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: 27

Answer

$\iint_S curl F \cdot dS=0$

Work Step by Step

Divergence Theorem: $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} $ where, $div F=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}$ Now, we have $\iint_S curl\overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_E div (curl F)dV $ and $\iint_S curl\overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_E div (curl F)dV=\iiint_E (0) dV $ This gives: $div (curl F)=0$ Hence, the result has been verified such that $\iint_S curl F \cdot dS=0$
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