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: 29


\begin{align*} \mathbf{div(F)} &=0\\ \text{curl} \mathbf{(F)} &=(\sin x) j + \left(\cos x-e^y \right) k \end{align*}

Work Step by Step

Given $$\mathbf{F}=\left\langle e^{y}, \sin x, \cos x\right\rangle$$ 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 ( e^{y})}{\partial x}+\frac{\partial (\sin x) }{\partial y}+\frac{\partial ( \cos x)}{\partial z}\\ &=0 \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}\\ e^{y}& \sin x& \frac{z}{x} \end{array} \\ &= \left( \frac{\partial( \cos x)}{\partial y}- \frac{\partial(\sin x)}{\partial z}\right)i - \left( \frac{\partial( \cos x )}{\partial x}- \frac{\partial(e^{y})}{\partial z}\right)j + \left( \frac{\partial(\sin x)}{\partial x}- \frac{\partial( e^{y} )}{\partial y}\right) k\\ &= \left( 0 \right)i - \left( -\sin x \right)j + \left(\cos x-e^y \right) k\\ &=(\sin x) j + \left(\cos x-e^y \right) k \end{align*}
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