Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 16: Integrals and Vector Fields - Section 16.2 - Vector Fields and Line Integrals: Work, Circulation, and Flux - Page 957: 52


Circulation $=0$

Work Step by Step

The circulation can be found as: $\oint F \cdot dr =\oint(xi+y j+zk) \cdot ( dx i+dy j +dz k) ...(1)$ Here, $ \oint F \cdot dr=\oint x dx +y dy +z dz$ We can see that for this circulation of flux, the initial point and ending point are same at the points $(a,b,c)$. Thus, equation (1) becomes: $\oint F \cdot dr =\int_{a}^a x dx + \int_{b}^b y dy +\int_{c}^c z dz \\=[\dfrac{x^2}{2}]_a^a +[\dfrac{y^2}{2}]_b^b+[\dfrac{z^2}{2}]_c^c\\=0+0+0\\=0$
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