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: 11

Answer

a) Conservative b) $16$

Work Step by Step

a) WThe 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 $\dfrac{\partial a}{\partial y}=\dfrac{\partial b}{\partial x}= 2x$ Thus,we have the vector field $F$ is conservative. For all the three paths in the given graph the end points are same , so the line integral is same for all the three paths. b) consider $f(x,y)=x^2y+g(y) \implies f_y(x,y)=x^2+g'(y)$ and $g(y)=k$ Now,we have $f(x,y)=x^2y+k$ Thus, $\int_C F \cdot dr =f(3,2)-f(1,2)=(18+k)-(2+k)=16$
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