Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - 16.5 Curl and Divergence - 16.5 Exercises - Page 1149: 19

Answer

There does not exist such a vector field $G$.

Work Step by Step

Consider a vector field $G$ such that $div [curl (G)]=0$ $F=A i+B j+C k $ and $div F=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}$ Given: $curl G=\lt x \sin y, \cos y, z-xy \gt$ $div[curl(G)]=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}=\sin y-\sin y+1 $ $\implies div [curl (G)]=1 \ne 0$ Thus, we get there does not exist such a vector field $G$.
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