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: 31


$\iint_S f \cdot n dS=\iiint_E (\nabla f) dV$

Work Step by Step

Divergence Theorem: $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} $ where, $div F=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}$ $\iint_S fc \cdot n dS=\iiint_Ediv (fc) dV$ This implies that $\iint_S fc \cdot n dS=\iiint_E f( \nabla \cdot c) +(\nabla f) \cdot c dV$ and $\implies \iint_S fc \cdot n dS=\iiint_E f(0) +(\nabla f) \cdot c dV$ $\implies \iint_S fn \cdot c dS=\iiint_E (\nabla f) \cdot c dV$ and $\iint_S f \cdot n dS=\iiint_E (\nabla f) dV$ Hence the result has been verified.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.