Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 16 - Section 16.9 - The Divergence Theorem - 16.9 Exercise - Page 1146: 21

Answer

The divergence is positive for the points above the $x$-axis and negative for the points below the $x$-axis.

Work Step by Step

Divergence Theorem: $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $ Here, $S$ shows a closed surface and $E$ is the region inside that surface. $div F=\dfrac{\partial p}{\partial x}+\dfrac{\partial q}{\partial y}+\dfrac{\partial r}{\partial z}=y+2y=3y$ When the net flow of water is towards the point, then the divergence at that point is negative and when the net flow of water is outwards, then the divergence at that point is positive. When there is no net flow of water inwards or outwards, the divergence at that point will be zero. Hence, we can conclude that the divergence is positive for the points above the $x$-axis and negative for the points below the $x$-axis.
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