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: 21

Answer

The vector field $F$ is irrotational.

Work Step by Step

The vector field $F$ will be irrotational and conservative when $curl F=0$ When $F=ai+bj+ck$, then we have $curl F=[c_y-b_z]i+[a_z-c_z]j+[b_x-a_y]k$ We are given that $F(x,y,z)=f(x) i+g(y) j+h(z) k$ Here, we have $curl F= \nabla \times F=[\dfrac{\partial h(z)}{\partial y}-\dfrac{\partial g(y)}{\partial z}]i+[\dfrac{\partial f(x)}{\partial z}-\dfrac{\partial h(z)}{\partial x}]j+[\dfrac{\partial g(y)}{\partial x}-\dfrac{\partial f(x)}{\partial y}]k=0$ Hence, the vector field $F$ is irrotational.
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