Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 17 - Line and Surface Integrals - 17.1 Vector Fields - Exercises - Page 919: 28


\begin{align*} \mathbf{div(F)}&= \frac{-y}{x^2}+ \frac{1}{z}+ \frac{1}{x} \\ \text{curl} \mathbf{(F)}&= \left( \frac{y}{z^2} \right)i - \left( \frac{-z}{x^2} \right)j + \left(\frac{-1}{x} \right) k \end{align*}

Work Step by Step

Given $$\mathbf{F}=\left\langle\frac{y}{x}, \frac{y}{z}, \frac{z}{x}\right\rangle$$ Since \begin{align*} \mathbf{div(F)}&=\frac{\partial \mathbf{F}_x}{\partial x}+\frac{\partial \mathbf{F}_y }{\partial y}+\frac{\partial \mathbf{F}_z}{\partial z}\\ &=\frac{\partial ( \frac{y}{x})}{\partial x}+\frac{\partial (\frac{y}{z}) }{\partial y}+\frac{\partial ( \frac{z}{x})}{\partial z}\\ &= \frac{-y}{x^2}+ \frac{1}{z}+ \frac{1}{x} \end{align*} and \begin{align*} \text{curl} \mathbf{(F)}&=\begin{array}{|||} i&j &k\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ \frac{y}{x}& \frac{y}{z}& \frac{z}{x} \end{array} \\ &= \left( \frac{\partial( \frac{z}{x})}{\partial y}- \frac{\partial(\frac{y}{z})}{\partial z}\right)i - \left( \frac{\partial( \frac{z}{x} )}{\partial x}- \frac{\partial(\frac{y}{x})}{\partial z}\right)j + \left( \frac{\partial( \frac{y}{z})}{\partial x}- \frac{\partial( \frac{y}{x} )}{\partial y}\right) k\\ &= \left( \frac{y}{z^2} \right)i - \left( \frac{-z}{x^2} \right)j + \left(\frac{-1}{x} \right) k\\ \end{align*}
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