Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 16 - Vector Calculus - 16.9 Exercises - Page 1158: 14

Answer

$4 \pi R^5$

Work Step by Step

$div F=\dfrac{\partial p}{\partial x}+\dfrac{\partial q}{\partial y}+\dfrac{\partial r}{\partial z}=(x^2+y^2+z^2) \cdot \lt x,y,z \gt=3(x^2+y^2+z^2)+2(x^2+y^2+z^2)=5(x^2+y^2+z^2)=5\rho^2 $ $Flux=\int_{0}^{2 \pi}\int_0^{\pi} \int_{0}^{R} 5\rho^2 dv$ $=\int_{0}^{2 \pi}\int_0^{\pi} \int_{0}^{R} 5\rho^2 \times \sin \phi d \rho d\phi d \theta$ and $Flux=\int_{0}^{2 \pi}\int_0^{\pi} R^5 \times \sin \phi \times d \rho d\phi d \theta=\int_{0}^{2 \pi} R^5 [(-\cos \phi)_0^{\pi} d \theta$ $=\int_{0}^{2 \pi} R^5 [-[\cos \pi-\cos 0]) d \theta$ $=\int_{0}^{2 \pi} 2 R^5 d \theta$ $=2( 2\pi) R^5$ $=4 \pi R^5$
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