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: 24

Answer

$\dfrac{4\pi}{3}$

Work Step by Step

Divergence Theorem: $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} $ where, $div F=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}=\dfrac{\partial (2)}{\partial x}+\dfrac{\partial (2)}{\partial y}+\dfrac{\partial z}{\partial z}=1$ In the Divergence Theorem the integral $\iiint_E div F dV$ defines the volume of the region $E$ and $E$ lies inside a sphere with radius $1$. Volume of the region E: $\iiint_E dV=\dfrac{4\pi(1)^3}{3}=\dfrac{4\pi}{3}$ This implies that $\iint_S F \cdot n dS=\iiint_Ediv \overrightarrow{F}dV =\dfrac{4\pi}{3}$
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