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 - 16.9 Exercises - Page 1158: 20

Answer

a) $P_1$ is a source and and $P_2$ is a sink. b) $P_1$ is a source and and $P_2$ is a sink.

Work Step by Step

The Divergence Theorem states that $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $ Here, $S$ is a closed surface and $E$ is the region inside that surface. $div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}$ a) We can see at the point $P_1$ that the vectors that end near that point are shorter than the vectors that start near that point and thus the net flow is outwards, so $P_1$ is a source. On the other hand, we can see at the point $P_2$ that the vectors that end near that point are longer than the vectors that start near that point and thus the net flow is inwards, so $P_2$ is a sink. Hence, $P_1$ is a source and $P_2$ is a sink. b) $div F=\dfrac{\partial x}{\partial x}+\dfrac{\partial y^2}{\partial y}=1+2y$ We can see that the y-value of $P_1$ is positive thus, we have div F $\gt 0$ and so $P_1$ is a source. We can also see that the y-value of $P_2$ is less than $-1$, thus, we have div F $\lt 0$ and so $P_2$ is a sink. Hence, $P_1$ is a source and $P_2$ is a sink.
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