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: 35

Answer

$\oint_C D_n g ds=0$

Work Step by Step

Consider the First Green Identity: $\oint_C F \cdot n ds=\iint_D div F(x,y) dA$ When $\nabla^2 g=0$ This implies that $\oint_C \nabla g \cdot n ds=0$ Since, $F=\nabla g$ $\oint_C \nabla g \cdot n ds=\iint_D div (\nabla g) dA$ or, $\oint_C \nabla g \cdot n ds=\iint_D \nabla \cdot (\nabla g) dA=\iint_D \nabla^2 g dA=\iint_D (0) dA=0$ As per the statement, $D_ng$ is defined as $\nabla g \cdot n$ This implies that $\oint_C D_n g ds=0$ (proved) Thus, the result is proved.
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