University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 15 - Section 15.7 - Stokes' Theorem - Exercises - Page 896: 18



Work Step by Step

Applying Stoke's Theorem, we have $\oint F \cdot dr=\iint _S (\nabla \times F) \cdot n d\sigma$ $r_{\phi}=\lt 2 \cos \phi \cos \theta, 2 \cos \phi \sin \theta,-2\sin \phi \gt$ and $r_{\theta}=\lt -2 \sin \phi \sin \theta, 2 \sin \phi \cos \theta,0 \gt$ Then, we have $\iint _S (\nabla \times F) \cdot n d\sigma=\int _{0}^{\pi/2}\int_{0}^{2 \pi} (-16 \sin^ 2 \phi \cos \phi (\cos \theta +\sin \theta) -2 \sin \theta +2 \cos 2 \phi \sin \theta )d\phi d\theta $ This implies that $\int _{0}^{\pi/2}(-16 \sin^ 2 \phi \cos \phi (\cos \theta +\sin \theta) -2 \sin \theta +2 \cos 2 \phi \sin \theta )d\phi =0 $
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