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: 9


$f(x,y)=y^2 \sin x+x\cos y+k$

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 $\dfrac{\partial a}{\partial y}=\dfrac{\partial b}{\partial x}=2y\cos x-\sin y$ Thus, the vector field $F$ is conservative. Here, we have $f(x,y)=y^2 \sin x+x\cos y+g(y)$ $\implies f_y(x,y)=2y \sin x-x\sin y+g'(y)$ and $g(y)=k$; where $k$ is a constant. Thus, $f(x,y)=y^2 \sin x+x\cos y+k$
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