Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 16 - Vector Calculus - 16.5 Exercises - Page 1122: 36

Answer

$\iint_D |\nabla f|^2 dA=0$

Work Step by Step

Since, we have $\iint_Df \nabla^2 g dA=\oint_C f(\nabla g) \cdot n ds-\iint_D \nabla f \cdot \nabla g dA$ $\iint_D f \nabla^2 f dA=\oint_C f(\nabla f) \cdot n ds-\iint_D \nabla f \cdot \nabla f dA$ ...(a) When $f$ is harmonic then $\nabla^2 f=0$ on $D$, then we have $\iint_Df \nabla^2 f dA=0$ Also, when $f(x,y)=0$ on the curve $C$, then we have $\iint_Cf( \nabla f) \cdot n ds=0$ Equation (a) becomes: $\iint_D \nabla f \cdot \nabla f dA=0$ Hence, we get $\iint_D |\nabla f|^2 dA=0$
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