$F$ is Not conservative.

Work Step by Step

When $F(x,y)=pi+qj$ is a conservative field, then throughout the $D$, then we have $\dfrac{\partial p}{\partial y}=\dfrac{\partial q}{\partial x}$ Here, $p$ and $q$ represents the first-order partial derivatives on a domain $D$. Let us consider $p=(xy+y^2)$ and $q=(x^2+2xy)$ The first-order partial derivatives are: $p_x=x+2y$ and $q_x=2x+2y$ Here, we can see that $\dfrac{\partial A}{\partial y} \neq \dfrac{\partial Q}{\partial x}$ Hence, $F$ is not conservative.
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