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 1150: 27


$div (F \times G)=G \cdot curl F-F \cdot curl G$

Work Step by Step

Plug $F=F_1i+F_2j+F_3k; G=G_1i+G_2j+G_3k$ $div (F \times G)=\nabla \cdot (F \times G)$ $div (F \times G)=\dfrac{\partial}{\partial x}[G_3F_2-G_2F_3]-\dfrac{\partial}{\partial y}[G_3F_1-G_1F_3]+\dfrac{\partial}{\partial z}[G_2F_1-G_1F_2]$ or, $=[(G_1)(\dfrac{\partial F_3}{\partial y}-\dfrac{\partial F_2}{\partial z})-(G_2)(\dfrac{\partial F_3}{\partial x}-\dfrac{\partial F_1}{\partial z})+(G_3) (\dfrac{\partial F_2}{\partial x}-\dfrac{\partial F_1}{\partial y})]-[(F_1) (\dfrac{\partial G_3}{\partial y}-\dfrac{\partial G_2}{\partial z})-(F_2) (\dfrac{\partial G_3}{\partial x}-\dfrac{\partial G_1}{\partial z})+(F_3) (\dfrac{\partial G_2}{\partial x}-\dfrac{\partial G_1}{\partial y})]$ or, $=(curl F) \cdot G-F \cdot (curl G)$ Hence, $div (F \times G)=G \cdot curl F-F \cdot curl G$ (Proved)
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