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: 10


Conservative and $f(x,y)=x \ln y+y \ln x+k$;

Work Step by Step

The vector field $F(x,y)=ai+bj$ is known as conservative field throughout the domain $D$, when we have $\dfrac{\partial a}{\partial y}=\dfrac{\partial b}{\partial x}$ $a$ and $b$ represents the first-order partial derivatives on the domain $D$. From the given problem, we get $\dfrac{\partial a}{\partial y}=\dfrac{\partial b}{\partial x}= \dfrac{\partial Q}{\partial x}=\dfrac{1}{y}+\dfrac{1}{x}$ Thus, the vector field $F$ is conservative. Here, we have $f(x,y)=x \ln y+y \ln x+g(y)$ $\implies f_y(x,y)=xy^{-1}+g'(y)$ Here, $g(y)=k$; where $k$ is a constant. Thus, $f(x,y)=x \ln y+y \ln x+k$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.