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: 37


$$\operatorname{div}(\nabla f \times \nabla g)=0$$

Work Step by Step

Since \begin{align*} \nabla f&=\frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}+ \frac{\partial f}{\partial z} \\ \nabla g&= \frac{\partial g}{\partial x}+ \frac{\partial g}{\partial y}+ \frac{\partial g}{\partial z} \end{align*} Then \begin{align*} \operatorname{div}(\nabla f \times \nabla g)&=\begin{array}{|||} \frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ \frac{\partial f}{\partial x}& \frac{\partial f}{\partial y}& \frac{\partial f}{\partial z}\\ \frac{\partial g}{\partial x}& \frac{\partial g}{\partial y}& \frac{\partial g}{\partial z} \end{array} \\ &= \frac{\partial}{\partial x}\left( \frac{\partial f}{\partial y} \frac{\partial g}{\partial z}- \frac{\partial f}{\partial z} \frac{\partial g}{\partial y} \right) - \frac{\partial}{\partial y}\left( \frac{\partial f}{\partial x} \frac{\partial g}{\partial z}- \frac{\partial f}{\partial z} \frac{\partial g}{\partial x}\right) + \frac{\partial}{\partial z}\left( \frac{\partial f}{\partial x} \frac{\partial g}{\partial y}- \frac{\partial f}{\partial y} \frac{\partial g}{\partial x} \right) \\ &= \left( \frac{\partial f}{\partial y} \frac{\partial^2 g}{\partial z\partial x}+ \frac{\partial^2 f}{\partial y\partial x} \frac{\partial g}{\partial z}- \frac{\partial f}{\partial z} \frac{\partial^2 g}{\partial y\partial x} - \frac{\partial f}{\partial z} \frac{\partial ^2g}{\partial y\partial x} \right) - \left( \frac{\partial f}{\partial x} \frac{\partial^2 g}{\partial z\partial y}+\frac{\partial^2f}{\partial x\partial y} \frac{\partial g}{\partial z}- \frac{\partial ^2f}{\partial z\partial y} \frac{\partial g}{\partial x}- \frac{\partial f}{\partial z} \frac{\partial^2 g}{\partial x\partial y}\right) + \left( \frac{\partial ^2f}{\partial x\partial z} \frac{\partial g}{\partial y}+\frac{\partial f}{\partial x} \frac{\partial^2g}{\partial y\partial z}- \frac{\partial^2 f}{\partial y\partial z} \frac{\partial g}{\partial x}- \frac{\partial f}{\partial y} \frac{\partial ^2g}{\partial x\partial z} \right) \\ &=0 \end{align*}
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