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 1135: 23



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 $D$. Consider $f(x,y)=\dfrac{x^3}{4}+g(y)$ and $f_x(x,y)=x^3$ and $f_y(x,y)=y^3$ $\implies f_y(x,y)=y^3+g'(y)$ and $g(y)=k$ Thus, we get $f(x,y)=\dfrac{x^4}{4}+\dfrac{y^4}{4}+k$; $k$ is a constant. Hence, $W=\int_C F \cdot dr =f(2,2)-f(1,0)=\dfrac{1}{4}(32-1)=\dfrac{31}{4}$
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