Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 16 - Vector Calculus - 16.3 Exercises - Page 1106: 11

Answer

a) The vector field $F$ is conservative. For all three paths in the given graph, the end points are same, so the line integral is the same for all three paths. b) $16$

Work Step by Step

a) When $F(x,y)=Ai+Bj$ is a conservative field, then throughout the domain $D$, we get $\dfrac{\partial A}{\partial y}=\dfrac{\partial B}{\partial x}$ $a$ and $b$ are the first-order partial derivatives on the domain $D$. Here, we have $\dfrac{\partial A}{\partial y} = \dfrac{\partial B}{\partial x}=2x$ Thus, the vector field $F$ is conservative. For all three paths in the given graph, the end points are same, so the line integral is the same for all the three paths. b) Now, $f(x,y)=x^2y+g(y)$ [g(y) : A function of y] $f_y(x,y)=x^2+g'(y)$ Here, $g(y)=k$ Thus, $f(x,y)=x^2y+k$ Now, $\int_C F \cdot dr =f(3,2)-f(1,2)=(18+k)-(2+k)=16$
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