Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - 16.9 The Divergence Theorem - 16.9 Exercises - Page 1186: 19

Answer

$P_1$ is negative and at the point $P_2$ is positive.

Work Step by Step

Divergence Theorem: $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} $ where, $div F=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}$ The divergence will be negative when the net flow of water towards the inside and when the net flow of water is outwards, then the divergence at that point gets positive. Also, The divergence will be zero when there is net flow of water either inwards or outwards. We found that at the point $P_1$ that the net flow of water is inwards, thus, the divergence is negative and the divergence at point $P_2$ is positive. Hence, we get $P_1$ is negative and at the point $P_2$ is positive.
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