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: 32


$F=-W k$

Work Step by Step

Divergence Theorem: $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $ $div F=\dfrac{\partial p}{\partial x}+\dfrac{\partial q}{\partial y}+\dfrac{\partial r}{\partial z}$ As per the given problem, we have $\iint_S fc \cdot n dS=\iiint_Ediv (fc) dV$ $\implies \iint_S (fc) \cdot n dS=\iiint_E f( \nabla \cdot c) +(\nabla f) \cdot (c) dV$ $\implies \iint_S fc \cdot n dS=\iiint_E f(0) +(\nabla f) \cdot (c) dV$ This yields: $\iint_S fn \cdot c dS=\iiint_E (\nabla f) \cdot c dV$ and $\iint_S (f \cdot n) dS=\iiint_E (\nabla f) dV$ Also, $\nabla P=\dfrac{\partial (p)}{\partial x}i+\dfrac{\partial (p)}{\partial y}j+\dfrac{\partial (p)}{\partial z}k=\rho g k$ Since, $F=-\iiint_E (\nabla P) dV=-(\rho g k) \iiint_E dV$ Here, $\rho$ shows the density of the liquid and the integral $\iiint_E dV$ shows the volume of the solid. But the weight of the liquid is: $W=\rho g v$ Thus, we get $F=-W k$
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