Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - 16.9 The Divergence Theorem - 16.9 Exercises - Page 1186: 30


$\iint_S (f \nabla g-g \nabla f) \cdot n ds=\iiint_E (f \nabla^2g-g \nabla^2f) dV$

Work Step by Step

We know that $D_nf=(\nabla f) \cdot n$ $\iint_S (f \nabla g) dS=\iiint_E div (f \nabla g) dV=\iint_E \nabla (F \nabla g) dV$ Also, $F=\nabla g$ Then , we have $\iint_S (f \nabla g) dS=\iiint_E div (f \nabla g) dV$ or, $\iint_E \nabla (F \nabla g) dV=\iiint_E f(\nabla \cdot ( \nabla g) +\nabla f \cdot (\nabla g) dV$ ...(a) and $\iint_S (f \nabla g) \cdot n dS=\iiint_E (f \nabla^2g+\nabla f \cdot \nabla g)dV$ ...(b) Now, From equations (a) and (b), we have $[\iiint_E (f \nabla^2g+\nabla f \cdot (\nabla g) )dV]-[\iiint_E g ( \nabla^2f)+\nabla g \cdot (\nabla f) )dV=\iiint_E f \nabla^2g-g ( \nabla^2f) dV$ $\iint_S (f \nabla g-g \nabla f) \cdot n ds=\iiint_E (f \nabla^2g-g \nabla^2f) dV$ (Verified)
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