Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 16 - Section 16.5 - Curl and Divergence - 16.5 Exercise - Page 1110: 24


$curl (F+G)=curl F+curl G$

Work Step by Step

A vector field $F$ is conservative if and only if $curl F=0$ Let us consider that $F=ai+b j+c k$ Then, we have $curl F=[c_y-b_z]i+[a_z-c_z]j+[b_x-a_y]k$ Plug $F=a_1i+b_1j+c_1z; G=a_2i+b_2j+c_2k$ This implies that $curl (F+G)=curl [(a_1+a_2)i+(b_1+b_2)j+(b_3+c_3)k$ Now, use the distributive property of the cross product. This gives: $curl (F+G)=\nabla \times F+\nabla \times G=curl F+curl G$
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